Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 2 / 14A (Quantum Mechanics)

For a one-dimensional wavefunction, the probability current isA plane wave $D{e}^{ikx}$ carries current $(\hslash k/m)|D{|}^2$ to the right, while $D{e}^{-ikx}$ carries the negative of this current. Thus the asymptotic terms are interpreted as an incident wave of amplitude $A$, a reflected wave of amplitude $B$, and a transmitted wave of amplitude $C$. Their currents areCurrent conservation for a real potential gives $j_{\mathrm{in}}+j_{\mathrm{ref}}=j_{\mathrm{tr}}$, and thereforewith $P_{\mathrm{ref}}+P_{\mathrm{tr}}=1$.

A normalizable bound state has $E<0$. Writingits decaying asymptotic behaviour is

Write $T=\tanh (ax)$ and $S={\mathrm{sech}}^2(ax)$. Usingtwo differentiations ofgiveThus it solves the time-independent Schrodinger equation in the stated Pöschl-Teller potential, with

As $x\to -\infty$ and $x\to +\infty$, respectively,There is no left-moving term, so this is a reflectionless potential:The incident and transmitted amplitudes have equal modulus becauseHence

Now set $k=i\lambda$ with $\lambda >0$. ThenAt $+\infty$ this decays for every positive $\lambda$. At $-\infty$, however, the factor tends to $\lambda -a$ while $e^{-\lambda x}$ grows. Normalizability therefore requires $\lambda =a$. In that casewhich decays at both ends. The bound-state energy is