Let a quantum particle be subject to a one dimensional step potential barrier $V$ such that:

$$V(x)=\begin{cases}0, \ x<0 \\ V_0, \ x>0\end{cases}$$

where the particle's energy is $E>V_0>0$. The particle will be free in all its domain, since its energy is always superior to that of the barrier. From the particle's probability density continuity equation:

$$\frac{\partial |\Psi|^2}{\partial t}+\vec{\nabla}\cdot \vec{J}=0$$

we deduce said probability density will behave as a fluid. Up to this point, everything makes sense. Nonetheless, my professor then goes on to calculate the flux $\vec{J}_-$ in the region $x<0$ and $\vec{J}_+$ for $x>0$, and states this is a flux of particles, not just one. Also, when calculating the transmission and reflection coefficients, he claims these are valid for a beam of particles (for instance, if $R=1/2$, he states "half of the particles present in the beam will be reflected, and the rest will be transmited"). However, since we obtained the reflection and transmission coefficients for a single particle, what sense does it make to speak about beams of particles?

My best guess is what is really reflected / transmitted is the probability density (since that is what behaves like a fluid), and therefore if we shoot many particles to the step potential, the fraction of particles reflected will tend to the reflection coefficient.


1 Answer 1


Your last statement is basically correct. For a single particle, the current $\vec{J}$ is measuring a current of probability density $|\psi|^2$. However, to physically interpret what such a probability means, we typically imagine saying that we will repeat the experiment with many particles prepared in the same initial state, and then the relative probabilities for different outcomes translate into relative numbers of particles that we observe following those outcomes.

A beam of particles is (at least in an idealized sense) a collection of independent particles with the same initial momentum. Now, if you want to be more precise, you could build in some complicating factors, such as the fact that there is likely some distribution of momentum for the particles in the beam, and there may be some interactions between the particles in the beam. However, to zeroth order, a good beam of particles in a scattering experiment will behave as independent particles with the same initial state, so you can analyze its behavior by calculating what will happen to one particle and then interpreting the resulting probabilities for "out states" in terms of what happens to numbers of particles in the beam.

Mathematically, we're essentially assuming that the wavefunction for $N$ particles factors into $N$ one particle wavefunctions $$ \Psi(x_1, x_2, \cdots, x_N) = \psi(x_1) \psi(x_2) \cdots \psi(x_N) $$ and solving for $\psi(x)$ (which we are assuming is the same for all $N$ particles since the particles have the same initial state and encounter the same potential).


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