Solutions Homework 6
Physics 486, Fall 2014
University of Illinois at UrbanaChampaign
September 26, 2014
1
Problem 1: Harmonic Oscillator in Real Space
(a)
i t (x,t) = h 1/2
h
ma2 2it max it
e
+
e
(
UIUC Physics 486: Solutions for for HW 10
December 2, 2014
Problem 1: Matrix Elements of Ylm
(a): Using the relations, Lx = 1 (L+ + L ), and Ly =
2
1
2i (L+
L ), we can see that
Ylm  Lx  Yl m =
1
2
Homework 12
November 14, 2014
1. Grovers Algorithm
In this problem you will go carefully through the steps of Grovers algorithm
that we saw in class generalized to M marked elements. Assume the elemen
Homework 11
November 7, 2014
1. Additional degeneracy in the hydrogen atom
(a)Explicitly verify that
A=
1
( L
p
2m
L p)
e2
r
r
commutes with the coulomb Hamiltonian.
(b) Write A+ =
on 300 .
p
1/ 2(
Solution to HW11
December 4, 2014
Problem I: Additional degeneracy in the hydrogen
atom
(a)
Introduce two operators:
1
i
K = K = (L r r L) = L r ir = [L2 , r]
2
2
1
i
M = M = (L p p L) = L p ip = [L2
Homework 13
November 20, 2014
1. Particle in a box
For a particle in a box that extends from a to a, try (within the box), =
(x a)(x + a) and calculate E . There is no paramter to vary, but you will s
UIUC Physics 486: Solutions for for HW 4
October 12, 2014
Problem 1: Evaluating operators
(a):
X =
dxdydz (x) x  y y y  z (z)
= dxdydz (x)(y x)y(z y)(z)
= dxdy (x)(y x)y(y)
= dx x(x)2
2
+
= dx xe
Homework 14
December 8, 2014
1
1. Time evolve the state (0) = 3 iY00 + ei/4 Y11 + Y11 r
where the radial function rr = (r r0 ) for time t = under the action of the Hamiltonian H = 2 . What is (
Solutions Homework 13
Physics 486, Fall 2014
University of Illinois at UrbanaChampaign
December 4, 2014
1
Problem I: Particle in a box
E=
H =
h2
2m
x=+a
x=a
H

(x)xx (x)dx =
x=+a
 =
h2
2
2m
Solution to HW14
December 8, 2014
Problem 1
2
The energy of state Ylm (r r0 ) is El = l(l + 1)/r0 . So E0 = 0 and
2 . The state evolves by multiying a factor eiEl t , therefore,
E1 = 2/r0
1
() = iY
Solutions Homework 12
Physics 486, Fall 2014
University of Illinois at UrbanaChampaign
November 28, 2014
1
Problem 1: Grovers Algorithm
a) We know that i j = i j . Then:
 =
OO =
1
1
1
i j = N i
Solutions Homework 9
Physics 486, Fall 2014
University of Illinois at UrbanaChampaign
October 24, 2014
1
Problem I: Commutators
(a)
[y, px ] f (x, y) = y(ix ) f (x, y) + ix (y f (x, y) = iy fx (x, y)
Homework 9
October 23, 2014
1. Commutators
In class, we worked through a number of commutators. In these problems you
are going to explicitly compute the commutators. You may start by assuming
that [,
Physics 486: HW1
September 2, 2014
Problem 1: Identifying Quantum States
The following are legal quantum states. True or false?
(1) 0.1uz + i0.9dz
(2) 0.010 + 0.251 + 0.162
(3)
1
385
10
x=1
x2 x
Homework 6
October 1, 2014
1. Harmonic Oscillator in Real Space
(a) Show that
(x, t) =
m
1/4
exp
m
2
x2 +
a2
i t
(1 + e2it ) +
2axeit
2
m
satises the timedependent Schrodinger equation for the harm
Homework 4
September 17, 2014
1. Evaluating Operators
Dene the Hermitian operator X as
= xxihxdx
X
and the operator P as
P =
and the state  i =
e
kx2
ppihpdp
dxxi
Dene hxpi = eipx and remember
Solution to HW5
September 30, 2014
Problem I: Trying to Communicate Faster than the
Speed of Light
(a)
The two states are 0 and 1 , with probability 50% for each state.
(b)
The density matrix is
1
2
Homework 5
September 25, 2014
Problem 1: Trying to Communicate Faster then the Speed
of Light
In this problem, we are going to look at an explicit example where we will try
to use EPR pairs to communi
HW7
October 9, 2014
1. Normalizing states
(a) Assuming cfw_0 , 1 , 2 are a basis, normalize the state  = 0.30 + 0.71
(b) Normalize the state
cos(x) if /2 x /2
(x) =
0
otherwise
2. Unitary and H
Homework 8
October 17, 2014
1. A Pendulum
A pendulum consists of a massless rod of length l = 50 cm with one end attached to a rigid support so that it is free to pivot about this point and the
other
Solution to HW8
November 3, 2014
Problem I: A Pendulum
(a)
=
g
= 4.43Hz
l
(1)
(b)
1
E = m 2 A2 = 4.91 104 J
2
(2)
(c)
n
E
= 1.06 1030
h
(3)
(d)
= 4.65 1034 J
h
(4)
(e)
x
2A
= 9.43 1032 m
n
1
(5)
(f
Homework 10
October 29, 2014
Problem 1: Matrix Elements of Ylm
(a) Compute the matrix elements of Lx and Ly in the basis given by the spherical
harmonics. That is, calculate
Ylm Lx Yl m
and
Ylm Ly
Solutions Homework 3
Physics 486, Fall 2014
University of Illinois at UrbanaChampaign
September 18, 2014
1
Problem I: Semiinnite square well
(a) In the region 0 x L, the potential is V = 0. Thus, we