# De Broglie Wavelength - Compound Particles, Particle Systems

Can someone describe the physical processes which distinguish between separate and single particles when dealing with a collection of particles in the context of the De Broglie wavelength?

The De Broglie wavelength is inversely proportional to the momentum of a "particle".

Assume "separate" particles A and B (e.g. separated but moving with the same speed and direction) and having momentum a and b respectively, thus each have wavelengths proportional to 1/a and 1/b.

Loosely speaking one can look at the system of particles A and B, and it has a momentum a+b, which implies that the "system" (if it can be seen as a "compound' particle) has a wavelength proportional to 1/(a+b).

The interference outcomes based on these two different wavelengths contradict one another.

So given any context it would seem physically speaking there is a mechanism which physically differentiates between "separate particles" which have De Broglie wavelengths based on the particles separately and "compound" particles which have De Broglie wavelengths based on the sum of momenta of the individual particles.

When do separate particles of a system physically become "one particle" (according to De Broglie) and why?

What physical mechanism makes this distinction in reality? Nuclear forces? Covalent bonds? Are ionic bonds enough? Does the size of the compound particle matter or only the magnitude of the binding forces?

vanhees71
Gold Member
2021 Award
Two particles never become one particle, because a two-particle state is definitely different from any one-particle state, because the two-particle state has definite particle number 2 and the one-particle state definite particle number 1. Since the particle number is an observable these states are even orthogonal to each other and thus definitely exclusive. Perhaps, I don't understand the question correctly, but that's pretty self-evident, right?

Two particles never become one particle, because a two-particle state is definitely different from any one-particle state, because the two-particle state has definite particle number 2 and the one-particle state definite particle number 1. Since the particle number is an observable these states are even orthogonal to each other and thus definitely exclusive. Perhaps, I don't understand the question correctly, but that's pretty self-evident, right?

One would assume so.

What does it mean for a multi-particle system like an atom or a molecule to have a single De Broglie "wavelength"?

My question is essentially why physically (and how and when) does analyzing particles separately become inapplicable to the system?

vanhees71
Gold Member
2021 Award
I still don't understand the question. If you have a composite system you still can look at the motion of its center of mass (non-relativistically) or its center of momentum (relativistically) and you can associate a de Broglie wavelength with that motion.

I still don't understand the question. If you have a composite system you still can look at the motion of its center of mass (non-relativistically) or its center of momentum (relativistically) and you can associate a de Broglie wavelength with that motion.

Is this then an unwarranted assumption?
The interference outcomes based on these two different wavelengths contradict one another.

Two cases have to be distinguished

1) A and B are different particles. In this case the interference patterns of A, the one of B and the one of AB are different and in order to measure a pattern you will have to build an apparatus which measures the pattern of the particle of your choice, but not all at once.

2)A and B are identical particles(for simplicity let's assume their momenta are the same).
If there is no bonding between the particles, both particles will just exhibit the interference of a single free particle(because that's what they are after all). Their dynamics is determined by seperate hamiltonians H_A and H_B(which have the same form). This is why in the double-slit experiment, the intensity of the beam only makes the pattern brighter, but doesn't change it.
When there is interaction between the particles, a third term H_int(A,B) has to be added in the hamiltonian that changes the physics, it 'tries to keep both particles together' so that they are no longer independent.
In the limit of very high interaction, A and B are closer to each other than their own wavelenghts. The interference patterns of A and B now interfere with each other precisely creating the interference pattern of AB, which would result from a single particle AB in the center-of-mass.

1) A and B are different particles. In this case the interference patterns of A, the one of B and the one of AB are different and in order to measure a pattern you will have to build an apparatus which measures the pattern of the particle of your choice, but not all at once.

This is tied to the heart of the question. WHEN do you analyze a set of different particles (e.g. an electron and a proton) as two particles each having their own wavelength but NOT having a collective wavelength and when do you analyze the set of different particles as a single particle (a hydrogen atom) having a collective third wavelength, but not having their own wavelengths.

In the limit of very high interaction, A and B are closer to each other than their own wavelenghts. The interference patterns of A and B now interfere with each other precisely creating the interference pattern of AB, which would result from a single particle AB in the center-of-mass.

I'm not quite sure what you mean by "interference patterns" as such interfering "with each other". I understand that particles can interfere with each other which gives rise to an interference pattern, but the concept of interference patterns interfering with each other eludes me.

