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Explore Some Sins in Physics Didactics

Introduction

There are many sins in physics didactics. Usually, they occur, because teachers, professors, textbook or popular-science-book writers, etc. try to simplify things more than possible without introducing errors in reasoning, or they copy old-fashioned methods of explaining an issue, leading to the necessity to “erase” from the students’ heads what was hammered in in a careless way before. Some examples are the introduction of a velocity-dependent mass in special relativity, which is a relic from the very early years after Einstein’s ground-breaking paper of 1905, or the use of Bohr’s atomic model as an introduction to quantum theory, which provides not only quantitatively but even qualitatively wrong pictures about how an atom is understood nowadays in terms of “modern quantum theory”. In this blog, I like to address some of the questionable cases of physics didactics. Of course, this is a quite subjective list of “sins”.

For each case, I’ll first give a rather non-technical review, which should be understandable by a high-school student. Then I’ll give a more technical description of the point of view of contemporary (theoretical) physics.

The photoelectric effect and the abuse of the notion of photons

Particularly seductive is quantum theory to the well-intentional teacher. This has several reasons. First of all, it deals with phenomena at atomic or even subatomic scales that are not within our daily experience, and this realm of the natural world can be described only on quite abstract levels of mathematical sophistication. So it is difficult to teach quantum theory in the correct way, particularly on an introductory level, let alone on a level understandable to laypeople.

In this article, I address readers who are already familiar with modern nonrelativistic quantum theory in terms of the Dirac notation.

Historical development

Often introductory texts on quantum theory start with a heuristic description of the photoelectric effect, inspired by Einstein’s famous paper on the subject (1905). There he describes the interaction of light with the electrons in a metallic plate as the scattering of “light particles”, which have an energy of ##E=\hbar \omega## and momentum##\vec{p}=\hbar \vec{k}##, where ##\hbar## is the modified Planck constant, ##\omega## the frequency of monochromatic light, and ##\vec{k}## the wave number.

To kick an electron out of the metal one needs to overcome its binding energy ##W##, and the conservation of energy thus implies that the kicked-out electrons have maximal energy of \begin{equation} \label{1} E=\hbar \omega-W, \end{equation} and this formula is often demonstrated by letting the photo-electrons run against an electric field, which just stops them, and measuring the corresponding stopping voltage as a function of the light’s frequency ##\omega## nicely confirms Einstein’s Law.

After Planck’s discovery and statistical explanation of the black-body-radiation law in 1900, this work of Einstein’s started the true quantum revolution. Planck’s derivation was already mind-puzzling enough because he realized that he had to assume that electromagnetic radiation of frequency ##\omega## can only be absorbed in energy portions of the size ##\hbar \omega##. In addition, he had to apply a pretty strange method to count the number of microstates for the given macroscopic situation of radiation at a fixed temperature in a cavity in order to use Boltzmann’s famous relation between the entropy and this number of microstates, which in fact was written down first by Planck himself in explicit terms: ##S=k_{\text{B}} \ln \Omega##, where ##\Omega## is the number of microstates.

Although already this was breaking with the classical picture, and Planck tried to “repair” this radical consequences of his own discovery till the very end of his long life, Einstein’s paper was much clearer about how deep this departure from the principles of classical physics indeed was. First of all Einstein (re)introduced the idea of a particle nature of light, which was abandoned pretty much earlier due to the findings of wavelike phenomena like interference effects as in Young’s famous double-slit experiment, demonstrating the refraction of light. Finally, Maxwell’s theory about electromagnetism revealed that light might be nothing else than waves of the electromagnetic field, and H. Hertz’s experimental demonstration of electromagnetic waves with the predicted properties, lead to the conviction that light indeed is an electromagnetic wave (in a certain range of wavelengths, the human eye is sensitive to).

Second, Einstein’s model (which he carefully dubbed a “heuristic point of view” in the title of the paper) introduced wave properties into the particle picture. Einstein was well aware that this “wave-particle duality” is not a very consistent description of what’s going on on the microscopic level of matter and its interaction with the electromagnetic field.

Nevertheless, the wave-particle duality of electromagnetic radiation was an important step towards the modern quantum theory. In his doctoral dissertation, L. de Broglie introduced the idea that wave-particle duality may be more general and may also apply to “particles” like the electron. For a while, it was not clear what the stuff in vacuum tubes might be, particles or some new kind of wave field, until in 1897 J. J. Thomson could measure that the corresponding entity indeed behaves like a gas of charged particles with a fixed charge-mass ratio by studying how it was moving in electro- and magnetostatic fields.

All these early attempts to find a consistent theory of the microcosm of atoms and their constituents were very important steps towards the modern quantum theory. Following the historical path, summarized above, the breakthrough came in 1926 with Schrödinger’s series of papers about “wave mechanics”. Particularly he wrote down a field-equation of motion for (nonrelativistic) electrons, and in one of his papers, he could solve it, using the famous textbook by Courant and Hilbert, for the stationary states (energy eigenstates) of an electron moving in the Coulomb field of the much heavier proton, leading to an eigenvalue problem for the energy levels of the hydrogen atom, which were pretty accurate, i.e., only lacking the fine structure, which then was thought to be a purely relativistic effect according to Sommerfeld’s generalization of Bohr’s quantum theory of the hydrogen atom.

Now the natural question was, what the physical meaning of Schrödinger’s wave function might be. Schrödinger himself had the idea that particles have in fact a wavy field-like nature and might be “smeared out” over finite regions of space rather than behaving like point-like bullets. On the other hand, this smearing was never observed. Free single electrons, hitting a photo plate, never gave a smeared-out pattern but always a point-like spot (within the resolution of the photo-plate, given by the size of the grains of silver salt, e.g., silver nitrate). This brought Born, applying Schrödinger’s wave equation to describe the scattering of particles in potential, to the conclusion that the square of the wave function’s modulus, ##|\psi(\vec{x})|^2##, gives the probability density to find an electron around the position ##\vec{x}##.

A bit earlier, Heisenberg, Born, and Jordan had found another “new quantum theory”, the “matrix mechanics”, where the matrices described transition probabilities for a particle changing from one state of definite energy to another. Heisenberg had found this scheme during a more or less involuntary holiday on the Island of Helgoland, where he moved from Göttingen to escape his hay-fever attacks, by analyzing the most simple case of the harmonic oscillator with the goal to use only observable quantities and not theoretical constructs like “trajectories” of electrons within an atom or within his harmonic-oscillator potential. Back home in Göttingen, Born quickly found out that Heisenberg had reinvented matrix algebra, and pretty rapidly he, Jordan, and Heisenberg wrote a systematic account of their new theory. Quickly Pauli could solve the hydrogen problem (also even before Schrödinger with his wave mechanics!) within the matrix mechanics.

After a quarter of a century of the struggle of the best theoretical physicists of their time to find a consistent model for the quantum behavior of microscopic particles, all of a sudden one had not only one but even two of such models. Schrödinger himself could show that both schemes were mathematically equivalent, and this was the more clear, because around the same time another young genius, Dirac, found another even more abstract mathematical scheme, the so-called “transformation theory”, by introducing non-commuting “quantum numbers” in addition to the usual complex “classical numbers”, which commute when multiplied. The final step for the complete mathematical resolution of this fascinating theory came with a work by von Neumann, who showed that states and observables can be described as vectors in an abstract infinite-dimensional vector space with a scalar product, a so-called Hilbert space (named after the famous mathematician) and so-called self-adjoint operators acting on these state vectors.

In the next section, we shall use this modern theory to show, what’s wrong with Einstein’s original picture and why it is a didactical sin to claim the photoelectric effect proves the quantization of the electromagnetic field and the existence of “light particles”, now dubbed photons.

Modern understanding of the photoelectric effect

Let us discuss the photoelectric effect in the most simple approximation, but in terms of modern quantum theory. From this modern point of view, the photoelectric effect is the induced transition of an electron from a bound state in the metal (or any other bound system, e.g., a single atom or molecule) to a scattering state in the continuous part of the energy spectrum. To describe induced transitions, in this case, the absorption of a photon by an atom, molecule, or solid, we do not need to quantize the electromagnetic field at all but a classical electromagnetic wave will do, which we shall prove now in some detail.

