Well wave functions and finite barrier decay

Energy levels and wave functions for n = 1, 2, 3

For an infinite well of width $L$: $E_n = \dfrac{n^2h^2}{8mL^2}$ and $\psi_n = \sqrt{\dfrac{2}{L}}\sin\dfrac{n\pi x}{L}$. So $E_2 = 4E_1$, $E_3 = 9E_1$, and $\psi_n$ has $n$ half-wavelengths with $(n-1)$ nodes inside the well.

         V=inf                      V=inf
n=3      |   ****             ****  |
         | **    *           *    **|
E3=9E1   |--------------------------|
         |          **   **         |
         |            ***           |
         |                          |
n=2      |    ******                |
         |  **      **              |
E2=4E1   |--------------------------|
         |               **      ** |
         |                 ******   |
         |                          |
n=1      |        ***********       |
         |   *****           *****  |
E1       |--------------------------|
         |                          |
         0                          L

The levels get wider apart as $n$ increases ($\Delta E \propto 2n+1$). $\psi$ is zero at both walls because the electron cannot enter a region of infinite potential.

Exponential decay inside a finite barrier

Let an electron of energy $E$ meet a barrier of height $V_0 > E$ that starts at $x = 0$.

 V(x)
  ^          ________________
  |    V0   |
  |  E -----|- - - - - -
  |  wave   |  decaying
  |  ~~~~~~ |\__
  |_________|___\___________> x
  Region I  0  Region II

Region II ($x > 0$, $V = V_0$): the time-independent Schrödinger equation is

$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V_0\psi = E\psi \quad\Rightarrow\quad \frac{d^2\psi}{dx^2} = \frac{2m(V_0 - E)}{\hbar^2}\psi = \alpha^2 \psi$$

where

$$\alpha = \frac{\sqrt{2m(V_0 - E)}}{\hbar}$$

is real and positive because $V_0 > E$. The general solution is

$$\psi_{II}(x) = C e^{-\alpha x} + D e^{\alpha x}$$

The term $De^{\alpha x}$ grows without limit as $x \to \infty$, so $\psi$ could not be normalised and $|\psi|^2$ would not be a valid probability. Therefore $D = 0$ and

$$\psi_{II}(x) = C e^{-\alpha x}, \qquad |\psi_{II}|^2 = |C|^2 e^{-2\alpha x}$$

The constant $C$ is fixed by matching $\psi$ and $d\psi/dx$ to the oscillating wave $Ae^{ikx} + Be^{-ikx}$ of region I at $x = 0$.

Result: the wave function inside the barrier is not zero (the electron can penetrate it), but it decays exponentially with distance. The penetration depth is $\delta = 1/\alpha$; it is smaller for a heavier particle or a higher barrier ($V_0 - E$ larger). This finite penetration is the basis of tunnelling through thin barriers.