(a) The potential $V(x)$ for a particle of mass $m$ in one dimension is such that $V \rightarrow 0$ rapidly as $x \rightarrow \pm \infty$. Let $\psi(x)$ be a wavefunction for the particle satisfying the time-independent Schrödinger equation with energy $E$.

Suppose $\psi$ has the asymptotic behaviour

$$\psi(x) \sim A e^{i k x}+B e^{-i k x} \quad(x \rightarrow-\infty), \quad \psi(x) \sim C e^{i k x} \quad(x \rightarrow+\infty)$$

where $A, B, C$ are complex coefficients. Explain, in outline, how the probability current $j(x)$ is used in the interpretation of such a solution as a scattering process and how the transmission and reflection probabilities $P_{\mathrm{tr}}$ and $P_{\text {ref }}$ are found.

Now suppose instead that $\psi(x)$ is a bound state solution. Write down the asymptotic behaviour in this case, relating an appropriate parameter to the energy $E$.

(b) Consider the potential

$$V(x)=-\frac{\hbar^{2}}{m} \frac{a^{2}}{\cosh ^{2} a x}$$

where $a$ is a real, positive constant. Show that

$$\psi(x)=N e^{i k x}(a \tanh a x-i k)$$

where $N$ is a complex coefficient, is a solution of the time-independent Schrödinger equation for any real $k$ and find the energy $E$. Show that $\psi$ represents a scattering process for which $P_{\text {ref }}=0$, and find $P_{\mathrm{tr}}$ explicitly.

Now let $k=i \lambda$ in the formula for $\psi$ above. Show that this defines a bound state if a certain real positive value of $\lambda$ is chosen and find the energy of this solution.