The bound electron has of course to be quantized, and we use the abstract Dirac formalism to describe it. We shall work in the interaction picture of time evolution throughout, with the full bound-state Hamiltonian, \begin{equation} \label{2} \hat{H}_0=\frac{\hat{\vec{p}}^2}{2 \mu}+V(\hat{\vec{x}}), \end{equation} which we have written in terms of an effective single-particle potential, leading to bound states ##|E_n,t \rangle##, where ##n## runs over a finite or countable infinite number (including possible degeneracies of the energy spectrum, which don’t play much of a role in our treatment) and a continuous part ##|E ,t\rangle## with ##E \geq 0##. It is important to note that in the interaction picture the eigenvectors of operators that represent observables are time dependent, evolving with the unperturbed Hamiltonian, which is time-independent in our case, according to \begin{equation} \label{2b} |o,t \rangle=\exp \left [\frac{\mathrm{i}}{\hbar} (t-t_0) \hat{H}_0 \right ] |o,t_0 \rangle. \end{equation} For the eigenvectors of the unperturbed Hamiltonian this implies \begin{equation} \label{2c} |E,t \rangle=\exp \left [\frac{\mathrm{i}}{\hbar} (t-t_0) E \right ]|E,t_0 \rangle. \end{equation} The operators which represent observables themselves move accordingly as \begin{equation} \label{2d} \hat{O}(t)=\exp \left [\frac{\mathrm{i}}{\hbar} (t-t_0) \hat{H}_0 \right ] \hat{O}(t_0) \exp \left [-\frac{\mathrm{i}}{\hbar} (t-t_0) \hat{H}_0 \right ]. \end{equation} The classical radiation field is for our purposes best described by an electromagnetic four-vector potential in the non-covariant radiation gauge, i.e., with \begin{equation} \label{3} A^0=0, \quad \vec{\nabla} \cdot \vec{A}=0. \end{equation} Then the electromagnetic field is given by \begin{equation} \label{4} \vec{E}=-\frac{1}{c} \partial_t \vec{A}, \quad \vec{B}=\vec{\nabla} \times \vec{A}. \end{equation} This field is coupled to the particle in the minimal way, i.e., by substitution of \begin{equation} \label{5} \hat{\vec{p}} \rightarrow \hat{\vec{p}}+\frac{e}{mc} \hat{\vec{A}} \quad \text{with} \quad \hat{\vec{A}}=\vec{A}(t,\hat{\vec{x}}) \end{equation} in (\ref{2}). For a usual light wave we can assume that the corresponding field is very small compared to the typical field the electron “feels” from the binding potential. Thus we can restrict ourselves to the leading linear order in the perturbation ##\vec{A}##. We can also assume that a typical electromagnetic wave has much larger wavelengths than the dimensions of the typical average volume the electron is bound to within the atom, i.e., we can take \begin{equation} \label{6} \hat{\vec{A}} \simeq \vec{A}(t)=\vec{A}_0 \cos(\omega t)=\frac{\vec{A}_0}{2} [\exp(\mathrm{i} \omega t)+\exp(-\mathrm{i} \omega t)]. \end{equation} Then ##\vec{A}## is a pure external c-number field and commutes with ##\hat{\vec{p}}##. To linear order the perturbation (“interaction”) Hamiltonian thus reads \begin{equation} \label{7} \hat{H}_{\text{I}}=\frac{e}{mc} \vec{A} \cdot \hat{\vec{p}}. \end{equation} Now in the interaction picture the equation of motion for the state vector of the electron reads \begin{equation} \label{8} \mathrm{i} \hbar \partial_t |\psi(t) \rangle=\hat{H}_{\mathrm{I}} |\psi(t) \rangle. \end{equation} The formal solution is the time-ordered exponential [see any good textbook on quantum theory, e.g., J. J. Sakurai, Modern Quantum Mechanics, 2nd Edition, Addison Wesley (1994)], \begin{equation} \label{9} |\psi(t) \rangle=\hat{C}(t,t_0) |\psi(t_0) \rangle, \quad \hat{C}(t,t_0) = \mathcal{T} \exp \left [-\frac{\mathrm{i}}{\hbar} \int_{t_0}^{t} \mathrm{d} t’ \hat{H}_{\text{I}}(t’) \right ]. \end{equation} In leading order the exponential reads \begin{equation} \label{10} \hat{C}(t,t_0) = 1-\frac{\mathrm{i}}{\hbar} \int_{t_0}^{t} \mathrm{d} t’ \hat{H}_{\text{I}}(t’). \end{equation} Now we want to evaluate the transition probability that the electron which is assumed to have been at time ##t_0## in a bound state ##|\psi(t_0) \rangle=|E_n \rangle## to be found in a scattering state ##|E \rangle##. The corresponding transition-probability amplitude is given by \begin{equation} \label{11} a_{fi}=\langle E,t_0|\hat{C}(t,t_0)|E_n \rangle = -\frac{\mathrm{i}}{\hbar} \int_{t_0}^t \mathrm{d} t’ \langle E|\hat{V}_{\mathrm{I}}(t’)|E_n,t_0 \rangle. \end{equation} For the matrix element, because of (\ref{7}), we only need \begin{equation} \label{12} \langle E,t_0|\hat{\vec{p}}(t’)|E_n,t_0 \rangle = \exp \left (\mathrm{i} \omega_{fi} t’ \right) \langle E,t_0|\hat{\vec{p}}(t_0)|E_n,t_0 \rangle, \end{equation} where we have used the time evolution (\ref{2d}) for the momentum operator and the abbreviation ##\omega_{fi}=[E-E_n]/\hbar##.

Plugging this into (\ref{11}) we find \begin{equation} \begin{split} \label{13} a_{fi} &=-\frac{\alpha}{2 \hbar} \left [\frac{\exp[\mathrm{i} (\omega_{fi}-\omega) (t-t_0)]-1}{\omega_{fi}-\omega}+ \frac{\exp[\mathrm{i} (\omega_{fi}+\omega) (t-t_0)]-1}{\omega_{fi}+\omega} \right] \\ &= -\frac{\mathrm{i} \alpha}{\hbar} \left [\exp[\mathrm{i} (\omega_{fi}-\omega)(t-t_0)/2] \frac{\sin[ (\omega_{fi}+\omega)(t-t_0)/2]}{\omega_{fi}-\omega} +(\omega \rightarrow -\omega) \right], \end{split} \end{equation} where \begin{equation} \label{13b} \alpha=\vec{A}_0 \cdot \langle E,t_0|\hat{\vec{p}}(t_0)|E,t_0 \rangle \end{equation}

Now we are interested in the probability that the electron is excited from a bound state with energy ##E_i##,
\begin{equation}
\label{14}
\begin{split} P_{fi} = |a_{fi}|^2 =& \frac{\alpha^2}{\hbar^2}\frac{\sin^2[(\omega_{fi}-\omega)(t-t_0)]}{(\omega_{fi}-\omega)^2} \\ & + \frac{\alpha^2}{\hbar^2} \frac{\sin^2[(\omega_{fi}+\omega)(t-t_0)]}{(\omega_{fi}+\omega)^2} \\ &+ \frac{2 \alpha^2}{\hbar^2} \cos(\omega t) \frac{\sin[(\omega_{fi}-\omega)(t-t_0)]}{\omega_{fi}- \omega}\frac{\sin[(\omega_{fi}+\omega)(t-t_0)]}{\omega_{fi}+ \omega}. \end{split} \end{equation} For ##t-t_0 \rightarrow \infty## we can use \begin{equation} \label{15} \frac{\sin[(t-t_0) x)}{x} \simeq \pi \delta(x), \quad \frac{\sin^2[(t-t_0) x]}{x^2} \simeq \pi (t-t_0)\delta(x). \end{equation} Thus, after a sufficiently long time the transition rate, becomes \begin{equation} \label{16} w_{fi} = \dot{P}_{fi} \simeq \frac{\alpha^2}{\hbar^2} \delta(\omega_{fi}-\omega). \end{equation} This shows that the transition is only possible, if \begin{equation} \label{17} \omega_{fi} = \omega \; \Rightarrow \; E=E_i+\hbar \omega. \end{equation} Now ##E_i=-W<0## is the binding energy of the electron in the initial state, i.e., before the light has been switched on. This explains, from a modern point of view, Einstein’s result (\ref{1}) of 1905, however without invoking any assumption about “light particles” or photons.

We note that the same arguments, starting from Eq. (18), hold for ##\omega_{fi}<0## and ##\omega=-\omega_{fi}##. Then one has \begin{equation} \label{18} E_f=E_i-\hbar \omega, \end{equation} which describes the transfer of an energy ##\hbar \omega## from the electron to the radiation field due to the presence of this radiation field. This is called stimulated emission. Again, we do not need to invoke any assumption about a particle nature of light.

Where this feature truly comes into the argument can be inferred from a later work by Einstein (1917): One can derive Planck’s black-body-radiation formula (1900) only under the assumption that despite the absorption and stimulated emission of energy quanta ##\hbar \omega## of the electromagnetic field, there is also a spontaneous emission, and from a modern point of view, this can indeed only be explained from the quantization of the electromagnetic field (in addition to the quantization of the electron). Then indeed, for the free quantized electromagnetic field, there is a particle-like interpretation, leading to a consistent picture of the electromagnetic field, interacting with charged particles, Quantum Electrodynamics.

Interesting reading:
http://arxiv.org/abs/1309.7070
http://arxiv.org/abs/1203.1139

 

Read my next article: https://www.physicsforums.com/insights/relativistic-treatment-of-the-dc-conducting-straight-wire/

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  1. ZapperZ says:

    [QUOTE=”stevendaryl, post: 5105095, member: 372855″]This article is suggesting that the photo-electric effect doesn’t actually prove anything about the quantization of the electromagnetic field; the quantization of energy levels of matter is sufficient to explain it. So does ANYTHING prove the quantization of the E&M field? I guess not, because Feynman’s “absorber theory” reformulates QED so that there are no additional degrees of freedom in the E&M field.
    .[/QUOTE]

    But there is a problem here because for metals, the conduction bands are not “quantized” states, as if there are no discrete energy levels. The article cited photoemission from atoms and solids.

    And yes, there are plenty of other experiments that show the photon’s presence, including the Thorn’s which-way experiment that I cited. Read the paper.

    Zz.

  2. stevendaryl says:

    [QUOTE=”atyy, post: 5104475, member: 123698″]OK, but this doesn’t mean one should not teach old quantum theory first. It just means we don’t say that the photoelectric effect with large numbers of coherent photons cannot be explained without quantization of the EM field.

    If we only teach absolutely correct things, then we also cannot teach QM (first quantized language), because it is not relativistic.[/QUOTE]

    I agree. I think that it is important to separate empirical results from the theoretical models developed to explain those results. But I think it’s okay to teach old models, as long as we make it clear that they are just models, which are at best approximations.

