PHYS 424 Exam 1October 12, 20011. (a) Find all the states |n> and energies En for the potential V = 0 if 0 < x < a, and V = ¥ otherwise. (b) If |y> = Sn an |n> what is the average value of the energy in the state |y> ? Justify your answer. (c) What is the average value of x3 in state |n>? Justify this answer as well. You need not explicitly evaluate integrals once they are in the form xn sinp cosq. 2. Consider the step function potential, V(x) = 0 if x £ 0, and V(x) = V0 if x> 0. Calculate the reflection coefficient for a free particle incident from the left with an energy E = 2 V0. Without further calculation, determine the transmission coefficient. 3. The solution to the time-independent Schroedinger equation for the harmonic oscillator is written |n> = h(x) exp( -x2/2) hn (x) = Sq aqxq Justify the use of this form of the solution, and in particular the details of the exponential factor. |
PHYS 424 Exam 1October 15, 20011. (a) Find all the states |n> and energies En for the potential V = 0 if 0 < x < a, and V = ¥ otherwise. (b) If |y> = Sn an |n> what is the average value of the energy in the state |y> ? Justify your answer. (c) What is the average value of p3 in state |n>? Justify this answer as well. You need not explicitly evaluate integrals once they are in the form xn sinp cosq. 2. Consider the step function potential, V(x) = 0 if x £ 0, and V(x) = V0 if x> 0. Calculate the reflection coefficient for a free particle incident from the left with an energy E = V0 / 2 and say why your calculation must be (or must not be) correct.
3. Suppose the Hamiltonian matrix is
(g f)
H = ( ) i.e. a square matrix
(f g)
(a) Find the eigenvalues of the time-dependent Shroedinger equation with this Hamiltonian and show that the eigenvectors are proportional to
(1) ( 1)
( ) and ( )
(1) (-1)
Normalize the eigenvectors.
(b) Find |Y(t) > if |Y(0)> is
(0)
( )
(1)
|
PHYS 424 Exam 1November 4, 20021. Given the Hermitian matrix ( 1 -2i ) ( ) (2i 1 )one of whose eigenvectors is ( 1 ) ( ) (-i )Find the other eigenvector and both eigenvalues. 2. Consider the finite square well, whose potential energy is and V(x) = V0 elsewhere. C sin (kx) + D cos (kx) for -a £ x £ a F e -kx + G e kx for x³ a (b) Determine k and k. (c) Considering unnormalized, odd solutions only, state but do not attempt to solve or simplify all conditions on the constants A ... G. Justify each condition. (d) How will the normalization condition be satisfied? [Explain what you would do to satisfy the condition but do not attempt to write down any specific equations.] 3. The angular momentum operators satisfy the commutation relations [Lx , Lz] = - i hbar Ly [Ly , Lz] = i hbar Lx |