Assignment No. 6
Physics 4P51
Due Monday, November 4, 2013
1. Represent the components sx , sy , sz of a spin-1/2 particle in the eigenbasis cfw_|+ , |
of sz and express them in terms of Pauli matrices.
2. Show that the eigenvector of sn n correspondin
Assignment No. 1
Physics 4P51
Due Wednesday, September 17, 2013
1. (a) Show that the normalized eigenfunctions of the Hamiltonian for a particle in
one-dimensional box of length L
2 2
= h d + V (x) ,
H
2m dx2
with
+ if x 0 or x L
0
if 0 < x < L
V (x) =
a
Assignment No. 7
Physics 4P51
Due Thursday, November 11, 2013
1. The ammonia molecule NH3 possesses an electric dipole moment d because the valence electrons are clustered closer to the nitrogen nucleus than to the protons. As a
result, even though the mo
Assignment No. 3
Physics 4P51
Due Monday, October 7, 2013
1. The adjoint A of operator A is dened by
(, A ) = (A, ) = (, A)
for arbitrary and from H. Show that
(a) (A ) = A;
(b) (A + B) = A + B ;
(c) (AB) = B A ;
2. If A and B are Hermitian show that
Assignment No. 2
Physics 4P51
Due Monday, September 30, 2013
1. The inner (or scalar) product in a linear space H of complex functions (r) is dened
by
| (, ) = d3 r (r)(r) .
Convince yourself that
(a) | 0 with the equal sign valid only if (r)=0 (everywher
Assignment No. 9
Physics 4P51
Due Monday, November 25, 2013
1. Consider again the ammonia molecule NH3 from Problem 1, Assignment No. 7.
The Hamiltonian of the ammonia molecule in the absence of an external electric eld
is
H0 = 0 (|1 1| + |2 2|) V |1 2| V
Assignment No. 8
Physics 4P51
Due Monday, November 18, 2013
1
1. If at t=0 the electron spin 2 was in state |sn + , where the direction of unit vector
n is given by = and =0 (i.e. n is in xz-plane at an angle relative to the
z-axis), calculate sy = (t)|y
Assignment No. 5
Physics 4P51
Due Tuesday, October 22, 2013
1. Prove that
(a)
tr(AB) = tr(B A)
even when [A, B] =0, i.e. two operators always commute underneath the trace.
(b)
tr(AB C) = tr(B C A) = tr(C AB) ,
i.e. the trace is invariant under the c
Assignment No. 4
Physics 4P51
Due Thursday, October 21, 2013
1. The orbital angular momentum of a particle is dened by
= p =
l r
i j k
x y z
px py pz
,
so that x = y pz z py , etc.
l
Using the results from previous assignments show that
(a) x , y and