Franz Schwabl Quantum Mechanics 4th Edit · Problem 5.6

The Central Potential I

Physics — Quantum Mechanics · angular-momentum-eigenfunction-rotation

Problem

(a) Find the eigenfunction

\psi

\boldsymbol{L}^2

and

L_x

with eigenvalues

2\hbar^2

and

\hbar

, respectively. (Hint: Represent the eigenfunctions of

\boldsymbol{L}^2

and

L_z

of interest in terms of Cartesian coordinates, and determine

\psi

by a rotation through

\pi/2

.) (b) Express

\psi

as a linear combination of eigenfunctions of

\boldsymbol{L}^2

and

L_z