  3. stevendaryl says:

    This article is suggesting that the photo-electric effect doesn’t actually prove anything about the quantization of the electromagnetic field; the quantization of energy levels of matter is sufficient to explain it. So does ANYTHING prove the quantization of the E&M field? I guess not, because Feynman’s “absorber theory” reformulates QED so that there are no additional degrees of freedom in the E&M field.

    On the other hand, it seems strange to treat matter (fermions) completely different than gauge particles, when their physics is so similar.

  4. Ken G says:

    I think a few key points are getting lost here. There are two things that [B]vanhees71[/B] never implied, and I never implied them either: 1) that the photoelectric effect was thought to “prove” light was quanta (we all know science doesn’t prove, but we use the word loosely sometimes, that was never the issue), and 2) that Einstein was to be blamed for some incorrect interpretation of his experiment. The whole point, it seems to me, relates to [I]how we teach the significance of the photoelectric effect[/I]. Because the Nobel was awarded for it, and because it was awarded because that experiment was initially thought to demonstrate the photon nature of light, that’s the way it still gets taught. It seems to me [B]vanhees71[/B] is merely pointing out that we don’t need to teach it that way, just because it was once thought about that way, and just because a Nobel committee saw it that way. This isn’t about the history of discovery, it is about what are the actual ramifications of that experiment, given what we now know, and how history can follow some ironic turns that need to be ironed out in hindsight. I think that’s a valid point, and the objections being raised are somewhat extraneous to that point.

  5. ZapperZ says:

    [QUOTE=”atyy, post: 5104750, member: 123698″]I agree with your general point that old quantum theory should be taught, but aren’t multiphoton effects also explained without quantizaton of the electromagnetic field? I think the formalism is similar to that in vanhees71’s blog post, except that one has to go to higher orders in the perturbation expansion, eg. [URL]http://cua.mit.edu/8.421_S06/Chapter9.pdf[/URL].[/QUOTE]

    I don’t know. It looks like it is employing the dipole transition matrix for each transition due to photon absorption. That smells very much like it already assumes the photon model.

    BTW, here is a reference that I have on an example of multiphoton photoemission. Look at Eq. 1 and how it manifests itself as the slope of the charge with light intensity.

    [URL]http://qmlab.ubc.ca/ARPES/PUBLICATIONS/Articles/multiphoton.pdf[/URL]

    Zz.

  6. atyy says:

    [QUOTE=”ZapperZ, post: 5104360, member: 6230″]5. I also find it unfair that we apply modern quantum theory to reexamined the naive photoelectric effect, and yet we ignore modern EXPERIMENTS that have expanded the photoelectric effect as a more generalized photoemission phenomenon. If Einstein had access to high-powered laser, the quantum effect of light will be even more apparent via the multiphoton photoemission. I am aware that this is not within the scope of the thread’s derivation, but this point should be mentioned.[/QUOTE]

    I agree with your general point that old quantum theory should be taught, but aren’t multiphoton effects also explained without quantizaton of the electromagnetic field? I think the formalism is similar to that in vanhees71’s blog post, except that one has to go to higher orders in the perturbation expansion, eg. [URL]http://cua.mit.edu/8.421_S06/Chapter9.pdf[/URL].

  7. atyy says:

    [QUOTE=”vanhees71, post: 5104328, member: 260864″]I don’t understand what you mean by “particle space”. Quantum theory is about quanta, not particles nor classical fields, no matter in which of the many equivalent ways you express it.[/QUOTE]

    Well, these things are called “particles” by convention, because in the classical limit the classical particle is recovered.

    [QUOTE=”vanhees71, post: 5104272, member: 260864″]NO, this I don’t buy! You must not teach highschool students misleading stuff (in fact, we were told “old quantum theory” also before the modern theory was taught in highschool, and our (btw. really brillant) teacher said, before starting with the modern part that we should forget the quantum theory taught before, and she was right so.[/QUOTE]

    OK, but this doesn’t mean one should not teach old quantum theory first. It just means we don’t say that the photoelectric effect with large numbers of coherent photons cannot be explained without quantization of the EM field.

    If we only teach absolutely correct things, then we also cannot teach QM (first quantized language), because it is not relativistic.

    But if we teach QFT (second quantized language), we will also find there is only a low energy effective theory. So there has to be some non-perturbatively defined regularization, eg. the lattice, which basically means we go back to QM

    But if we start from lattice theory instead, we run into problems with chiral fermions.

    So at present we have a theory that is only perturbatively defined by some presumably asymptotic expansion, but we have nothing to which it is asymptotic to, so we have no theory. So we have no laws of physics. Which basically proves Many-Worlds is correct. Because in Many-Worlds, all possibilities happen, so what we observe must happen in at least one world. So we should basically teach MWI and the anthropic principle, since that is the only interpretation that is proven to capture all observations with perfect consistency. :biggrin:

  8. Dadface says:

    [QUOTE=”vanhees71, post: 5104272, member: 260864″]NO, this I don’t buy! You must not teach highschool students misleading stuff (in fact, we were told “old quantum theory” also before the modern theory was taught in highschool, and our (btw. really brillant) teacher said, before starting with the modern part that we should forget the quantum theory taught before, and she was right so.

    Of course, in highschool, you cannot teach the abstract Dirac/Hilbert-space notation and also not time-dependent perturbation theory, but you can completely omit misleading statements referring to the “old quantum theory”. At highschool we learnt modern quantum theory in terms of wave mechanics. I don’t know, how the schedule looks in the UK, but in Germany, usually one has a modul about classical waves before entering the discussion of quantum theory, and thus you can easily argue in the usual heuristic way to introduce first free-particle non-relativistic “Schrödinger waves”, but telling right away the correct Born interpretation. This gains you time to teach the true stuff and not waste it for outdated misleading precursor theories that are important for the science historian only (although history of science makes a fascinating subject in itself, and to a certain extent it should also be covered in high school).

    I don’t understand the 2nd question. Of course, the energy eigenvalue ##E## and the frequency of the corresponding eigenmode of the Schrödinger field are related by ##E=hbar omega=h f##, where ##omega=2 pi f## and ##hbar=h/(2 pi)##. Usually nowadays one doesn’t use the original Planck constant ##h## but ##hbar##, because you don’t need to write some factors of ##2 pi## when using ##omega## instead of ##f##.[/QUOTE]
    Vanhees there really are practical difficulties for any high school teacher in presenting the subject as you suggested. Just look at the relevant section of the Cambridge International AS/A level syllabus. Quantum theory is 25 out of 26 different topics. In addition to covering all of the topics teachers need to teach experimental and practical skills and do numerous other things such as incorporating social, environmental, economic and other aspects into their lesson plans. And, of course, there is the continuing amount of meetings and paperwork to contend with. Taking everything into account, the time teachers have to cover photoelectricity is very limited. Quantum theory is just one small part of a very large syllabus.
    Have a look at the syllabus requirements and you will see exactly what it is teachers have to teach. To do otherwise would jeopardise the chances of their students. I don’t see anything wrong in teaching a subject as the syllabus demands and then informing the students that the subject is far more developed than what has been taught so far. Most of them know that anyway.

  9. ZapperZ says:

    I find the discussion in this thread very confusing and difficult to follow. This is because in some cases, one appeals to the historical context of the derivation, but then one switches to present-day knowledge and criticize the former. I don’t get it.

    Still, let’s get a few things out of the way:

    1. Very much like the use of “relativistic mass”, is it still news that the basic, simple, historical photoelectric effect is not a “proof” of the existence of photons or quantized electromagnetic field? The paper by [URL=’http://people.whitman.edu/~beckmk/QM/grangier/Thorn_ajp.pdf’]J.J. Thorn et al.[/URL] has been cited many times in this forum (do a check if you don’t believe me). In it, the status of the photoelectric effect has been clearly stated as far as the idea of photons is concerned. This paper was published in 2004, and this idea has existed even way before that (see the citation). Are we all just slow to catch on?

    2. Is there such a thing as a “proof” in physics? So is the problem here the photoelectric effect description, or overzealous teachers or writers who somehow stated such a word without realizing the fallacy of it?

    3. Note that the classical derivation, using modern quantum theory, arrived at the [B]same[/B] mathematical expression for the photoelectric effect that Einstein described. So Einstein’s insight on the phenomenon gave the same mathematical formalism without any knowledge of the quantum phenomenon of solids and before the existence of modern QM. This leads to his [B]interpretation[/B] that this is due to a quantized light based on what was known back then. How is this not any different than our current situation with quantum mechanics itself where we all agree on the formalism, but many of us differ in its interpretation?

    4. Because of #3, it is a valid reason to award Einstein with the Nobel Prize, because the mathematical description is still valid (and note that this is coming from someone who had previously written about [URL=’https://www.physicsforums.com/threads/violating-einsteins-photoelectric-effect-model.765714/’]ways to violate Einstein photoelectric effect description[/URL] based on newer experiments). Note that the [URL=’http://www.nobelprize.org/nobel_prizes/physics/laureates/1921/index.html’]Nobel citation for Einstein’s prize[/URL] read:

    [quote=Nobel Foundation] .. for his services to theoretical physics, and for his discovery of the law of the photoelectric effect.[/quote]

    i.e. the mathematical description of the photoelectric effect. It says nothing about the quantized light. As far as I can tell, nothing that has been uncovered here contradicts that.

