I DeBroglie wavelength & particle-wave duality

[jtbell]

$\Psi$ at all points in an orbital oscillates together, in step with each other, according to the position-independent time dependence $e^{−i(E_0/ℏ)t}$

I'd better modify this. I'm feeling nit-picky this morning. $\psi_{nlm}(r,\theta,\phi)$ can be either positive or negative. In regions where it is negative, $\Psi_{nlm}(r,\theta,\phi,t)$ oscillates in the opposite sense because of that minus sign, even though the time-dependent factor is the same. Some orbital diagrams indicate this by using e.g. different colors for the "positive" and "negative" regions.

This is analgous to a standing wave on a string. In my example above, alternating sections of the string oscillate in opposite directions because $f(x) = \cos(kx)$ can be either positive or negative, even though the same $g(t) = \cos(\omega t)$ applies everywhere.

 

[vanhees71]

I'm not sure, but aren't orbital diagrams defined with $|\psi|^2$ rather than $\psi$, which is complex anyway, so you'd need to draw two orbital diagrams, one for the real and one for the imaginary part, but what does this tell me? The wave function itself is not observable but only the probability distribution $|\psi|^2$.

 

[Blue Scallop]

I don't know, what's your problem and why you call probability waves "invisible". They are measurable and the probabilistic content is measured all the time.

But aren't probability waves the wave function itself? And I said the wave function was invisible because it didn't really exist in our world.. remember that 2 properties of a particle has already 6 dimensional wave function.. which occurs in configuration space.. and we have been taught repeatedly that wave function was not objective.. so we never imagine the wave function can form standing wave in the orbital even when not measured or observed.. or if it is continuously measured... what serves as the observer... potential in the nucleus or the neighboring electrons? i'll reflect on yours and jtbell excellent arguments below and contemplate on it for a day. Thanks a lot.

The basic principles of QT are as follows:

(a) There is a Hilbert space. Pure states are represented by rays in this Hilbert space or equivalently by a projection operator $\hat{P}_{\psi}=|\psi \rangle \langle \psi|$ with a normalized vector $|\psi \rangle$ ($|\psi \rangle$ is determined up to an unimportant phase factor, thus the states are represented by rays not the vectors themselves)
(b) Observables are represented by essentially self-adjoint operators defined on a dense subspace of the Hilbert space.
(c) When an observable $A$ is measured precisely, the value is an eigenvalue of the self-adjoint operator $\hat{A}$ that represents it
(d) If $|a,\beta \rangle$ are a complete orthonormal set of eigenvectors of $\hat{A}$ to the eigenvalue $a$, then the probability to find the value $a$ when measuring $A$ on a system prepared in the state $\hat{P}_{\psi}$ is given by
$$P(a|\psi)=\sum_{\beta} \langle a,\beta|\hat{P}_{\psi}|a,\beta \rangle=\sum_{\beta} |\psi(a,\beta)|^2.$$

If you have observables with a continuous spectrum the sums become integrals and the eigenvectors are generalized distribution valued vectors.

That's it. According to the standard minimal interpretation there's no other meaning to the formal building plots of QT than that.

 

[vanhees71]

This is all philosophical gibberish (SCNR). The wave function has a probabilistic meaning and is thus measurable on ensembles of equally prepared systems. For me anything reproducibly measurable "exists" objectively (and nothing else!).

 

[Blue Scallop]

This is all philosophical gibberish (SCNR). The wave function has a probabilistic meaning and is thus measurable on ensembles of equally prepared systems. For me anything reproducibly measurable "exists" objectively (and nothing else!).

That's a great idea. But I wonder if Bohr believed what you believe. I'll ponder on it. It is mentioned in the book "Quantum: Einstein, Bohr and the Great Debate about the Nature of Reality":

"Once the momentum of particle A is measured, it is possible to predict accurately the result of a similar measurement of the momentum of particle B as outlined by EPR. However, Bohr argued that that does not mean that momentum is an independent element of B's reality. Only when an 'actual' momentum measurement is carried out on B can it be said to possesses momentum. A particle's momentum becomes 'real' only when it interacts with a device designed to measure its momentum. A particle does exist in some unknown but 'real' state prior to an act of measurement. In the absence of such a measurement to determine either the position or momentum of a particle, Bohr argued that it was meaningless to assert that it actually possessed either."

I'm thinking if there are ensembles of equally prepared system where no measurements were made.. then would the properties even exist. For a lone hydrogen atom with a nucleus and electron.. the electron exists because the nucleus observed it.. without the nucleus, maybe the electron would cease to exist as per Bohr belief? Note this is the heart of orthodox QM.

 

[vanhees71]

I have no clue what Bohr believed nor am I able to understand many of his papers on philosophical issues of QT. If you want to understand QT from the "founding fathers" avoid Heisenberg and Bohr and refer to Born, Pauli, and Dirac. They are the early "no-nonsense fraction" of the club.

As I've very often argued in this forum, the EPR paradox is entirely resolved when taking an epistemic view on the quantum state and abandon the collapse idea from QT, which is anyway very problematic. I also read the EPR paper as a critique of the flavor of interpretation including a collapse rather than a critique against minimally interpreted QT.

If there's a lonely hydrogen atom known to be in some state somewhere in the universe according to the known conservation laws for sure there'll exist an electron and a proton all the time. That's why things are for sure there, even if we are not looking.

 

[Blue Scallop]

Have you ever compared the mathematical description of a standing wave on e.g. a vibrating string with the mathematical description of a hydrogen orbital?

A standing wave (as opposed to a traveling wave) on a vibrating string can be described by an equation of the form $y(x,t) = f(x)g(t)$ e.g. $y(x,t) = \cos(kx)\cos(\omega t)$.

