Problems

Find the condition that must be satisfied by a spherically symmetric square well potential V(r)=-V₀, r<r₀; V(r)=0, r³r₀ if it is just barely deep enough to contain one bound state.

Solution:

If the potential only supports one bound state, this state must be an l=0 state. The radial Schroedinger equation then is

Let

Then from the previous problem we have , in regions where cotkr₀ < 0.

If we plot f(k)=|sinkr₀| and g(k)=k/k₀ we find solutions at the intersections of the two curves in regions where cotkr₀ < 0, i.e. . To only support one bound state with E nearly zero, we need .

The smallest possible value for V₀ is .

A particle of mass m is bound in a 2-dimensional isotropic oscillator potential with a spring constant k. Use the fact that

and write the Schroedinger equation for this system in both Cartesian and polar coordinates. Separate the equation in polar coordinates and solve the resulting equation in q. Write the resulting radial equation utilizing the q solution, but do not solve it. Demonstrate the connection between the q solution and its classical analog.

Solution:

is the Schroedinger equation in Cartesian coordinates,

is the Schroedinger equation in polar coordinates.

Let y=f(r)c(q). Then

Multiply by .

independent of q independent of r

The first term is a function of r only and the second term is a function of q only. Both terms must be equal to a constant and the sum of these constants must be zero.

since .

The q solutions are eigenfunctions of L_z with eigenvalues nh.

commutes with H, L_z is a constant of motion, quantum mechanically as well as classically. The resulting radial equation is

A particle of charge -e and mass m is under the influence of two stationary heavy nuclei, each with charge Ze positioned at z=±a. We shall further assume that the particle is spinless and non relativistic.

(a) What is the Hamiltonian and the Schroedinger equation of the system.

(b) Define the angular momentum operator along the z-direction, L_z, and show that its eigenvalues are good quantum numbers for all of the non-degenerate energy eigenstates. What are the possible eigenvalues of L_z?

(c) Define the parity operator P and show that parity is a good quantum number for all the non-degenerate energy eigenstates. What are the possible eigenvalues of P?

(d) Define the total angular momentum operator L and show that the eigenvalues of L² are not good quantum numbers for the energy eigenstates.

Solution:

(a)

is the Hamiltonian of the electron.

is the Schroedinger equation. In spherical coordinates we have

(b) We know [L²,L_z]=0, therefore [H,L_z]=0, since the L² term is the only one with a f dependence. If two operators commute, then a common eigenbasis can be found. An eigenstate with a non degenerate eigenvalue of H is also an eigenstate of L_z. The possible eigenvalues of L_z are

[P,L²]=0, since changing and does not change the operator L².

[P,V(r)]=0, since changing and does not change the operator V(r).

Therefore [P,H]=0. As above, an eigenstate with a non-degenerate eigenvalue of H is also an eigenstate of P. The eigenvalues of P are ±1.

(d)

No common eigenbasis of L² and exists.

Two particles of mass m₁ and m₂ are separated by a fixed distance r. There center of mass is fixed at the origin of the of the coordinate system and they are free to rotate about their center of mass. (The system is a "rigid rotator".)

(a) Write down the Hamiltonian of the system.

(b) Find the eigenvalues and eigenfunctions of this Hamiltonian. What is the separation between adjacent levels? What is the degeneracy of the eigenvalues?

Solution:

	(a) for two interacting particles. Here p_r² =0, therefore .
	(b) The eigenfunctions of H are the eigenfunctions of L². These are the spherical harmonics, . The eigenvalues are . It is customary to set , and write . The separation between adjacent levels is , it increases linear with l. Each energy eigenvalue is (2l+1) fold degenerate.