    5. I also find it unfair that we apply modern quantum theory to reexamined the naive photoelectric effect, and yet we ignore modern EXPERIMENTS that have expanded the photoelectric effect as a more generalized photoemission phenomenon. If Einstein had access to high-powered laser, the quantum effect of light will be even more apparent via the multiphoton photoemission. I am aware that this is not within the scope of the thread’s derivation, but this point should be mentioned.

    Zz.

  10. Ken G says:

    To me, the crucial insight of any “wave theory” is simply the importance of interference. So the problem with “wave/particle duality” is only in how we came to understand waves, historically, in terms of the interference in actual observables (displacements, pressures, etc.). But when Huygens realized that wave propagation was an interference among many different processes going on at the same time, and Feynman discovered how to think about it as a superposition of path integrals, it seems to me we should have generalized what we mean by “waves” to include complex fields that show the same behavior, from which real (observable) fields can just be obtained by matching real initial conditions by use of the complex conjugate (which is how real waves are often analyzed anyway). Take away those real initial conditions, and you have a complex wave theory. Could not such a thing be formalized in terms very similar to “new” quantum field theory? In other words, maybe the problem was not old quantum mechanics, but old wave theory.

    Also, I think the main problem with “wave/particle duality” is that it is often explained like “sometimes it acts like a wave, sometimes like a particle.” That makes it sound confusing and downright schizophrenic. But there’s no need to describe it like that, the wave aspects are consistent, the particle aspects are consistent. What works for me is to say that particles are “told what to do” by wave mechanics. Even trajectories are things that short-wavelength waves do just fine, so there never was anything “different” about what particles do, it was always wave mechanics we just had no reason to see it that way. So to me, the photoelectric effect looks simply like the requirement that if you will get a big response out of an electron (like knocking it out of a metal or forcing a transition in an atom) with a very weak field, you need to find a way to repeat over and over that tiny energy coupling between the field and the electron, in a resonant way, until you accumulate the big response. Like if your house was on springs, and you wanted to raise it an inch, you’d just very gradually bounce it at the resonant frequency until the amplitude was an inch. So that kind of process picks out power from the driving field at the necessary resonant frequency. Behind all that lovely formal mathematics, there is still something quite simple that is [I]physically[/I] going on– the particle is picking out a particular mode from the field, because that is the mode that produces constructive interference in all the possible ways the necessary energy transfer can occur, none of which would independently have sufficient amplitude to be of any consequence, a la Huygens.
    [I][/I]
    So it seems[I] that[/I] would have been the greater insight from the photoelectric effect and all types of stimulated emission, and possibly spontaneous emission too (in a kind of radiation reaction force sense). That’s what I take from[B] vanhees71[/B]’s argument– Einstein thought he was discovering the photon, but that discovery would have to wait– he was really discovering the quantum mechanics of the electron and he didn’t know it! No wonder he never liked quantum mechanics so much…

  11. vanhees71 says:

    I don’t understand what you mean by “particle space”. Quantum theory is about quanta, not particles nor classical fields, no matter in which of the many equivalent ways you express it. Of course, you can treat the photoeffect also with single photons. For that you have to quantize the electromagnetic field. The only difference at this order is that for the excited bound states there’s a transition probability from an excited (bound) state to a lower state under emission of (one or more) photons, spontaneous emission, and that’s why Planck’s Law shows the necessity for the quantization of the em. field, as Einstein has figured out in 1917 from another semiclassical argument within old quantum theory. To get the correct radiation law, he had to assume spontaneous emission, and that was later explained by Dirac when introducing the formalism for non-conserved “particle numbers” in terms of creation and annihilation operators.

  12. atyy says:

    [QUOTE=”vanhees71, post: 5104284, member: 260864″]It’s a bit strange to me to say the Schrödinger waves are in Hilbert space. It’s simply a scalar complex valued field which describes waves, but just in the mathematical sense. It’s not like the waves of a measurable field quantity like, e.g., the density of air or water (sound waves) or the electromagnetic field, but its physical meaning is given by the Born rule.

    What the quantum mechanics is supposed to describe is, of course, given by the choice of the problem. If you have a single particle, you start by defining observables by assuming some operator algebra, heuristically taken from some classical analgous situation. Of course, you cannot derive this in a mathematical sense from anything, but you have to more or less guess it. What helps you, are symmetries and (Lie-)group theory to guess the right operator algebra. The self-adjoint operators live on a Hilbert space, and the rays in this Hilbert space represent the (pure) states. Then, if ##|psi rangle## is a normalized representant of such a ray, a wave function is given wrt. a complete basis, related to the determination of a complete set of compatible observables, ##|o_1,ldots,o_n rangle##, i.e.,
    $$psi(o_1,ldots,o_n)=langle o_1,ldots o_n|psi rangle.$$
    That’s it. In my opinion there’s no simpler way to express quantum theory than this. Admittedly it’s very abstract und unintuitive, but that’s the only way we have found so far to adequately describe (pretty comprehensively) the phenomena in terms of a pretty self-consistent mathematical scheme.[/QUOTE]

    Yes, it’s a bit strange, but that it’s not wrong shows that there is nothing wrong with wave-particle duality. Again in non-rigourous QFT, the Fock space is still a particle space. Then if we take the Wilsonian viewpoint and accept a lattice regularization, the lattice is again QM, which is a particle space.

  13. vanhees71 says:

    [QUOTE=”atyy, post: 5104270, member: 123698″]Well, it’s a wave in Hilbert space, which is “particle space” (I’ll explain that below).

    In QM, the Hilbert space is a particle space because if we have one particle, then we write ψ(x). If we have two particles then we write basis functions that are ψ[SUB]m[/SUB](x[SUB]1[/SUB])ψ[SUB]n[/SUB](x[SUB]2[/SUB]). So we still have particles, it just so happens they don’t have definite position and momentum at all times.

    The way I was taught had the quantum mechanical axioms quite early, but we still got the old quantum theory. Lecture 1 was dimensional analysis to motivate the introduction of Planck’s constant. Lecture 2 was old quantum theory, including all the wonderful thermodynamics. Lecture 3 was the quantum mechanical axioms (state is a ray, etc).

    I’m not sure either, I too think it has to do with the symmetry. The Bohr-Sommerfeld quantization somehow has a notion of integrability in it, and the hydrogen atom is integrable in some sense. (Actually, I never know exactly what a quantum integrable system is, since integrability is really a classical concept). Anyway, the semiclassical quantization is still useful, for example, to explain phenomena like “scars” [URL]http://www.ericjhellergallery.com/index.pl?page=image;iid=22[/URL].

    So I would still like to know if the QM calculation you used as a simple “old quantum theory” interpretation without Einstein’s photons, closer to Planck’s view. Could we say that sonehow the wave has to be of a certain frequency because of a resonance effect?[/QUOTE]

    It’s a bit strange to me to say the Schrödinger waves are in Hilbert space. It’s simply a scalar complex valued field which describes waves, but just in the mathematical sense. It’s not like the waves of a measurable field quantity like, e.g., the density of air or water (sound waves) or the electromagnetic field, but its physical meaning is given by the Born rule.

    What the quantum mechanics is supposed to describe is, of course, given by the choice of the problem. If you have a single particle, you start by defining observables by assuming some operator algebra, heuristically taken from some classical analgous situation. Of course, you cannot derive this in a mathematical sense from anything, but you have to more or less guess it. What helps you, are symmetries and (Lie-)group theory to guess the right operator algebra. The self-adjoint operators live on a Hilbert space, and the rays in this Hilbert space represent the (pure) states. Then, if ##|psi rangle## is a normalized representant of such a ray, a wave function is given wrt. a complete basis, related to the determination of a complete set of compatible observables, ##|o_1,ldots,o_n rangle##, i.e.,
    $$psi(o_1,ldots,o_n)=langle o_1,ldots o_n|psi rangle.$$
    That’s it. In my opinion there’s no simpler way to express quantum theory than this. Admittedly it’s very abstract und unintuitive, but that’s the only way we have found so far to adequately describe (pretty comprehensively) the phenomena in terms of a pretty self-consistent mathematical scheme.

  14. vanhees71 says:

    [QUOTE=”Dadface, post: 5104105, member: 157128″]Nice article but I would like to point out a practical difficulty in presenting the photoelectric effect as described here.

    In the UK photoelectricity is first introduced to AS level students who are 16 to 17 years old. The AS specification is such that there will be about 16 hours of teaching time available to teach the whole of the quantum theory unit photoelectricity being just one of four topics studied. Subtract the time taken for lab sessions, class discussions settling in time and so on and the teacher will be left with a maximum of about two hours to cover the subject.

    In addition to timing problems, students (and probably most teachers whose maths is rusty) would not have the necessary maths expertise to follow this more detailed approach to the subject. I would suggest carry on teaching it as it is and inform the students that there are more sophisticated approaches to the subject but these are beyond the scope of AS.

    I would be grateful if someone could clarify the following:
    It seems to be suggested that there is a better alternative to the equation E = hf . Have I been reading this correctly? If so I can’t see that there is a better way of calculating, for example, the frequency of the photons produced in electron positron annihilation.[/QUOTE]

    NO, this I don’t buy! You must not teach highschool students misleading stuff (in fact, we were told “old quantum theory” also before the modern theory was taught in highschool, and our (btw. really brillant) teacher said, before starting with the modern part that we should forget the quantum theory taught before, and she was right so.

