5.2: Solution
- Page ID
- 14775
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As stated before the continuity arguments for the derivative of the wave function do not apply for an infinite jump in the potential energy. This is easy to understand as we look at the behaviour of a low energy solution in one of the two outside regions (I or III). In this case the wave function can be approximated as
\[e^{ \pm k r}, \quad k=\sqrt{\frac{2 m}{\hbar^2} V_0},\]
which decreases to zero faster and faster as V 0 becomes larger and larger. In the end the wave function can no longer penetrate the region of infinite potential energy. Continuity of the wave function now implies that ϕ( a)=ϕ(− a)= 0.
Defining
\[\kappa=\sqrt{ \frac{2 m}{\hbar^2} E},\]
we find that there are two types of solutions that satisfy the boundary condition:
\[\phi_{2 n+1}(x)=\cos \left(\kappa_{2 n+1} x\right), \phi_{2 n}(x)=\sin \left(\kappa_{2 n} x\right) .\]
Here
\[\kappa_l=\frac{\pi l}{2 a} .\]
We thus have a series of eigen states \(\phi_l(x), l=1, \cdots, \infty\). The energies are
\[E_l=\frac{\hbar^2 \pi^2 l^2}{8 a^2}\]
These wave functions are very good to illustrate the idea of normalisation. Let me look at the normalisation of the ground state (the lowest state), which is
We need to require
\[\int_{-\infty}^{\infty}\left|\phi_0(x)\right|^2 d x=1\]
where we need to consider the absolute value since \(A_1\) can be complex. We only have to integrate from \(-a\) to \(a\), since the rest of the integral is zero, and we have
\[\begin{aligned}
\int_{-\infty}^{\infty}\left|\phi_0(x)\right|^2 d x & \\
& =|A|^2 \int_{-a}^a \cos ^2\left[\frac{\pi x}{2 a}\right] d x \\
& =|A|^2 \frac{2 a}{\pi} \int_{-\pi 2}^{\pi 2} \cos ^2\lceil y\rceil d y \\
& =|A|^2 \frac{2 a}{\pi} \int_{-\pi 2}^{\pi 2} \frac{1}{2}(1+\cos \lceil 2 y[]) d y \\
& =|A|^2 \frac{2 a}{\pi} \pi
\end{aligned}\]
Here we have changed variables from \(x\) to \(y=\frac{\pi x}{2 a}\). We thus conclude that the choice
\[A = \sqrt{ {1\over2a}}\]
leads to a normalised wave function.