If a non-interacting particle behaves like an undamped wave, can an interacting particle behave like a damped wave?

Question

As a standard textbook introduction to quantum mechanics, there are often examples, such as single particle in a box, described as waves. I'd like to better understand problems involving more particles and their chaotic time-dependent dynamics resulting from interactions.

I already know, the single-particle wavefunction does not apply to these situations (because we don't know its boundary conditions), but I'm not sure whether the notion of wave phenomena is completely abandoned when describing them. Perhaps there is a possibility to describe a single particle, that takes part in a many-body chaotic dynamics, as a damped wave propagation? Number of particles of this type may change as the wave amplitude decreases over time, however, composite particle formation accounts for conserved quantities so that fundamental laws of physics aren't violated, but rather hidden.

Could such a description capture the information loss associated with the heat loss of condensed matter, as its temperature is lowered?

If so, is there a phase transition associated with the single particle wave becoming overdamped?

Edit:

One of the answers so far has pointed out that my last two paragraphs were a

word salad

so, let me add some context. For example, on the low temperature side of the phase diagram of $^3He$ , there is a region of pressures where lowering temperature leads from solid to liquid. In a solid, every atom could be modeled almost like a particle in a box. By

the information loss associated with the heat loss of condensed matter, as its temperature is lowered

I meant the position information of the $^3He$ particle relative to its nearest neighbors. However, very similar, yet, more complicated phenomena happen within electrically conductive solids. That's why I need to ask.

Now you see, there is a phase transition in reality. I also asked:

is there a phase transition associated with the single particle wave becoming overdamped?

That was meant as a question about theoretical models.

Answer 1

In many-body quantum physics one introduces a concept of quasiparticle, which is a pole of the Green's function, and whose energy may have a negative part, which makes it behave as a damped wave.

To go into a bit more detail:

• firstly, when dealing with many identical particles, we cannot define a wave function for each particle, but only a common wave function describing all of them, which in addition has symmetric/antisymmetric property in respect to interchanging any two particles - depending on whether we deal with bosons or fermions. For non-interacting particles such wave functions can be obtained from single-particle wave functions via appropriate (anti-)symmetrization procedures (see Slater determinant and Permanent), but for interacting particles one needs to solve man-particle Schrödinger equation for the common wave function.
• Since the many-particle wave functions are unwieldy to work with, one usually works in terms if second quantization techniques and Green's functions. Note however that these Greeen's functions are actually Green's functions of the appropriate Schrödinger equations (for a conserved number of particles.) In particular, a particle is a pole of a one-particle Green's function. E.g., a retarded Green's function in momentum and frequency space, for non-interacting case, may have form: $$ G_0(\mathbf{k},\omega)=\frac{1}{\omega -\epsilon_\mathbf{k}/\hbar+i0^+} $$
• Once the interactions are included, the Green's functions usually can be obtained only via approximations - e.g., summing diagrammatic series or via path integral techniques. Yet, some general properties of these Green's functions can be determined - notably, like as functions of complex variable, they may have branch cuts and poles, i.e., a Green's function may have form $$ G(\mathbf{k},\omega)=\frac{1}{\omega -E_\mathbf{k}/\hbar+i\Gamma_{\mathbf{k}}/\hbar} $$
  Image: treating a pole in Green's function (taken from Fetter&Walecka)
  Comparing this with the previous expression, we may liken such a pole to a particle with complex energy, i.e., a decaying particle: $$ \psi(\mathbf{x},t)\propto e^{i\mathbf{k}\cdot\mathbf{x}-i(E_\mathbf{k}-i\Gamma_{\mathbf{k}})t/\hbar}= e^{i\mathbf{k}\cdot\mathbf{x}-iE_\mathbf{k}t/\hbar-\Gamma_{\mathbf{k}}t/\hbar} $$ Now, we can easily see that this is only approximately a particle - e.g., the normalization is not preserved. Indeed, after all we are really dealing with a complex excitation of a many-particle system, so what resembles a particle is really a collective movement of many particles, which doe snot always resemble a single particle. Still, representation in terms of various quasiparticles - electrons/holes, phonons, plasmons, magnons, polarons, excitons, etc.s has proven to be very successful. (Some of these can be conceptualized in simpler ways, but they are really all quasiparrticles.)

One of the spectacular successes of quasiparticles is Landau Fermi Liquid theory, which allows to describe properties of metals, as if they were collections of non-interacting particles, even though this is clearly not true due to the strong Coulomb forces:
Image: Fermi-like distribution function of Landau quasiparticles (image source):

Answer 2

No.

The wave function $\psi(x)$ tells you about the probability of finding the particle at $x$. The particle must be somewhere, so the total probability of finding it must be $1$. That is $\int\psi^*(x)\psi(x)dx = 1$.

This must remain true in all future times (unless the particle is destroyed. For example, a photon may be absorbed.) So the wave can diminish in amplitude in one spot only if it increases elsewhere. This is not what a damped wave does.

An interaction like absorption is not described by the wave evolving as described by the Schrodinger equation. This is a measurement. Before the measurement the particle is in one state. Afterward it is in a different state. You don't get an answer for what happens during the measurement.

The Schrodinger equation does describe the evolution of a particle that evolves in a potential. This is how an electron interacts with a nucleus in an atom. The Force of attraction of the nucleus is represented as a potential. For an atom, we are interested in the orbitals, the unchanging eigenstates of the Hamiltonian. For your question, you might want to consider how an electron evolves in a square well potential.

You might also find this Veritasium video useful. It describes interacting propagating particles on the way to the Many Worlds interpretation. Parallel Worlds Probably Exist. Here’s Why

Answer 3

I already know, the single-particle wavefunction does not apply to these situations (because we don't know its boundary conditions)

The conclusion is correct, but the reasoning is totally off. Why do you think we would have difficulties with the boundary conditions?

but I'm not sure whether the notion of wave phenomena is completely abandoned when describing them.

Quantum systems are always treated with waves. Why would we abandon perfectly good stuff?

Perhaps there is a possibility to describe a single particle, that takes part in a many-body chaotic dynamics, as a damped wave propagation?

Only for particle decay and creation.

Could such a description capture the information loss associated with the heat loss of condensed matter, as its temperature is lowered?

If so, is there a phase transition associated with the single particle wave becoming overdamped?

You have to ask better questions. These are word salad.

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It seems like you want to model heat phænomena in condensed matter systems. You can just consult the standard textbooks on the topic. It is part of the standard physics curriculum. Einstein and Debye's amazing work on the topic is always being taught, and it is built on top of single particle wave approximations, yet is remarkably good approximations nonetheless.

Answer 4

Many body quantum systems can sometimes be described by damping interactions:

https://arxiv.org/abs/2103.06646

In general copying information out of a quantum system will suppress interference: this is called decoherence. Many decoherence models feature damping, see Section 4 of this paper:

https://arxiv.org/abs/1911.06282