# DeBroglie equation applied to atoms & molecules: not so obvious

1. Jun 29, 2014

### dsoodak

One of the first things about QM we were taught in my undergraduate physics program is the deBroglie relation:
λ = h/p

Now, it makes sense that that this might hold for all elementary particles, especially since the evidence generally seems to suggest that the commonly observed forms of matter and energy are basically made of the same stuff.

However, it doesn't logically follow that the same holds true for things like atoms and molecules (or even protons). This would suggest that all the constituents somehow "know" that they are part of a larger system and adjust (and sync up) their wavelengths accordingly.

The way I would expect this to be approached is to treat an atom (or maybe just start with something simpler like an electron-positron pair) as a multi-particle system, then calculate where all the individual particles would end up when shoved through a 2-slit experiment. Then the professor would say: "Notice that when you express this in terms of the center of mass of the system, you get the same equation as you would for a single particle whose mass is the sum of the parts". If you skip this step, you have an over-defined mathematical equation.

So...can anyone point me to this derivation? I assume that I was not given it because it is too complicated to teach undergraduates. We already know its true from experimental evidence, but it seems like SOMEONE would have double checked the math...

Dustin Soodak

2. Jun 29, 2014

### Staff: Mentor

Ideas like that are common in beginner treatments, but in fact are not correct.

Since Dirac came up with the transformation theory in 1927 ideas like wave particle duality etc etc were outmoded, and seen to actually be counter productive to understanding QM - but such is not usually pointed out to start with.

I think you need to see the real conceptual core:
http://www.scottaaronson.com/democritus/lec9.html

Once you understand that then it will likely be easier to see how multi particle systems are handled.

Here is a correct analysis of the double slit experiment from QM principles:
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

Its got nothing to do with wave particle duality etc. Nor does it change with composite systems. Its purely got to to with the laws of QM.

Thanks
Bill

Last edited: Jun 29, 2014
3. Jun 30, 2014

### zincshow

One way to think of this:

"All particles can exhibit wave properties. A particle that expands and contracts as it moves, will behave differently if it hits another particle when expanded compared to when it is contracted. In 1924, a physicist named Louis deBroglie proposed that the electron would exhibit a wave-like nature based on the electrons kinetic energy. His theory, together with the Davisson-Germer experiment done in 1927, established that an electron, accelerated by an electric field does have wave properties.

Based on deBroglie's calculations, an electron, accelerated through a field of 54 volts, has a wavelength of 0.167 nanometers. Based on this wavelength, the electron takes 5.6 attoseconds to complete one expansion/contraction cycle. When these electrons are shot at a surface of nickel atoms where the spacing between atoms is a similar size, the electrons show a recoil pattern that allowed the spacing between the nickel atoms to be calculated.
"

4. Jun 30, 2014

### Staff: Mentor

See the FAQ:
'So there is no duality – at least not within quantum mechanics. We still use the “duality” description of light when we try to describe light to laymen because wave and particle are behavior most people are familiar with. However, it doesn’t mean that in physics, or in the working of physicists, such a duality has any significance.'

In QM it has been known for a long time quantum objects are neither particles or waves. Statements like 'All particles can exhibit wave properties' are extremely misleading. Strictly speaking it is partly true - they can exhibit wave like properties in certain situations - but there are questions such as waves of what, and the space they propagate in is an abstract Hilbert space. Also the delayed choice experiments casts serious doubt on even this view:
http://en.wikipedia.org/wiki/Wheeler's_delayed_choice_experiment
'Our results demonstrate that the viewpoint that the system photon behaves either definitely as a wave or definitely as a particle would require faster-than-light communication. Because this would be in strong tension with the special theory of relativity, we believe that such a viewpoint should be given up entirely'

Quantum objects are neither particle or wave - even sometimes considering them as waves is problematical - they are quantum stuff that is not amenable to such simple pictures.

Thanks
Bill

5. Jul 17, 2014

### dsoodak

First of all, thanks for all the responses. The links provide a nice variety of different ways to approach this subject.

In order to clarify my question, I will use one of the links:
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

In several equations the term "p" for momentum is included. My question is how you can take a multi-particle system such as a proton, neutron, or in some cases whole atoms, and assume that this system's eventual detection can be predicted using the same equation as is used for a single elementary particle with the sum of the masses (-binding energy correction) of the particles of which it consists.

It seems like the way you would approach this is to reduce the total states of a 2 particle system from (Using the notation in www.scottaaronson.com/democritus/lec9.html)
(|00>+|01>+|10>+|11>)
to
|00>+|11>
to indicate that particle 1 and particle 2 must always end up in the same place (final positions are |0> and |1>) since they are physically connected to each other.

This is NOT, evidently, how the world actually works.

However, I think that it deserves some explanation, even if that explanation just a reference to a overcomplicated calculation of the wave function of something like an electron-positron pair.