$f(x)$ gives the overall "envelope" or "shape" of the wave, which does not change with time. $g(t)$ gives an oscillating time-dependence which is the same for every point on the wave. The amplitude of the oscillation at each point x is $f(x)$. The oscillations at all points are in step ("in phase") with each other.

[If you're not already acquainted with the above, I suggest you study standing waves in classical mechanics.]
The wave function for a single hydrogen orbital with quantum numbers n,l,m is $\Psi_{nlm}(r,\theta,\phi,t) = \psi_{nlm}(r,\theta,\phi) e^{-i(E_n/\hbar)t}$.

$\psi_{nlm}(r,\theta,\phi)$ gives the overall "envelope" or "shape" of the orbital, which does not change with time, and is analogous to $f(x)$ in the standing wave above. $e^{-i(E_n/\hbar)t} = \cos[(E_n/\hbar)t] + i \sin[(E_n/\hbar)t]$ gives an oscillating time-dependence which is the same for every point of the orbital. The amplitude of the oscillation at each point $r,\theta,\phi$ is $\psi_{nlm}(r,\theta,\phi)$. The oscillations at all points are in step ("in phase") with each other.

This description is completely analogous with a standing wave on a string, except for the number of spatial dimensions and the use of complex numbers in the time-dependent part.

You can find tables of $\psi_{nlm}$ for different sets of quantum numbers, in many textbooks and on many web pages, e.g. here:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydwf.html

So e.g. for the ground state (n,l,m = 1,0,0) we have $$\Psi_{100}(r,\theta,\phi) = \frac{1}{\sqrt{\pi} a_0^{3/2}} e^{-r/a_0} e^{-i(E_0/\hbar)t}$$ where $a_0$ is the Bohr radius and $E_0$ is the ground-state energy of hydrogen.

The pictures that you see in textbooks (like the ones you showed above) are an attempt to show the "shapes" of the probability distributions $\left| \Psi_{nlm}(r,\theta,\phi,t) \right|^2 = \left| \psi_{nlm}(r,\theta,\phi) \right|^2$. They are analogous to pictures of the "envelope" of a standing wave on a string. To be complete, they should include some kind of "shading" or "fuzziness" to indicate the variations in probabilty density with position.

There are no smaller waves "hidden" inside these diagrams, as you attempted to draw in one of your diagrams. $\Psi$ at all points in an orbital oscillates together, in step with each other, according to the position-independent time dependence $e^{-i(E_0/\hbar)t}$.

Many thanks for your detailed message.. are you a minimal ensembler statistician like Vanhees71 or are you a Copenhagenist or others? I know they are standing waves of probability. But how do you measure the standing waves of probability? Wave functions are supposed not to be objective. For example. Let's say there is probability of rain or winning the lottery, and you assign probability waves to them, how do you measure the probability wave of your winning the lottery or whether it would rain tomorrow?

 

[Blue Scallop]

I have no clue what Bohr believed nor am I able to understand many of his papers on philosophical issues of QT. If you want to understand QT from the "founding fathers" avoid Heisenberg and Bohr and refer to Born, Pauli, and Dirac. They are the early "no-nonsense fraction" of the club.

As I've very often argued in this forum, the EPR paradox is entirely resolved when taking an epistemic view on the quantum state and abandon the collapse idea from QT, which is anyway very problematic. I also read the EPR paper as a critique of the flavor of interpretation including a collapse rather than a critique against minimally interpreted QT.

If there's a lonely hydrogen atom known to be in some state somewhere in the universe according to the known conservation laws for sure there'll exist an electron and a proton all the time. That's why things are for sure there, even if we are not looking.

Only less than half of physicists believe in the minimal statisticial interpretation in which single system doesn't exist and you need ensembles for QT to make sense.. and even much less believe Bell's Theorem is not real.

Therefore I'd like to hear the opinions of Copenhagenists or Many Worlders or even the Bohmians.

Do you also believe like vanhees71 the probabilistic waves or orbitals are measurable and the probabilistic content is measured all the time? How do you measure the orbitals? Are the orbitals being measurable is dependent on what interpretation you are holding?

 

[Nugatory]

Only less than half of physicists believe in the minimal statisticial interpretation in which single system doesn't exist and you need ensembles for QT to make sense.

A reliable reference supporting this claim would be nice.
(It's also worth considering whether whoever did the tabulation here was counting "shut up and calculate" and "do I care?" as MSI supporters).

 

[vanhees71]

Where have I said that Bell's theorem is not real? It's confirmed by high-precision experiments with outrageous significance. It's among the most convincing empirical findings for the correctness of QT and for me it's the more convincing for the minimal statistical interpretation.

 

[Blue Scallop]

Where have I said that Bell's theorem is not real? It's confirmed by high-precision experiments with outrageous significance. It's among the most convincing empirical findings for the correctness of QT and for me it's the more convincing for the minimal statistical interpretation.

As one of the most powerful proponent of the ensemble interpretation. I'd like to ask you about behavior of single system.

In the case of a hydrogen atom with a nucleus and one electron. Let's say the quantum states are just mathematical. But what part is mathematical. Is the electron the mathematical entity and the nucleus too. Or neither and only how the electron position appears? In the ensemble interpretation, how do you imagine an atom then as single system?

Conventionally, the electron wave functions is standing wave around the nucleus and when measured, it collapses into being. But for the ensemble intepretation where the collapse is not real. How then does the electron behave in the nucleus for a single system?

I'd just like an idea how you view it. Thanks.

 

[vanhees71]

All you can say is the probability for finding the electron at a given place when you measure its position. There's no other information encoded in the wave function than that.