Franz Schwabl Quantum Mechanics 4th Edit · Problem 5.7

The Central Potential I

Physics — Quantum Mechanics · orbital-angular-momentum-integer-eigenvalues

Problem

Show that the eigenvalues

l

of the orbital angular momentum are non-negative integers by showing that the eigenvalues of

L_z

are integers. Hint: Express by means of the transformation

x_i = \sqrt{\frac{\hbar}{2m\omega}}\left(a_i + a_i^\dagger\right),

p_i = -\,\mathrm{i}\sqrt{\frac{\hbar m\omega}{2}}\left(a_i - a_i^\dagger\right)

(with arbitrary

m

and

\omega

) the operator

L_z

in terms of creation and annihilation operators. Bring

L_z

by means of a linear transformation to new annihilation operators

b_1

and

b_2

into the form

L_z = \hbar\left(b_2^\dagger b_2 - b_1^\dagger b_1\right).

(See also Appendix C.)