Where did the de Broglie waves go?

[wofsy]

As a beginner in quantum mechanics I have already lost my way on de Broglie waves.
A free particle is a linear superposition of de Broglie waves and satisfies the Shroedinger equation with zero potential.

Do the component de Broglies waves have physical meaning or are they mathematical formalisms?

Now I turn on a potential and the particle is no longer free. Where did the de Broglie waves go? Do they tranform according to some physical process? Is there an analogue of de Broglie wave that describes the components of the wave function in the presence of this potential? or is the de Broglie wave just a fiction used to motivate the Shroedinger equation?

 

[malawi_glenn]

We have been discussing the DeBroigle waves in quite many threads this week.

 

[0xDEADBEEF]

Yes they have physical meaning and yes they are formalisms. Technically plain waves occupy the whole universe though, hence turning a potential off and on is more problematic than you think. Now that philosophy is out of the way: Any particle's wave function can be expressed as a superposition of free waves, even that of of a bound particle. It's called Fourier transform. But if you catch a particle in a potential, only a certain combination of waves will produce a valid eigenfunction. And to be very inexact, a free particle that you catch in a trap will emit energy to get rid of the waves that don't fit and finally end up in the ground state.

 

[wofsy]

so ... with a potential the dispersion relation for each de Broglie wave includes the potential in the energy of the particle.

when the potential gets turned on, the waves with the wrong energy are shed somehow and leaving only the eigen-energy waves in the packet. I guess this means that those component de Broglie waves that have the wrong momentum are dissipated.

Is this right?

So i guess the combination of the de Broglie waves splits into two or more pieces one of which has components with the same eigen-energy.

As a side questio, I know that de Broglie arrived at his waves using the Lorenz transformation on a particel at rest. Does the same derivation work when there is a potential?

 

[0xDEADBEEF]

so ... with a potential the dispersion relation for each de Broglie wave includes the potential in the energy of the particle.

when the potential gets turned on, the waves with the wrong energy are shed somehow and leaving only the eigen-energy waves in the packet. I guess this means that those component de Broglie waves that have the wrong momentum are dissipated.

1)Is this right?

2)So i guess the combination of the de Broglie waves splits into two or more pieces one of which has components with the same eigen-energy.

3) As a side questio, I know that de Broglie arrived at his waves using the Lorenz transformation on a particel at rest. Does the same derivation work when there is a potential?

The way we discuss this is not very common, so I doubt that there will be great insights.

1) In the new potential forget anything from before about the Energy of the waves. The waves that form the Eigenfuctions in a new potential are pretty random. It's just a mathematical fact, that you can piece waves together to form any kind of wavefunction. They do not separate according to their Energy, but according to their overlap with the Eigenstates.

2) If you look at what happens instantaneously, the wavefunction stays the same, after a change of the potential. In a new potential it can just be regarded as the combination of eigenfunctions of the new potential. Over time the particle will tend to end up in the ground state. We can go backwards again and the ground state can be reinterpreted as a combination as de Broglie waves.
It seems that you would like to see the waves as something stable, that has a certain energy and a certain momentum, and they come to the potential or leave it. This is mostly wrong. The de Broglie waves are the only wavefunctions with a defined momentum, that is true, but apart from that they don't really help the understanding. They are the eigenfunctions of a particle when the potential is 0 everywhere, for another potential another set of Eigenfunctions is used. You can express any wavefunction in any set of eigenfunctions and you can convert freely between them.
So de Broglie waves are stable in a flat potential. In a different potential the eigenfunctions of that potential are stable. Don't stay fixated on the plain waves.

3) I don't know, and I didn't check the original derivation, but starting with a particle in a fixed location at rest sounds problematic, to reconcile with QM anyway.

 

[wofsy]

Ok. Now we are getting somewhere. The de Broglie waves are formalisms only - they are just the components of the Fourier decomposition of the wave function.

It seems to me that there is a logic gap in all of this because the motivation for the Shroedinger equation seems to be the de Broglie wave, an idea derived from Einstein's relations for light quanta. De Broglie imagines such a particle in its rest frame then transforms it via a Lorenz tranformation to get his formula. OK. this then seems to be something physically real - yet we are saying here that it is purely a formalism.

then for unexplained reasons we are suddenly allowed to superpose these deBroglie waves and there seems to be no physical intuition for that - we just do it.

The when we add a non-constant potential, the de Brogie waves go away all together. We just modify the Shroedinger equation for the for the free particle with a potential term and zoom along. Where is the phsical intuition for that?

 

[atyy]

What about the Bohr model - the usual justification is to model the electron as a wave and say the circumference is an integer number of wavelengths for a "standing wave". Would you count this circular wave as a de Broglie wave?

 

[0xDEADBEEF]

Sorry, this will possibly sound snappy...

Ok. 1) [...] The de Broglie waves are formalisms only - they are just the components of the Fourier decomposition of the wave function.
[...] OK. this then seems to be something physically real - yet we are saying here that it is purely a formalism.

