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Some Sins in Physics Didactics

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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 lay people.

In this blog 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 a 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 break through 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 where 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 a 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 quarter of a century of 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

 

 

vanhees71 works as a postdoctoral researcher at the Goethe University Frankfurt, Germany. His research is about theoretical heavy-ion physics at the boarder between nuclear and high-energy particle physics, particularly the phenomenology of heavy-ion physics to learn about the properties of strongly interacting matter, using relativistic many-body quantum field theory in and out of thermal equilibrium.

Short CV:
since 2011 Postdoc at the Institute for Theoretical Physics at the Goethe University Frankfurt and Research Fellow at the Frankfurt Institute of Advanced Studies (FIAS)
2008-2011 Postdoc at the Justus Liebig University Giessen
2004-2008 Postdoc at the Cyclotron Institute at the Texas A&M University, College Station, TX
2002-2003 Postoc at the University of Bielefeld
2001-2002 Postdoc at the Gesellschaft für Schwerionenforschung in Darmstadt (GSI)
1997-2000 PhD Student at the Gesellschaft für Schwerionenforschung in Darmstadt (GSI) and Technical Univeresity Darmstadt

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  1. Ken G
    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.

  2. vanhees71
    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 ;-).

  3. rude man
    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.

  4. rude man
    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.

  5. Septim
    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

  6. Ken G
    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…

  7. vanhees71
    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.

  8. Ken G
    Ken G 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!

    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.

    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.

    Yes, she tops the list of Nobel snubs: [URL]http://www.scientificamerican.com/slideshow/10-nobel-snubs/[/URL]

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