    Of course, in highschool, you cannot teach the abstract Dirac/Hilbert-space notation and also not time-dependent perturbation theory, but you can completely omit misleading statements referring to the “old quantum theory”. At highschool we learnt modern quantum theory in terms of wave mechanics. I don’t know, how the schedule looks in the UK, but in Germany, usually one has a modul about classical waves before entering the discussion of quantum theory, and thus you can easily argue in the usual heuristic way to introduce first free-particle non-relativistic “Schrödinger waves”, but telling right away the correct Born interpretation. This gains you time to teach the true stuff and not waste it for outdated misleading precursor theories that are important for the science historian only (although history of science makes a fascinating subject in itself, and to a certain extent it should also be covered in high school).

    I don’t understand the 2nd question. Of course, the energy eigenvalue ##E## and the frequency of the corresponding eigenmode of the Schrödinger field are related by ##E=hbar omega=h f##, where ##omega=2 pi f## and ##hbar=h/(2 pi)##. Usually nowadays one doesn’t use the original Planck constant ##h## but ##hbar##, because you don’t need to write some factors of ##2 pi## when using ##omega## instead of ##f##.

  15. atyy says:

    [QUOTE=”vanhees71, post: 5104038, member: 260864″]This is exactly, why I consider the teaching of old quantum theory in the beginning of quantum physics courses as a sin. First of all in quantum theory there is no wave-paticle duality. In my opinion it’s the greatest overall achievement of modern quantum theory to have get rid of such contradictory pictures about nature. Formally you are right, and the Schrödinger equation has some features of a classical (wave) field, but that’s only a mathematical analogy. That any partial differential equation in a space-time with the symmetries and admitting a causality principle (orientation of time) we usually assume (Galilei or Poincare symmetry for Newtonian and special relativistic physics, respectively) has similar forms than any other is very natural, because these symmetries are restrictive enough to determine at least a large part of the form such equations should take. Nevertheless, the Schrödinger-wave function is not a classical field and thus does not describe any kind of classical waves. You don’t observe waves, when you consider a single particle but a single particle being detected somewhere in space, and these particles are not “smeared” out in a continuum like way, as a superficial interpretation of the Schrödinger wave function as a classical field would suggest. The right interpretation was given by Born: It describes the probability amplitudes to find a particle at a place, i.e., its modulus squared is the position-probability distribution for particles. [/QUOTE]

    Well, it’s a wave in Hilbert space, which is “particle space” (I’ll explain that below).

    [QUOTE=”vanhees71, post: 5104038, member: 260864″]Also I don’t see, why you say the Hilbert-space formalism describes particles. The abstract Hilbert-space formalism consists of abstract mathematical objects, which have the advantage that you cannot mix them up with the hand-waving associations of the old quantum theory.[/QUOTE]

    In QM, the Hilbert space is a particle space because if we have one particle, then we write ψ(x). If we have two particles then we write basis functions that are ψ[SUB]m[/SUB](x[SUB]1[/SUB])ψ[SUB]n[/SUB](x[SUB]2[/SUB]). So we still have particles, it just so happens they don’t have definite position and momentum at all times.

    [QUOTE=”vanhees71, post: 5104038, member: 260864″]That’s why I think, the ideal way to teach quantum theory is to start right away with the basis-free formulation in (rigged) Hilbert space. Sakurai’s textbook shows a clever way, how to do this without getting into all mathematical subtleties of unbound operators. The aim in teaching quantum theory must be to introduce the students to the fact that the micro-world is “unintuitive”, and that’s so, because our senses and brains are not made primarily to comprehend or describe the microcosm but to survive in a macroscopic world, which behaves quite “classically”, although the underlying “mechanism” is of course quantum. [/QUOTE]

    The way I was taught had the quantum mechanical axioms quite early, but we still got the old quantum theory. Lecture 1 was dimensional analysis to motivate the introduction of Planck’s constant. Lecture 2 was old quantum theory, including all the wonderful thermodynamics. Lecture 3 was the quantum mechanical axioms (state is a ray, etc).

    [QUOTE=”vanhees71, post: 5104038, member: 260864″]To teach this, you don’t need old-fashioned precursor models, of which the worst is the Bohr-Sommerfeld model of the atom, because that’s not even right in any sense of the modern theory. I never figured out, why the model gets the hydrogen energy levels right. The reason must ly in the very high symmetry of the hydrogen-atom hamiltonian (an SO(4) for the bound energy eigenstates, a Galilei symmetry for the zero-modes, and a SO(1,3) for the other scattering states). The same holds, of course true, for the N-dimensional harmonic oscillator which always has a SU(N) symmetry. For the harmonic oscillator it’s clear, why it looks so classical also in quantum theory: The equations of motion of the operators representing observables in the Heisenberg picture are linear and thus the expectation values obey exactly the classical equations of motion.[/QUOTE]

    I’m not sure either, I too think it has to do with the symmetry. The Bohr-Sommerfeld quantization somehow has a notion of integrability in it, and the hydrogen atom is integrable in some sense. (Actually, I never know exactly what a quantum integrable system is, since integrability is really a classical concept). Anyway, the semiclassical quantization is still useful, for example, to explain phenomena like “scars” [URL]http://www.ericjhellergallery.com/index.pl?page=image;iid=22[/URL].

    So I would still like to know if the QM calculation you used has a simple “old quantum theory” interpretation without Einstein’s photons, closer to Planck’s view. Could we say that somehow the wave has to be of a certain frequency because of a resonance effect?

  16. Dadface says:

    Nice article but I would like to point out a practical difficulty in presenting the photoelectric effect as described here.

    In the UK photoelectricity is first introduced to AS level students who are 16 to 17 years old. The AS specification is such that there will be about 16 hours of teaching time available to teach the whole of the quantum theory unit photoelectricity being just one of four topics studied. Subtract the time taken for lab sessions, class discussions settling in time and so on and the teacher will be left with a maximum of about two hours to cover the subject.

    In addition to timing problems, students (and probably most teachers whose maths is rusty) would not have the necessary maths expertise to follow this more detailed approach to the subject. I would suggest carry on teaching it as it is and inform the students that there are more sophisticated approaches to the subject but these are beyond the scope of AS.

    I would be grateful if someone could clarify the following:
    It seems to be suggested that there is a better alternative to the equation E = hf . Have I been reading this correctly? If so I can’t see that there is a better way of calculating, for example, the frequency of the photons produced in electron positron annihilation.

  17. vanhees71 says:

    [QUOTE=”atyy, post: 5103912, member: 123698″]
    Wave particle duality: Still true in QM where the Hilbert space is particles, and the Schroedinger equation is waves. Also true in non-rigourous QFT where there are Fock space particles and waves via the Heisenberg picture equations of motion for the operators.
    de Broglie relations: Still true for the relativistic free quantum fields[/QUOTE]
    This is exactly, why I consider the teaching of old quantum theory in the beginning of quantum physics courses as a sin. First of all in quantum theory there is no wave-paticle duality. In my opinion it’s the greatest overall achievement of modern quantum theory to have get rid of such contradictory pictures about nature. Formally you are right, and the Schrödinger equation has some features of a classical (wave) field, but that’s only a mathematical analogy. That any partial differential equation in a space-time with the symmetries and admitting a causality principle (orientation of time) we usually assume (Galilei or Poincare symmetry for Newtonian and special relativistic physics, respectively) has similar forms than any other is very natural, because these symmetries are restrictive enough to determine at least a large part of the form such equations should take. Nevertheless, the Schrödinger-wave function is not a classical field and thus does not describe any kind of classical waves. You don’t observe waves, when you consider a single particle but a single particle being detected somewhere in space, and these particles are not “smeared” out in a continuum like way, as a superficial interpretation of the Schrödinger wave function as a classical field would suggest. The right interpretation was given by Born: It describes the probability amplitudes to find a particle at a place, i.e., its modulus squared is the position-probability distribution for particles.

    Also I don’t see, why you say the Hilbert-space formalism describes particles. The abstract Hilbert-space formalism consists of abstract mathematical objects, which have the advantage that you cannot mix them up with the hand-waving associations of the old quantum theory. That’s why I think, the ideal way to teach quantum theory is to start right away with the basis-free formulation in (rigged) Hilbert space. Sakurai’s textbook shows a clever way, how to do this without getting into all mathematical subtleties of unbound operators. The aim in teaching quantum theory must be to introduce the students to the fact that the micro-world is “unintuitive”, and that’s so, because our senses and brains are not made primarily to comprehend or describe the microcosm but to survive in a macroscopic world, which behaves quite “classically”, although the underlying “mechanism” is of course quantum.

    To teach this, you don’t need old-fashioned precursor models, of which the worst is the Bohr-Sommerfeld model of the atom, because that’s not even right in any sense of the modern theory. I never figured out, why the model gets the hydrogen energy levels right. The reason must ly in the very high symmetry of the hydrogen-atom hamiltonian (an SO(4) for the bound energy eigenstates, a Galilei symmetry for the zero-modes, and a SO(1,3) for the other scattering states). The same holds, of course true, for the N-dimensional harmonic oscillator which always has a SU(N) symmetry. For the harmonic oscillator it’s clear, why it looks so classical also in quantum theory: The equations of motion of the operators representing observables in the Heisenberg picture are linear and thus the expectation values obey exactly the classical equations of motion.

  18. bhobba says:

    The thing to realise about Einstein isn’t that he got things right or wrong – they are important of course – its that he could see deeper into things than anyone else.

    There is another great thinker – Von Neumann. Although called a mathematician, and he was one of the greatest that ever lived, he really was much much more than that – he was a polymath. He had, like Feynman, the mind of a magician, but even Feynman said he was above him. Those exposed to him said he was the only human ever that was fully awake. His technical mathematical brilliance was very great – well above Einstein, who, while a competent mathematician, was nowhere close to that class.

