Tunneling

A beam of particles of mass $m$ and kinetic energy $E<eV_0$ enter from the right, traveling from right to left, toward a potential function $V(x)$ of shape shown in Fig. 3.

(a)

What is the momentum $p$ of the incoming particles?

(b)

Taking $k=p/\hbar$, show that the wavefunction

$$\Psi_>(x) = e^{-ik(x-a)} + \underline{r} e^{ik(x-a)}$$

satisfies the Schr $\ddot{\rm o}$dinger's equation for $x>a$, while the wavefunction

$$\Psi_<(x) = \underline{t} e^{-ikx}$$

satisfies the Schr $\ddot{\rm o}$dinger's equation for $x<0$. What is the physical meaning of $\underline{r}$ and $\underline{t}$?

Note: In lecture on Dec. 2, we will show that the Schrödinger equation (equation of motion for the wave function $\Psi(x)$) is

$$-\frac{\hbar^2}{2m} \frac{d^2 \Psi(x)}{dx^2} + V(x) \Psi(x) = E \Psi(x).$$

HINT: Recall problem set # 8, problem 1.

(c)

Show that the wavefunction

$$\Psi_{\rm in}(x) = Ce^{-\alpha x} \,+\, De^{\alpha x}$$

satisfies the Schr $\ddot{\rm o}$dinger's equation for $0<x<a$ for any values of $C, D$. Determine $\alpha$.

(d)

Write down the equations that should be satisfied by the wavefunctions at $x=0$ and $x=a$ in terms of the unknown coefficients $C$, $D$, $\underline{t}$, $\underline{r}$.
Note: In lecture on Dec. 2, we will show that the wave function $\Psi(x)$ and its derivative $d\Psi(x)/dx$ are always both continuous for finite potentials.

(e)

By solving the equations obtained in part (d), find the coefficients $\underline{r}$ and $\underline{t}$. What is the probability that the incoming particle will pass the barrier (or ``tunnel'' through it)?
HINT: It is easiest to first solve for $C$ and $D$ in terms of $\underline{t}$ from the equation at $x=0$, and then to substitute the result into the equation you get from the boundary conditions at $x=a$.

Figure 3: Tunneling.