In spherical coordinates ( from the axis, ; the azimuth, ) the angular momentum…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Orbital angular momentum, the spherical harmonics and rotations

Problem

In spherical coordinates (

\theta

from the

z

axis,

0\le\theta\le\pi

;

\phi

the azimuth,

0\le\phi<2\pi

) the angular momentum operators are

\hat L_z = -i\hbar\frac{\partial}{\partial\phi},\qquad \hat L^2 = -\hbar^2\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right],

with

\hbar

the reduced Planck constant, and the element of solid angle is

d\Omega = \sin\theta\,d\theta\,d\phi

. Consider the spherical harmonic

Y_1^{-1}(\theta,\phi) = \sqrt{\frac{3}{8\pi}}\,\sin\theta\,e^{-i\phi}.

((a)) Verify by integration over the whole sphere that

\int|Y_1^{-1}|^2\,d\Omega

has the value required of a normalised angular function. ((b)) Show that it is an eigenfunction of

\hat L_z

and give the eigenvalue as a multiple of

\hbar

. ((c)) Show that it is an eigenfunction of

\hat L^2

and give the eigenvalue as a multiple of

\hbar^2

. ((d)) Using the eigenvalues found, give the angle between the angular momentum vector and the

z

axis for this state, in degrees.