    The thing though is this, as great as Von Neumann was, as good as he was in seeing to the heart of a problem, and he was good, Einstein was better. He couldn’t match von Neumanns technical brilliance, but his ability to hone in on the essential issues was without peer. And that is the important thing, not mathematical brilliance, being fully awake, or any of the other attributes someone like Von Neumann had, and its what is required to make progress.

    Einstein got a number of things wrong, but really that’s just by the by, he was still able to see to the heart of things and hone in on what was important. He got the photoelectric effect wrong – but got right understanding it was essential to future progress. Here is what Poincare (he was also a noted polymath), along with Madam Curie, said of Einstein:

    ‘Herr Einstein is one of the most original minds that we have ever met. In spite of his youth he already occupies a very honorable position among the foremost savants of his time. What we marvel at him, above all, is the ease with which he adjusts himself to new conceptions and draws all possible deductions from them. He does not cling to classic principles, but sees all conceivable possibilities when he is confronted with a physical problem. In his mind this becomes transformed into an anticipation of new phenomena that may some day be verified in actual experience….The future will give more and more proofs of the merits of Herr Einstein, and the University that succeeds in attaching him to itself may be certain that it will derive honor from its connection with the young master’

    Thanks
    Bill

  19. atyy says:

    [QUOTE=”vanhees71, post: 5103703, member: 260864″]This is again a somewhat more philosophical than physical question. Strictly speaking both Einstein and Planck where wrong, but Einstein got the black-body law right around 1917. Concerning the photoeffect at the level of sophistication discussed in his 1905 paper you cannot distinguish between the light-quantum picture, i.e., that light consists of light particles in Einstein’s sense and knock out electrons in collision-like events, and Planck’s view that there’s a classical electromagnetic wave, but the absorption of em. field energy can only be in energy quanta of size ##hbar omega##.[/QUOTE]

    But why should only quanta of a certain size be absorbed? Can’t the qunatized system take a chunk out of a classical EM wave that has a frequency below the critical frequency? In the old quantum theory view, I do understand Einstein’s model, but I find it quite hard to understand Planck’s alternative.

    [QUOTE=”vanhees71, post: 5103703, member: 260864″]It’s also true that Einstein was well aware that the “old quantum theory” was far from satisfactory, and his long struggle with the “radiation problem” finally lead to the development of modern quantum theory. So one should not diminish Einstein’s and Planck’s achievements in “old quantum theory”, but I still consider it a sin to start quantum physics lectures with this old quantum theory, which introduces wrong pictures already on the qualitative level, which then you have to “unlearn” again. It’s unnecessary and confusing for the students. On the other hand, it’s also very important to have some insight into the historical developments to fully appreciate the modern picture of contemporary physics. I guess that 100 years later also this status of science may be seen as just a historical step to a better understanding of nature. That’s just the way science (hopefully!) works.[/QUOTE]

    But there are curious things about old quantum theory that make it seem “right”. For example, the de Broglie relations are relativistic. Many intuitions of old quantum theory are preserved in non-relativistic QM or relativistic QFT.

    Bohr model: stationary waves and boundary conditions determining discrete energy levels – still true in the Schroedinger equation

    Wave particle duality: Still true in QM where the Hilbert space is particles, and the Schroedinger equation is waves. Also true in non-rigourous QFT where there are Fock space particles and waves via the Heisenberg picture equations of motion for the operators.

    de Broglie relations: Still true for the relativistic free quantum fields

  20. vanhees71 says:

    This is again a somewhat more philosophical than physical question. Strictly speaking both Einstein and Planck where wrong, but Einstein got the black-body law right around 1917. Concerning the photoeffect at the level of sophistication discussed in his 1905 paper you cannot distinguish between the light-quantum picture, i.e., that light consists of light particles in Einstein’s sense and knock out electrons in collision-like events, and Planck’s view that there’s a classical electromagnetic wave, but the absorption of em. field energy can only be in energy quanta of size ##hbar omega##.

    Concerning black-body radiation, Planck’s derivation is interesting, because he used a then ad-hoc counting method of the microstates. Of course, nowadays we recognize this counting method as the correct one for bosons. As we also know today, the foundation of this counting rule is neither justifyable with classical fields nor with classical particles. Here Einstein in 1917 had the right insight by discovering spontaneous emission, which only about 10 years later could be derived from fundamental principles by Dirac in terms of modern quantum theory by introducing creation and annihilation operators for modes of em. fields, which is nothing else than field quantization, and indeed field quantization is the only way to make full sense of electromagnetic phenomena within relativistic quantum theory today.

    It’s also true that Einstein was well aware that the “old quantum theory” was far from satisfactory, and his long struggle with the “radiation problem” finally lead to the development of modern quantum theory. So one should not diminish Einstein’s and Planck’s achievements in “old quantum theory”, but I still consider it a sin to start quantum physics lectures with this old quantum theory, which introduces wrong pictures already on the qualitative level, which then you have to “unlearn” again. It’s unnecessary and confusing for the students. On the other hand, it’s also very important to have some insight into the historical developments to fully appreciate the modern picture of contemporary physics. I guess that 100 years later also this status of science may be seen as just a historical step to a better understanding of nature. That’s just the way science (hopefully!) works.

  21. atyy says:

    [QUOTE=”vanhees71, post: 5103299, member: 260864″]I’d say no, although you can doubt this in some sense: Of course, the photoeffect must also be describable in terms of QED. The setup, most similar to the semiclassical one in my article, is to use a free atom and a coherent state of the em. field as “initial state” and a free electron of momentum ##vec{p}##, another coherent state of the em. field, and a free proton in the final (asymptotic) state. You should get the same, or a very similar, result as in the semiclassical treatment. In some sense you can indeed say, Einstein’s picture is not that wrong, because the corresponding transition-matrix element describes the processes as absorption of one photon out of the coherent field (and even more, because it includes the change of the state of the em. field due to the interaction with the atom in 1st-order perturbation theory).

    Nevertheless, at this level of accuracy of the description and just making a measurement to demonstrate the validity of Eq. (1) from Einstein’s paper, does not “prove” the necessity of a quantization of the em. field, because there is this semiclassical calculation, leading to this formula (1).[/QUOTE]

    So in QM, Einstein is wrong and in QED Einstein is right. In old quantum theory, Einstein clearly can be right. But is there a way to make Einstein wrong in old quantum theory, along the lines of what Planck considered, where the quantization is in the energy levels of the electrons, not the electromagnetic field?

  22. LsT says:

    I, as a layman, can confirm that the fact that photoelectric effect is presented as if it is obvious that it prooves the quantization of light, has done no good for me. In my mind I was envisioning for some time now, if it is possible to “bounce out” losely fitted balls from a wall by resonance with sound waves, irrespective of intensity. And probably this is possible. So if it was not of vanhees71 post, maybe sometime I would post another incosistent question in this forum. Now I understand at least that there is a MUCH bigger story behind this. And it creates a motive for me to learn the math so I can “get it”. Of course if it is on the context of the history of QM in an academic course, it’s another matter.

  23. ddd123 says:

    [QUOTE=”vanhees71, post: 5102279, member: 260864″]It’s pretty funny with Nobel prizes anyway. A said case of negligence is Lise Meitner, who for sure should have gotten the prize together with Otto Hahn since she was the one who gave the correct interpretation of Hahn’s results in terms of fission of Uranium nuclei. Hahn didn’t have a clue! The reason seems to be that Siegbahn’s influence in the Nobel-prize decisions prevented the Nobel prize for Meitner, whom he didn’t like due to his antisemitic attitude.[/QUOTE]

    On that note, a more recent example is Cabibbo having been neglected for the CKM matrix work while awarding the Nobel to the KM part. I couldn’t find any explanation for that, political or otherwise, and the only hypothesis I could think of that it was because Cabibbo was catholic seems far fetched :)

  24. atyy says:

    I guess a better (?) reason for including the cosmological constant is not so much the actual measurement, but the Wilsonian viewpoint? Include all terms consistent with the basic assumptions. For some reason that seems to have become known as Gell-Mann’s totalitarian principle [URL]http://en.wikipedia.org/wiki/Totalitarian_principle[/URL].

    Edit: Hmmm, reading the Wikipedia article, it may be different. In Gell-Mann’s case, the motivation seems to have been basic probability, whereas in the Wilsonian case, the terms are generated automatically by the renormalization flow.

  25. atyy says:

    [QUOTE=”Ken G, post: 5103387, member: 116697″]And if you assert a different historical sequence, in which the quantum mechanics of an electron is discovered prior to the photoelectric effect, there is no Nobel prize there– it’s more like “ho hum, yes quantum mechanics works in other situations than just atoms.” No one even imagines the radiation is quantized, there’s just no need for it from that experiment. Einstein was right about the cosmological constant too, it seems, but we don’t teach students “scientists concluded there is a cosmological constant because it is needed to make the universe static, and modern observations of dark energy confirm that there is indeed a cosmological constant.” Instead we call it Einstein’s greatest blunder– even though he was right! [/QUOTE]

    It was his greatest blunder because he did not realise he was right!

  26. Ken G says:

    And if you assert a different historical sequence, in which the quantum mechanics of an electron is discovered prior to the photoelectric effect, there is no Nobel prize there– it’s more like “ho hum, yes quantum mechanics works in other situations than just atoms.” No one even imagines the radiation is quantized, there’s just no need for it from that experiment. Einstein was right about the cosmological constant too, it seems, but we don’t teach students “scientists concluded there is a cosmological constant because it is needed to make the universe static, and modern observations of dark energy confirm that there is indeed a cosmological constant.” Instead we call it Einstein’s greatest blunder– even though he was right!