6. Jul 19, 2014

### atyy

That's a great question. If quantum mechanics is a theory with degress of freedom {x}, and we study a theory with emergent degrees of freedom {y}, why should the emergent theory also be quantum mechanical? I can't answer it off the top of my head in technical detail for the systems you mention, but here are some things which answer similar questions.

1) Derivation of QM with fixed number of particles from quantum field theory (which is something like QM with an infinite number of particles): http://www.damtp.cam.ac.uk/user/tong/qft/two.pdf (section 2.8.1)

2) Heuristic derivation of a low energy theory of pions from QCD in which the degrees of freedom are not pions:
http://arxiv.org/abs/hep-ph/0703297
"The basic idea of an effective field theory is to treat the active, light particles as relevant degrees of freedom, while the heavy particles are frozen and reduced to static sources. The dynamics are described by an effective Lagrangian which is formulated in terms of the light particles and incorporates all important symmetries and symmetry-breaking patterns of the underlying fundamental theory."

3) Heuristic derivation of low energy quantum gravity from an unknown quantum theory of everything:
http://arxiv.org/abs/1209.3511
"Let us start by asking how any quantum mechanical calculation can be reliable. Quantum perturbation theory instructs us to sum over all intermediate states of all energies. However, because physics is an experimental science, we do not know all the states that exist at high energy and we do not know what the interactions are there either. So why doesn’t this lack of knowledge get in the way of us making good predictions?"

4) The Born-Oppenheimer approximation:
http://en.wikipedia.org/wiki/Born–Oppenheimer_approximation
http://www.cmth.ph.ic.ac.uk/people/p.haynes/thesis/node13.html
http://vergil.chemistry.gatech.edu/notes/bo/bo.pdf
http://www.eng.fsu.edu/~dommelen/quantum/style_a/bo.html

5) Density functional theory:
http://newton.ex.ac.uk/research/qsystems/people/coomer/dft_intro.html

Last edited: Jul 19, 2014
7. Jul 19, 2014

### akhmeteli

Previously, I offered the following view based on the following ideas of Duane (W. Duane, Proc. Natl. Acad. Science 9, 158 (1923)) and Lande (A. Lande, British Journal for the Philosophy of Science 15, 307 (1965)): the change of direction of motion of a particle in the interference experiment is determined by the momentum transferred to the screen, and this momentum corresponds to quanta (e.g. phonons) with spatial frequencies from the spatial Fourier transform of matter distribution of the screen. So I tend to make the following conclusion: when the mass of the incident particle increases, the momentum transferred to the screen remains the same, but the angle of deflection of the incident particle becomes smaller, as its momentum is greater. So the mass of the incident particle is in some sense an “external” parameter for the interference experiment.

8. Jul 19, 2014

### atyy

Just a note to illustrate that the de Broglie relation does make sense for large molecules http://arxiv.org/abs/1009.1569.

Interestingly, the abstract says "While the observation of Poisson's spot offers the advantage of non-dispersiveness and a simple distinction between classical and quantum fringes in the absence of particle wall interactions, van der Waals forces may severely limit the distinguishability between genuine quantum wave diffraction and classically explicable spots already for moderately polarizable objects and diffraction elements as thin as 100 nm. "

http://arxiv.org/abs/0903.1614 also has remarks about how the internal structure may affect these measurements.

"Large and thermally excited molecules often resemble small lumps of condensed matter. One consequence of this is that each individual many-body system may often be regarded as carrying along its own internal heat bath. This can determine the likelihood for exchange events between the quantum system and its environment, and thus a ect the molecular coherence properties."

"Complex, floppy molecules may undergo many and very different conformational state changes even while they pass the interferometer. Several electro-magnetic properties, for instance the electric polarizability or the dipole moment, will change accordingly. This, in turn, can affect both the molecular interaction with the diffraction elements as well as their probability to couple to external perturbations."

9. Jul 20, 2014

### atyy

I think these come pretty close to what you want. They start from the hydrogen atom, which is a system containing one proton and one electron, and derive that the centre of mass has a wave function which obeys the Schroedinger equation for a single free particle.
http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/HydrogenAtom.htm
http://aforrester.bol.ucla.edu/educate/Articles/Derive_HydrogenAtom.pdf

10. Jul 20, 2014

### dsoodak

This seems to be what I'm looking for (also: thanks for all the other interesting links everyone has posted on this thread).
Now I just have to wade through enough of it to get an intuitive understanding why it should come out so neatly...

11. Jul 20, 2014

### atyy

I'm unsure whether it comes out so neatly for everything, or whether it is only exact for the hydrogen atom. Of course, one only needs it to be approximate for more complex systems, but it would still be nice to know whether it is exact or approximate.

12. Jul 21, 2014

### dsoodak

That's exactly what I find to be so weird...

