Quantum Mechanics - Eigenvalues and Eigenstates of the Momentum Operator & Angular Momentum Operator

Описание: Eigenvalues and Eigenstates of the Momentum Operator and the Angular Momentum Operator

(1) We know that fp = e^{-ipx/hbar} is an eigenfunction of the momentum operator. Find the wave function psi given by
A psi = \int_{-infinity}^{infinity} exp(-ipx/hbar) exp(ip'x/hbar) dx,
where A is a nonzero constant.

(2) Show that in the eigenstate of lz, the average values of Ix and Iy are zero.

(3) The component of angular momentum along the direction at an angle theta with the z-axis (i.e., the direction vect{r}) is given by
l_theta = sin(theta) cos(phi) lx + sin(theta) sin(phi) ly + cos(theta) lz,
where (r, phi, theta) are the spherical coordinates.
Calculate, in the eigenstate of lz, the average values of sin(theta) cos(phi) lx, sin(theta) sin(phi) ly, and cos(theta) lz.