    So is the quanta of photons Einstein’s second greatest blunder, on the same grounds? In many ways, it kind of is. What if he had used the photoelectric effect to deduce the quantum mechanics in [B]vanhees71[/B]’s calculation, instead of hypothesizing photons? The latter takes away the need to explain why those frequency modes need to be present to get the necessary energy coupling to the electron, so can be viewed as an opportunity lost, akin to the Hubble law, if we are postulating 20-20 hindsight.

  27. vanhees71 says:

    I’d say no, although you can doubt this in some sense: Of course, the photoeffect must also be describable in terms of QED. The setup, most similar to the semiclassical one in my article, is to use a free atom and a coherent state of the em. field as “initial state” and a free electron of momentum ##vec{p}##, another coherent state of the em. field, and a free proton in the final (asymptotic) state. You should get the same, or a very similar, result as in the semiclassical treatment. In some sense you can indeed say, Einstein’s picture is not that wrong, because the corresponding transition-matrix element describes the processes as absorption of one photon out of the coherent field (and even more, because it includes the change of the state of the em. field due to the interaction with the atom in 1st-order perturbation theory).

    Nevertheless, at this level of accuracy of the description and just making a measurement to demonstrate the validity of Eq. (1) from Einstein’s paper, does not “prove” the necessity of a quantization of the em. field, because there is this semiclassical calculation, leading to this formula (1).

  28. atyy says:

    As zonde says, the issue is general. An observation cannot prove a theory. At best, it can prove a theory within a well-defined model class, eg. in classical Mendelian genetics or in Wilsonian renormalization where one considers the “space of all possible theories”.

    However, although the photoelectric effect does not prove that the electromagnetic field is quantized, now that we do know the electromagnetic field is quantized, can Einstein’s explanation be considered correct?

  29. Ken G says:

    [QUOTE=”zonde, post: 5102990, member: 129046″]Simply ban the word “proved” from your lexicon whenever you are talking about science. However when teaching something you want to present your subject as solid as possible so there is sort of conflict with inconclusive statements that science can make.

    In science cause for rejecting some model is falsification of it’s predictions. You could rather say that:
    Once we understood the quantum mechanics of the electron, we had no cause to reject (pathced) wave model because of the photoelectric effect.[/QUOTE]
    I agree with all of your more careful restatements, yet you are saying the same thing. We have taken [B]vanhees71[/B]’s points here.

  30. vanhees71 says:

    What I wanted to say is that one must not teach students “old quantum theory” as if it was still considered correct. The photoelectric effect, at the level of accuracy described in Einstein’s paper, does not show that the electromagnetic field is quantized, as shown by the standard calculation provided in my Insights article (the only thing, I’ve never found is the argument given there, why one can omit the interference term between the two modes with ##pm omega## of the em. field, which are necessarily there, because the em. field is real).

    I’ve not calculated the cross section to the end, because I thought that’s an unnecessary complication not adding to the point at the level of the (in my opinion false) treatment in introductory parts of many QM1 textbooks. You can do this quite easily yourself, using as an example the analytically known hydrogen wavefunctions for the bound state and a plane-wave free momentum eigenstate for the continuum state. Then you integrate out the angles and rewrite everything in terms of energy instead of ##vec{p}##. You can find the resul in many textbooks, e.g., Sakurai, where this example is nicely treated.

    Of course, what I’ve calculated is the leading-order dipole approximation. Perhaps one should ad a paragraph showing this explicitly, but I don’t know, whether one can add something to a puglished insight’s article. There are also some typos :-(.

    So here is the derivation. What we need is the right-hand side of Eq. (15), i.e., the matrix element in the Schrödinger picture (which coincides by assumption with the interaction picture at ##t=t_0##). First of all we note that in the interaction picture
    $$dot{hat{vec{x}}}=frac{1}{mathrm{i} hbar} [hat{vec{x}},hat{H}_0]=frac{1}{m} hat{vec{p}}.$$
    Thus we have
    $$langle E (t_0)|hat{vec{p}}(t_0)|E_n(t_0) rangle=frac{m}{mathrm{i} hbar} (E-E_n) langle E(t_0)|hat{vec{x}}|E rangle.$$
    Now if you plug this into (20) then due to the energy-conserving ##delta## distribution and making use of the fact that this piece relevant for the absorption (photoeffect) transition rate only comes from the positive-frequency piece ##propto exp(-mathrm{i} omega t)## in ##vec{A}##, you find that what enters is in fact
    $$alpha^2 propto |vec{E}_0 cdot langle E(t_0)|hat{vec{x}}(t_0) E_n(t_0) rangle|^2,$$
    and this is nothing else than the electric-field amplitude times the dipole-matrix transition matrix element.

    The whole calculation also shows that there’s no absorption of frequency modes of the em. field if ##hbar omega## is smaller than the binding energy of the initial state of the electron and that the rate of absorption processes is proportional to the intensity of the external field (for small fields so that perturbuation theory is still applicable).

    For those who like to print the article, I’ve put it on a new website, I’ve just created:

    [URL]http://fias.uni-frankfurt.de/~hees/pf-faq/[/URL]

  31. zonde says:

    [QUOTE=”Ken G, post: 5102594, member: 116697″]I see his argument as correct, so much so in fact that I am smacking my head and saying “doh” for ever repeating the everyday argument that the photoelectric effect proved that light had to come in quanta.[/QUOTE]
    Simply ban the word “proved” from your lexicon whenever you are talking about science. However when teaching something you want to present your subject as solid as possible so there is sort of conflict with inconclusive statements that science can make.

    [QUOTE=”Ken G, post: 5102594, member: 116697″]It was basically a coincidence stemming from the existence of a time period in which we did not understand the quantum mechanics of the electron, that we ever thought that way, so we don’t need to re-enter a mistaken mindset every time we bring up the photoelectric effect! [B]vanhees71 [/B]is saying that once we understood the quantum mechanics of the electron, [B]we had cause to reject Einstein’s explanation of the photoelectric effect[/B], but since quantum electrodynamics came along in short order, that rejection never actually happened.[/QUOTE]
    In science cause for rejecting some model is falsification of it’s predictions. You could rather say that:
    Once we understood the quantum mechanics of the electron, we had no cause to reject (pathced) wave model because of the photoelectric effect.

  32. Shyan says:

    [QUOTE=”Ken G, post: 5102984, member: 116697″]All I can say is, after reading this, I will from now on say that the photoelectric effect was incorrectly interpreted as evidence that the radiation field comes in quanta, when in fact it was merely evidence that getting an energy E into an electron often requires resonant coupling to some electromagnetic power at frequency E/h. A radiation field that doesn’t oscillate at that frequency is therefore not good at doing it. However, it turns out that radiation is regarded as quantized [I]anyway[/I].

    Incidentally, I’m not even sure you need to quantize the radiation field to get spontaneous emission. It seems to me a classical treatment of the radiation field can work for that as well, if you simply let the Fourier mode that perturbs the electron be the electromagnetic field that the electron itself creates, in the spirit of the bootstrap effect sometimes used to analyze the radiative reaction force. Which leaves us with the question– what is the best observational evidence that the radiation field needs to be quantized? The Compton effect? Even photon shot noise could conceivably be modeled as stochastic amplitude variations in a classical field, I would think. Maybe there’s even some way to get the Compton effect with a classical field, if such stochastic amplitude variations are included?[/QUOTE]
    I remember [USER=6230]@ZapperZ[/USER] once said that multiphoton photoemission and angle-resolved photoemission can only be explained in terms of photons.

  33. Ken G says:

    All I can say is, after reading this, I will from now on say that the photoelectric effect was incorrectly interpreted as evidence that the radiation field comes in quanta, when in fact it was merely evidence that getting an energy E into an electron often requires resonant coupling to some electromagnetic power at frequency E/h. A radiation field that doesn’t oscillate at that frequency is therefore not good at doing it. However, it turns out that radiation is regarded as quantized [I]anyway[/I].

    Incidentally, I’m not even sure you need to quantize the radiation field to get spontaneous emission. It seems to me a classical treatment of the radiation field can work for that as well, if you simply let the Fourier mode that perturbs the electron be the electromagnetic field that the electron itself creates, in the spirit of the bootstrap effect sometimes used to analyze the radiative reaction force. Which leaves us with the question– what is the best observational evidence that the radiation field needs to be quantized? The Compton effect? Even photon shot noise could conceivably be modeled as stochastic amplitude variations in a classical field, I would think. Maybe there’s even some way to get the Compton effect with a classical field, if such stochastic amplitude variations are included?

  34. Shyan says:

    [QUOTE=”strangerep, post: 5102904, member: 70760″]Have you studied the quantum optics textbook of Mandel & Wolf? They perform careful calculations along these lines, both for the semi-classical case, and also the full quantum case.[/QUOTE]
    Oh…So I guess the amount of agreement is satisfactory!

    But I still can argue that this isn’t a sin in education. Because in a QM course, photoelectric effect is described as a step in the historical development of QM. Historical development means what phenomena inspired scientists to suggest a particular theory. So as far as historical development is concerned, it doesn’t matter photoelectric effect actually proves the existence of photons or not, it just matters that Einstein thought as such. In fact no one could predict such a semi-classical description! So I think its not a sin.