This is tied to the heart of the question. WHEN do you analyze a set of different particles (e.g. an electron and a proton) as two particles each having their own wavelength but NOT having a collective wavelength and when do you analyze the set of different particles as a single particle (a hydrogen atom) having a collective third wavelength, but not having their own wavelengths.

measuring the different particles seperately gives you the most information and hence it is the more precise method. When you know the two particles are tightly bound, you can approximate the whole as a single particle.

I'm not quite sure what you mean by "interference patterns" as such interfering "with each other". I understand that particles can interfere with each other which gives rise to an interference pattern, but the concept of interference patterns interfering with each other eludes me.

Imagine a swimming pool with a wall inside that seperates the front and the back of the water. There are two vertical slits in the wall. Drop 1 stone at the front, and the wavefronts from the different slits interfere behind the wall. Now drop two stones at the front, all the waves in the water add up.

measuring the different particles seperately gives you the most information and hence it is the more precise method. When you know the two particles are tightly bound, you can approximate the whole as a single particle.

Imagine a swimming pool with a wall inside that seperates the front and the back of the water. There are two vertical slits in the wall. Drop 1 stone at the front, and the wavefronts from the different slits interfere behind the wall. Now drop two stones at the front, all the waves in the water add up.

I don't need to be reminded of undergrad experiments. (BTW good luck with your MSc studies!) But since you bring up the double slit experiment imagine throwing hydrogen atoms at a double slit one at a time, they form a pattern based on the separation of the slits, the distance to the screen, and the wavelength of the atom which is calculable from its momentum.

The pattern has a spacing z.

Imagine throwing the atoms closer together in time, in a pairwise pattern. One after another. Assume for argument sake if you throw them close enough in time they can be seen as a "system" having twice the mass as a single atom, this means each "bundle" has twice the momentum and hence half the wavelength.

The pattern now would have a spacing of z/2.

But the problem here is we have two completely different patterns and the only distinction between when we predict one pattern versus when we would predict the other pattern is arbitrary. We simply have to guess when the system can be treated as a single particle rather than two. This cannot be the case, as reality is not determined by guesses physicists make.

So what ARE the physical bases for the change in the wavelengths being separate and one value to there being a single collective one of a different value?

Thanks for the wishes first of all. If you really want a criterion for how much these two particles are bound i assume the two-body correlation function is the one you're looking for(maybe someone else can confirm this). Anyway, the difference between e.g. two H-atoms and one H2-molecule is that in the first case the atoms are independently moving, making it possible that A goes trough slit 1 and B through slit 2 simultanuously. If A and B are strongly bound, AB will go through a single slit, so it is not possible to detect the the two particles at a different slit.

Anyone here (sufficiently advanced) care to provide a definitive and explicit answer to this?

"
imagine throwing hydrogen atoms at a double slit one at a time, they form a pattern based on the separation of the slits, the distance to the screen, and the wavelength of the atom which is calculable from its momentum.

The pattern has a spacing z.

Imagine throwing the atoms closer together in time, in a pairwise pattern. One after another. Assume for argument sake if you throw them close enough in time they can be seen as a "system" having twice the mass as a single atom, this means each "bundle" has twice the momentum and hence half the wavelength.

The pattern now would have a spacing of z/2.

But the problem here is we have two completely different patterns and the only distinction between when we predict one pattern versus when we would predict the other pattern is arbitrary. We simply have to guess when the system can be treated as a single particle rather than two. This cannot be the case, as reality is not determined by guesses physicists make.

So what ARE the physical bases for the change in the wavelengths being separate and one value to there being a single collective one of a different value?"

Bump.

vanhees71
Gold Member
2021 Award
I can't help. As I said above, I don't understand the problem you have. Of course, de Broglie's original theory (his PhD thesis in fact, approved by Einstein) is just a predecessor to full quantum theory developed only very little later in 1925 by Heisenberg, Born, Jordan as "matrix mechanics" and Dirac as "transformation theory", i.e., the basis-free approach, and in 1926 by Schrödinger as "wave mechanics" (which just is the position representation of the QT of a non-relativistic single particle). So, maybe it's a wise advice to learn modern quantum theory, which is much more consistent than the original idea in de Broglie's thesis!

Bump.