PHYS 424 Final Exam

December 14, 2001

Ñ2 = (1/r2)/r (r2/r) + 1/(r2sinq) /q (sinq /q) + 1/(r2 sin2q) 2/f2

Y00 = 1/sqrt(4p)
Y10 = sqrt[3/(4p)] cos(q)
Y1-1 = + sqrt[3/(8p)] sin(q) e -if       Y11 = - sqrt[3/(8p)] sin(q) e +if

 

1. Consider the potential

V(x) = 0,         0 < x < a
¥ ,       x < 0 or x > a

(a) Show carefully that the states |n> = (2/a)1/2 sin(pnx/a) and the energies En = hbar2p2n2/a2.

(b) What is the time dependence of Yn(x,t)?

(c) What values of n are allowed? What is the degeneracy of the state |n>?

(d) Write down the matrix <n'|H|n> [at least enough of it that the form is apparent].

(e) In state |n> of this system, what values of p are allowed, and what is the probability of a measurement of pop giving each of these values?

 

2. If in the preceding problem, a term H' = b cos(2px/a) with b small is added to the Hamiltonian, what is the energy of each state to first order in the parameter b? [Hint: 2 sin A sin B = cos(A+B) - cos(A-B) and 2 cos A cos B = cos(A+B) + cos(A-B) ].

 

3. Given that the operators     S± = Sx ± iSy     obey     S±|s=1/2, sz=m>   =   hbar |s=1/2, sz=m±1>     whenever the state on the right-hand side of the relation exists, show that |s=1/2, sz=1/2> is an eigenstate of the operator S2 with eigenvalue (3/4)hbar2 .

4. For the Hamiltonian H = T + V = p2/(2m) + (1/2)mw2x2 use the operators a± = [1/sqrt(2m)] [px ± imwx] to show that <T> = <V> , where the expectation values are taken in an eigenstate of the Hamiltonian. [If you must refer to an integral or a matrix element that involves an integral to work this problem, name it and do not perform the integration. More credit will be awarded for a solution that does not need the specific value of any integral.]

 

5.Consider the potential

V(r) = 0,         0 < r < a
¥ ,       r > a
where r is the radial coordinate in spherical coordinates.

(a) Show that the l= 1 Y11 given in the box at the top of the exam gives the angular dependence of some of the states, and find the differential equation for the radial dependence of these l=1, ml=1 states.

(b) Without solving for the radial dependence find the best minimum value you can for the number of states degenerate with (having the same energy as) each one of these states.

(c) Is it guaranteed, likely, unlikely, or impossible that the actual degeneracy is higher than your minimum? Explain.

Last Revised 02/12/11