A Modern Approach to Quantum Mechanics · Problem 6.10

The One-Dimensional Harmonic Oscillator — John S. Townsend

Physics — Quantum Mechanics · schrodinger-equation-linear-potential

Problem

The one-dimensional time-independent Schr\"odinger equation for the potential energy discussed in Problem 6.9 is

\frac{d^2\psi}{dx^2} + \frac{2m}{\hbar^2}(E - a|x|)\psi = 0

Define

E = \varepsilon(\hbar^2 a^2/m)^{1/3}

and

x = z(\hbar^2/ma)^{1/3}

. (a) Show that

\varepsilon

and

z

are dimensionless. (b) Show that the Schr\"odinger equation can be expressed in the form

\frac{d^2\psi}{dz^2} + 2(\varepsilon - |z|)\psi = 0

\varepsilon

, beginning with

d\psi/dz = 0

z = 0

, to find the value of

\varepsilon

corresponding to the ground-state eigenfunction.