DeBroglie wavelength & particle-wave duality

[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.