Throughout, is the polar angle from the axis, the azimuth, and L_z = -i , L^2 = - ^2…

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

Throughout,

\theta

is the polar angle from the

z

axis,

\phi

the azimuth, and

\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].

A turn through

\phi_0

about the

z

axis is carried out by

\hat R(\phi_0) = e^{-i\phi_0\hat L_z/\hbar}

, defined by its exponential series. ((a)) A classical point particle has, at one instant,

\mathbf r = (0,\,-2.0,\,1.5)

nm and

\mathbf p = (3.0,\,1.0,\,0)\times10^{-24}

kg m s

^{-1}

. Calculate the components of

\mathbf L = \mathbf r\times\mathbf p

, in J s. ((b)) Show that

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

is an eigenfunction of

\hat L^2

and of

\hat L_z

, and state the eigenvalues as multiples of

\hbar^2

and of

\hbar

. ((c)) A rotation about the

z

axis multiplies this

Y_1^{-1}

i

. Find the smallest positive angle

\phi_0

, in degrees, for which this happens.