5

Suppose, just to set some context, that we are dealing with a classic rectangular potential barrier, in one dimension. Now consider the following two possibilities:

  • A particle comes from the left of the system (e.g. an electron), what are the solutions for the system, the wavefunction of the particle, the probability current, the transmission coefficient, etc.

  • A particle beam comes from the left of the system (e.g. a beam of electron), what are the solutions for the system, the wavefunction, the probability current, the transmission coefficient, etc.

My question is: what changes between these two scenarios? (Can we even define a wavefunction or a probability current for a particle beam?)

I have no problem treating a single particle, but I have no idea on how to treat a beam.

But from what I saw, from my lecture notes, it seems that the resolution is almost identical in these two cases, but I don't get why. Seems to me that a particle beam should be treated as a collection of particles; instead in my lecture notes it simply gets its wave function like it was a single particle.

Ruslan
  • 28,862
Noumeno
  • 4,474

3 Answers3

4

If we suppose that in the beam of the particles, individual particles don't interact with each other then we can treat the beam as an ensemble of particles. Then you don't need to worry about the solution for the beam, only the solution for one particle is needed to calculate the transmission or reflection probabilities.

Note: Ensemble interpretation of $\mathbf{j}$

$\mathbf{j}\cdot d\mathbf{S}$ is the rate at which probability flows past the area $d\mathbf{S}$. If we consider an ensemble of $N$ particles all in some state $\psi(\mathbf{r},t)$, then $N\mathbf{j}\cdot d\mathbf{S}$ particles will trigger a particle detector of area $d\mathbf{S}$ per second, assuming that $N$ goes to infinity and that $\mathbf{j}$ is the current associated with $\psi(\mathbf{r},t)$.

Reference

  • Principles of Quantum Mechanics R Shankar Section 5.4
Young Kindaichi
  • 11,746
2

I think it is the interpretation of a wave function of more particles that confuses you (it did me!). To what refers the probability amplitude of such a wave function? You can't say that it corresponds to the probability of finding one particle within some space interval (corresponding to an interval of the wave function). It corresponds to the probability of finding one of all particles in that interval. Then what is the probability for finding just one particle in that interval? Exactly the same. The chance of finding one of all particles in the interval (along the space where the wave function is non-zero) is one. Which particle you find can't be said in advance.

Deschele Schilder
  • 1
2

I agree with the answer by @YoungKindaichi. However, I would like to add some clarity: when talking about particles/beams incident on a barrier we are dealing with a scattering problem rather than with an eigenvalue problem. While the two types of the problems mainly differ by the boundary conditions used, the normalization use for wave function is another peculiarity worth mentioning:

  • in eigenvalue problems the wave function is normalized by probability - its integral equals to the number of particles confined in the region.
  • scattering problems usually deal with extended states, where normalization of probability is impractical (the integral diverges). Thus, one often resorts to normalizing the particle flux (either to unity or to the number of particles incident per unit time in a body angle/area).
Roger V.
  • 58,522