  35. strangerep says:

    [QUOTE=”Shyan, post: 5102876, member: 160907″][…] So I think someone should actually do some calculations using the semi-classical theory of the photoelectric effect […][/QUOTE] Have you studied the quantum optics textbook of Mandel & Wolf? They perform careful calculations along these lines, both for the semi-classical case, and also the full quantum case.

  36. Shyan says:

    I just see one loophole here. Its true that when we go to QM, it turns out that some phenomena that are impossible in CM, become possible. But we often find out that those strange phenomena have a relatively low probability to happen and that explains why we weren’t measuring them before. So I think someone should actually do some calculations using the semi-classical theory of the photoelectric effect and find the amplitude for the immediate emission of electrons(with some proper definition of immediate) and compare that with experiments. I think it may be too low to account for the experimental value. I’m not saying it will be, but I’m just thinking that only because such an explanation is possible, doesn’t give us the conclusion. Maybe such explanation is still inadequate!

  37. Ken G says:

    I can give an example of what I think [B]vanhees71[/B] is talking about, because I’ve taught students about the photoelectric effect, and this is what I used to say. I said that if light was just an electromagnetic wave, and not a particle, then you should be able to crank up the intensity of a red light until it is knocking off electrons out of the metal. The idea is, if it’s just the strength of the electric field that is jostling the electrons around, you should be able to compensate for low frequency by having a high intensity. But if you have to knock the electron out in a single “quantum event,” then you need enough energy per quantum of light, since you only get to use one such quantum before the metal has in some sense reabsorbed the electron.
    [B]
    vanhees71 [/B]is saying my explanation was a didactic sin– first of all, if you have a strong enough field, it could be a DC field and still get electrons out, so it’s just not true that low frequency couldn’t work. But what is really going on is that the field amplitude is always way too low to knock the electron out in a single period of the oscillation, so you need a kind of resonant accumulation of the effect, and that can be completely accomodated by a wave picture for the light. The need for a resonance, comes from the quantum mechanics of the electron, so is a “first quantization” issue, it does not require the light come in quanta, so is not a “second quantization” issue.

    I see his argument as correct, so much so in fact that I am smacking my head and saying “doh” for ever repeating the everyday argument that the photoelectric effect proved that light had to come in quanta. It was basically a coincidence stemming from the existence of a time period in which we did not understand the quantum mechanics of the electron, that we ever thought that way, so we don’t need to re-enter a mistaken mindset every time we bring up the photoelectric effect! [B]vanhees71 [/B]is saying that once we understood the quantum mechanics of the electron, we had cause to reject Einstein’s explanation of the photoelectric effect, but since quantum electrodynamics came along in short order, that rejection never actually happened. It’s a bit like Einstein’s cosmological constant, which did encounter a period of rejection, but it was not long lived! I think the case can be made that two of Einstein’s most famous suggestions, that light is quantized and that there is a cosmological constant, both turned out to be true for reasons other than the ones that motivated his suggestions! So not to take too much away from the Great One, but it could be concluded that on both those counts, he got lucky.

  38. zonde says:

    I don’t think that you show what you promise here:
    “In the next section we shall use this modern theory to show, what’s wrong with Einstein’s original picture and why it is a didactical sin to claim the photoelectric effect proves the quantization of the electromagnetic field and the existence of “light particles”, now dubbed [B]photons[/B].”
    What you show (I assume your mathematical argument is correct) is that modern wave model can accommodate quantized energy transfer in photoelectric effect.

    But Einstein’s model is certainly good for didactical purposes because – [B]in science it is important that proposed model gives testable prediction, that this prediction is tested and it is confirmed.[/B] In that sense explanation of photoelectric effect from perspective of photons is good example.
    But of course claiming that such confirmation [B]”proves”[/B] particular model can totally spoil positive side of such example. But this is very general objection and is not very specific to particular case.

  39. Ken G says:

    [QUOTE=”vanhees71, post: 5102279, member: 260864″]Interesting, where have you heard that the Nobel committee first wanted to give it for GR? I’ve never heard this, but only that they hesitated to give the prize for relativity at all. So there’s no Nobel for the discovery of GR at all![/quote]I don’t know what deliberations they had, I just mean that giving him the Nobel for the interpretation of the photoelectric effect could have proved disastrous if it had not turned out that light was quantized, merely the process of adding energy to the electromagnetic field inherited the required resonances from quantum mechanics. Then they might have felt they had made a mistake– only to be vindicated later by quantum field theory! I was commenting that something quite similar to that might have happened had they given him the Nobel for GR with a cosmological constant in it, since then Hubble’s observations would have made it look like they had been premature– only to be vindicated later by dark energy. It just shows our many ups and downs with all of Einstein’s great ideas.
    [quote]
    It’s pretty funny with Nobel prizes anyway. A said case of negligence is Lise Meitner, who for sure should have gotten the prize together with Otto Hahn since she was the one who gave the correct interpretation of Hahn’s results in terms of fission of Uranium nuclei. Hahn didn’t have a clue! The reason seems to be that Siegbahn’s influence in the Nobel-prize decisions prevented the Nobel prize for Meitner, whom he didn’t like due to his antisemitic attitude.[/QUOTE]Yes, she tops the list of Nobel snubs: [URL]http://www.scientificamerican.com/slideshow/10-nobel-snubs/[/URL]

  40. vanhees71 says:

    Interesting, where have you heard that the Nobel committee first wanted to give it for GR? I’ve never heard this, but only that they hesitated to give the prize for relativity at all. So there’s no Nobel for the discovery of GR at all!

    It’s pretty funny with Nobel prizes anyway. A said case of negligence is Lise Meitner, who for sure should have gotten the prize together with Otto Hahn since she was the one who gave the correct interpretation of Hahn’s results in terms of fission of Uranium nuclei. Hahn didn’t have a clue! The reason seems to be that Siegbahn’s influence in the Nobel-prize decisions prevented the Nobel prize for Meitner, whom he didn’t like due to his antisemitic attitude.

  41. Ken G says:

    It was as though they had given him the Nobel prize for general relativity including a built-in cosmological constant, then regretted it when universal expansion was discovered, then been vindicated when dark energy was inferred! Of course, if we ever discover a need for a lumineferous ether, we’ll be glad they gave it to him for the light-quantum hypothesis over special relativity…

  42. Septim says:

    Nice post together with the comprehensive mathematical treatment. Although I am a physics graduate I am having hard time grasping the mathematical part since my quantum mechanics and classical mechanics are a bit rusty. What should I particularly revise to get this?Thanks

  43. rude man says:

    Well, you said it was a relic from the early years of 1905.  Feynman taught the course in question at Caltech in the '60's.I'm aware Einstein later changed his mind but Feynman certainly did not.

  44. rude man says:

    " … the introduction of a velocity-dependent mass in special relativity, which is a relic from the very early years after Einstein’s ground-breaking paper of 1905. "The statement is incorrect.  See below.I have never liked the elimination of rest mass as a separate parameter.  It changes several formulae that were accurate before this change, not the least being E = mc^2 for a moving particle.If it was good enough for Richard Feynman it's good enough for me.  Reminder:  the milennial edition of  "The Feynman  Lectures on Physics" was issued just a year or two ago.  It includes significant revised material from earlier editions but the use of rest mass as a separate parameter  was retained.  And wisely so IMO.

  45. vanhees71 says:

    Exactly! Planck didn't like Einstein's "light quanta hypothesis". In contradistinction to that he was an immediate follower of Einstein's special relativity resolution of the puzzle concerning the lack of Galilei invariance of Maxwell electrodynamics, and he wanted to get Einstein to Berlin very much. Together with von Laue and other Berlin physicist he made Einstein an irresistable job offer, including the post of a director of the Kaiser-Wilhelm-Institut für Theoretische Physik, which consisted only of Einstein himself at the time, which meant minimal effort of time for him. In addition, and this was the most attractive feature of the offer for Einstein, he was free from any teaching duties but still being a professor at the University. For this, of course, Planck needed the agreement of the faculty, and in his letter of recommendation, he stated that Einstein was a genius, and one should not take it against him that he sometimes got over the line into speculation, particularly concerning his "light-quanta hypothesis".Ironically the opposite was true for the Nobel-prize committee. For them (both spacial and general) relativity was too speculative to ground his nomination for the prize, and they rather gave it for the light-quanta hypothesis. He got the prize for 1921 in 1922, and I guess the main reason was the discovery of the Compton effect, which convinced many physicists of the time about the reality of light quanta, then also dubbed with the modern name "photon". That's the more ironic, because at this time there was neither non-relativistic quantum theory nor quantum-field theory, which latter was introduced only in 1927/28 by Dirac and in 1929 by Jordan et al. So, in some sense you can say that Einstein got his Nobel for the only theory he discovered that has not survived (completely) the development of modern quantum theory. In my opinion if you have to name only one achievement of Einstein's to theoretical physics to justify his Nobel prize, then it's General Relativity. You could have awarded him for many other things, including his tremendous capability in statistical physics (already the 1905 Brownian Motion paper would have deserved the prize). Einstein, of course, well deserved the prize (if not him, who else?), but that it was given for his light quanta, is really funny ;-).

  46. Ken G says:

    Fascinating, so the photoelectric effect did not really demonstrate light was a particle, it merely showed that the electron cannot resonate with the radiation field unless there are frequency components present that can lift the electron past the work function.  IIRC, Planck derived his famous function using similar thinking, he didn't imagine the high frequencies were underoccupied because of light quanta, only because electrons could only give energy to the field in quantized bits.

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