Franz Schwabl Quantum Mechanics 4th Edit · Problem 3.6

The Uncertainty Relation

Physics — Quantum Mechanics · sech-squared-potential

Problem

Solve the time independent Schr\"odinger equation for the potential V(x)=−ℏ2/(ma2cosh⁡2xa) .V(x) = -\hbar^2 \Big/ \left( ma^2 \cosh^2\frac{x}{a} \right) \ . Hint: Introduce y=x/ay = x/a, make the ansatz ψ(y)=eikyφ(y) ,\psi(y) = \mathrm{e}^{\mathrm{i}ky} \varphi(y) \ , and substitute z=tanh⁡yz = \tanh y. The resulting equation can be solved in elementary fashion by a power series together with a condition for the series to terminate. Discuss the solutions that occur.

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