2) then for unexplained reasons we are suddenly allowed to superpose these deBroglie waves and there seems to be no physical intuition for that - we just do it.

3)The when we add a non-constant potential, the de Brogie waves go away all together. We just modify the Shroedinger equation for the for the free particle with a potential term and zoom along. Where is the phsical intuition for that?

1) This is really badly phrased, because it assumes a very negative view of the word formalism. Actually it reminds me of people who call the theory of relativity "just a theory". The problems are in the nature of quantum mechanics. Either you accept it all as real, or you fight with the reconciliation of the probabilistic nature of quantum mechanics with your concept of reality. You are not the first one who does this, but quantum mechanics doesn't care.
The decomposition and the reality of it could be exemplified with a piece of butter. Someone might call it a rectangular block of butter, and someone might say that it consists of two pieces of equal weight and demonstrates it but cutting it in the middle, and then somebody might say it's three pieces of equal weight cutting it twice. Here we decomposed the piece of butter into rectangular functions in different ways. If you want, you can call it a formalism, but nonetheless these pieces are real, and we can do physical things with them. Once we "change the potential" for our piece of butter, and say the left half is pulled to the left and the right half is pulled to the right, it makes more sense to speak of two pieces of butter than of one piece or three.

2) Superimposing is something you can simply do with waves, because of the differential equation that governs their behavior. If that equation is again a nasty formalism to you. Than you may call QM unintuitive or learn more about these equations till they feel natural.

3) Either you find some intuition in the piece of butter analogy or cannot help. I know that QM is very confusing in the beginning. There are a few different ways of looking at the same problem, and once you get introduced to one you try to cling to it. So when you see a new formalism, you want to understand it in terms of the old formalism, which is certainly possible, but one formalism is not better or more real then the other. Now I want to steal your de Broglie waves and you resist, because you try to make it to the base of your understanding, but if you look closely there was not much intuition in accepting de Broglie waves as real either.
Learn the problems in the beginning of the quantum mechanics textbooks, with the particle in the box, the harmonic oscillator, and the hydrogen atom as a base. Things to avoid in the beginning: de Broglie, Bohm, Path Integrals, QFT, density matrices, matrix mechanics, Sommerfeld formalism. Also the historical texts are terrible learning material. At least I find the intuitions that these people used to get their ideas more confusing then helpful.

 

[f95toli]

Well, the Bohr model is "wrong" in the sense that it is essentially just a semi-classical model, it can be useful and was important for the development of QM; but I don't think you can actually "justify" it in any way that is physically meaningful.

I don't thing de Broglie waves are "explicity" used much nowadays; the de Broglie formula is useful for e.g. making rough calculations of the smallest thing you can see with an electron microscope but that is about it. de Broglie's work was conceptually very important when QM was first developed but the "wave nature" of particles is so to speak already built into modern QM formalism so you don't need to think about it when doing calculations.

 

[atyy]

Well, the Bohr model is "wrong" in the sense that it is essentially just a semi-classical model, it can be useful and was important for the development of QM; but I don't think you can actually "justify" it in any way that is physically meaningful.

I don't thing de Broglie waves are "explicity" used much nowadays; the de Broglie formula is useful for e.g. making rough calculations of the smallest thing you can see with an electron microscope but that is about it. de Broglie's work was conceptually very important when QM was first developed but the "wave nature" of particles is so to speak already built into modern QM formalism so you don't need to think about it when doing calculations.

Yes, I was just wondering if the Bohr model waves would be considered de Broglie waves for the purposes of this thread. The Schroedinger wave equation is more fundamental and explains why the semi-classical approximations work.

 

[wofsy]

I have no problem with Fourier transforms but was hoping de Broglie's idea led to other physical intuitions that became quantum mechanics. I know that de Broglie himself did not think in terms of Fourier transforms but in terms of relativistic particles.

Maybe it really went like this - start with the usual wave equation e.g. the equation for a stretched string and solve for a normal mode with the de Broglie relations imposed - the space dependent part immediately becomes the time independent Shroedinger equation - the time dependent part - well that won't work because Einstein's energy relaion is violated so try only one time derivative and throw in a factor of i to avoid divergent exponentials. Voila. You have the time dependent Shroedinger equation.

One derivation I've seen starts with the assumption that states evolve according to a "Markov process" for amplitudes. A smoothness assumption gives the Shroedinger equation in general operator form. Then requiring conservation of probability implies that the operator must be Hermitian. This is nice because it shows the connection between the amplitudes and probabilities and the Shroedinger equation but it assumes the probability amplitude picture from the start.

 

[atyy]

I know that de Broglie himself did not think in terms of Fourier transforms but in terms of relativistic particles.

Apparently Schroedinger did consider what we call the "Klein-Gordon" equation. It is a relativistic equation, but there are problems interpreting it as a single particle equation. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Klein_Oskar.html