How do I figure out the probability of finding a particle between two barriers?

Question

Given a delta function $\alpha\delta(x+a)$ and an infinite energy potential barrier at $[0,\infty)$, calculate the scattered state, calculate the probability of reflection as a function of $\alpha$, momentum of the packet and energy. Also calculate the probability of finding the particle between the two barriers.

I start by setting up the standard equations for the wave function:

$$\begin{align}\psi_I &= Ae^{ikx}+Be^{-ikx} &&\text{when } x<-a, \\ \psi_{II} &= Ce^{ikx}+De^{-ikx} &&\text{when } -a<x<0\end{align}$$

The requirement for continuity at $x=-a$ means

$$Ae^{-ika}+Be^{ika}=Ce^{-ika}+De^{ika}$$

Then the requirement for specific discontinuity of the derivative at $x=-a$ gives

$$ik(-Ce^{-ika}+De^{ika}+Ae^{-ika}-Be^{ika}) = -\frac{2m\alpha}{\hbar^2}(Ae^{-ika}+Be^{ika})$$

At this point I set $A = 1$ (for a single wave packet) and set $D=0$ to calculate reflection and transmission probabilities. After a great deal of algebra I arrive at

$$\begin{align}B &= \frac{\gamma e^{-ika}}{-\gamma e^{ika} - 2ike^{ika}} & C &= \frac{2e^{-ika}}{\gamma e^{-ika} - 2ike^{-ika}}\end{align}$$

(where $\gamma = -\frac{2m\alpha}{\hbar^2}$) and so reflection prob. $R=\frac{\gamma^2}{\gamma^2+4}$ and transmission prob. $T=\frac{4}{\gamma^2+4}$.

Here's where I run into the trouble of figuring out the probability of finding the particle between the 2 barriers. Since the barrier at $0$ is infinite the only leak could be over the delta function barrier at $-a$. Would I want to use the previous conditions but this time set $A=1$ and $C=D$ due to the total reflection of the barrier at $0$ and then calculate $D^*D$?

Answer 1

Hints to the question(v5):

1. OP correctly imposes two conditions because of the delta function potential at $x=-a$, but OP should also impose the boundary condition $\psi(x\!=\!0)=0$ because of the infinite potential barrier at $x\geq 0$.

2. There is zero probability of transmission because of the infinite potential barrier at $x\geq 0$. (Recall that transmission would imply that the particle could be found at $x\to \infty$, which is impossible.)

3. Hence there is a 100 percent probability of reflection, cf. the unitarity of the $S$-matrix. See also this Phys.SE answer.

4. As OP writes, away from the two obstacles, one has simply a free solution to the time-independent Schrödinger equation, namely a linear combination of the two oscillatory exponentials $e^{\pm ikx}$. This solution is non-normalizable over a non-compact interval $x\in ]-\infty,0]$.

5. To make the wave function normalizable, let us truncate space for $x< -K$, where $K>0$ is a very large constant. So now $x\in [-K,0]$. One may then define and calculate the probability $P(-a \leq x\leq 0)$ of finding the particle between the two barriers via the usual probabilistic interpretation of the square of the wave function.

6. If we now let the truncation parameter $K\to \infty$, then we can deduce without calculation that this probability $P(-a \leq x\leq 0)\to 0$ goes to zero.

Answer 2

The probability of finding a particle in an interval $a<x<b$ is given by the integral $$\int_a^b \psi^* \psi \, dx ,$$ assuming that your wave function is properly normalised.

So in your case, you should calculate $$\frac{\int_{-a}^0 \psi_{II}^* \psi_{II} \,dx}{\int_{-\infty}^{-a} \psi_{I}^* \psi_{I} \,dx+\int_{-a}^0 \psi_{II}^* \psi_{II} \,dx} . $$

The numerator is the region you are interested in, the denominator takes care of the normalisation so that the probability will come out between 0 and 1. I'll leave it to you to calculate the integrals.