The masses of the neutron and proton are put into Schrodinger's equation as if they are fundamental particles, and it works perfectly even though each is actually 3 different particles all interacting with each other via 2 different forces (one of which isn't even an inverse square).

If THESE systems somehow magically work out, then there must be some general mathematical rule (maybe something to do with each particle's energy potential function being lowest in the vicinity of the other particles and the whole system being in its ground state most of the time). If there wasn't, then this would imply that there was something awfully suspicious going on.

13. Jul 21, 2014

### Staff: Mentor

That makes zero difference to the validity of the equation ie if they are fundamental objects or not.

As we have been discussing in another thread the validity of the Schroedinger equation is in fact a requirement of symmetry principles - see Ballentine - Chapter 3.

In treating the hydrogen atom the key insight is that its classical Hamiltonian translates to the quantum one. This is in fact a fundamental issue in QM - that in general classical Hamiltonians do not uniquely determine quantum ones.

Even our deepest formalism, the geometric approach, doesn't fully resolve that one.

Like I said symmetry considerations imply the quantum Hamiltonian has exactly the same form as the classical one. The general procedure is to interchange one for the other. Why is it valid? Well there are general theorems that show for expectation values they would have to be the same - so the answer is - simplicity. We do not postulate nature to be more complex than necessary unless the simple solution fails. It doesn't - so I guess our faith in the simplicity of nature worked here.

BTW if you wanted to analyse the hydrogen atom in terms of quarks etc that would mathematically involve the standard model and QFT - not QM.

The mathematical rule is dead simple. The quantum Hamiltonian corresponds to the classical one. In the case of the hydrogen atom the classical Hamiltonian is very easy - we have a light electron attracted via the coulomb force to a much heaver nucleus.

Exactly what holds the quarks in the nucleus together etc is irrelevant in this analysis any more than what holds electrons to the surface of objects is relevant to electrostatic experiments. Obviously something does it, and a deeper analysis will show what it is - but we do this in physics all the time - we abstract away inessentials. That's what's going on here - we abstracted away the inessential of what holds the quarks in protons together.

Thanks
Bill

Last edited: Jul 21, 2014
14. Jul 22, 2014

### atyy

As above, I can't answer in detail, but these are good questions, and hopefully some of the references in post #6 will help answer them. Here are a couple more references.

How do we get the proton mass from QFT?
http://arxiv.org/abs/1301.4905
http://theor.jinr.ru/~diastp/summer11/lectures/Jansen-1.pdf

How do we get the Schroedinger equation from QFT?
http://arxiv.org/abs/1402.5005

And in the spirit of your question, even after we get Schroedinger's equation, there are more mysteries. For example, electrons in a solid should interact by the Coulomb force. Yet band theory, which is so successful, seems to be the theory of single electrons! http://gdr-mico.cnrs.fr/UserFiles/file/Ecole/biermann_mico.pdf

I suspect it doesn't work out so nicely beyond the hydrogen atom, and is just some sort of approximation. If the wave function for the centre of mass were everything, then molecules with different internal structure but the same total mass would show the same de Broglie behaviour. But they don't: http://arxiv.org/abs/1405.5021. Actually, there is a flaw in my argument, so perhaps it does work out magically even for complex molecules.

Last edited: Jul 22, 2014
15. Jul 22, 2014

### Staff: Mentor

Indeed.

Take the simple explanation of the orbits of planets using Newtons laws. They are not really point particles - but aggregates of particles held together by gravitational forces. But what stops all those points clumping together into a single point. Its actually Pauli's exclusion principle - but how do we know if we include that the explanation doesn't break down. Without detailed calculations we don't know. Its simply what's reasonable and if you model it this way you get pretty good correspondence with observation.

Its really a fundamental issue with mathematical modelling in general - exactly what you can abstract away and what is crucial.

Thanks
Bill

16. Jul 22, 2014

### kith

I would say that a sufficient condition for this is that all solutions of the N-body Schrödinger equation can be built from product solutions of the form ψ(R,r) = χ(R)φ(r). Here, R is the center of mass coordinate and r is a shorthand notation for N-1 other coordinates. This method is called "separation of variables" and is possible if the Hamiltonian can be written as H = HR + Hr which I think is the case at least as long as there's no external potential.

/edit: So I think the basic question here is not specific to quantum mechanics.

Last edited: Jul 22, 2014
17. Jul 22, 2014

### Staff: Mentor

Neither do I.

I think it's a general issue with mathematically modelling anything. What you do is get rid of the inessential - and keep what's critical. I don't think there is any general answer other than what seems reasonable and what works.

The real issue here is students often don't think deeply about this stuff and sort of pick it up by osmosis so don't realize exactly what's going on.

It can require a bit of thought when taken to task about it.

Thanks
Bill