PH12b 2010 Solutions HW#7
1.
a) Our state is
p
j (dt)i =
and
h (dt)j =
p
dt^ j i
a
^
j1i + I
1
dt^y a j i
a ^
2
j0i
dt^y h j
a
^
h1j + I
1
dt^y a h j
a ^
2
h0j
then
h (dt)j
1
dt^y a
a ^
2
^
dt h j ay a j i h1j 1i + h j I
^ ^
(dt)i =
=
dt h j ay a j i + h
PH12b 2010 Solutions HW#3
1.
The Hamiltonian of this two level system is
b
H = Eg jgi hgj + Ee jei hej ;
where Eg < Ee : The experimentalist basis is
1
j i = p (jgi
2
1
j+i = p (jgi + jei) ;
2
jei) :
a) At t = 0 the state is j (0)i = j+i, then, the state
Ph 12b Quantum Physics
28 January 2010
Measurement
Our study of decoherence has indicated how a quantum system can exhibit classical behavior. Macroscopic
quantum systems become correlated with the environment. If we observe only the system, and do not mo
Ph 12b Final Exam Solution
J. Preskill Wednesday, 17 March 2010
1. Bound states of two narrow wells 30 total points
A particle with mass m moves in one dimension, governed by the potential
V (x) =
2
h
(x + a) + (x a) ,
m
where (x) denotes the Dirac -fun
Ph 12b Final Exam
Due: Wednesday, 17 March 2010, 5pm
This exam is to be taken in one continuous time interval not to exceed 4
hours, beginning when you rst open the exam.
You may consult the textbook Introductory Quantum Mechanics by Libo,
the textbook
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Ph 12b - Quantum Physics
21 January 2010
The density operator
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PH12b 2010 Solutions HW#5
1.
a) We solve the dierential equation in the following way
i
d
dx
i x
(x) = 0;
d
(x) = ( x + i ) (x) ;
dx
d (x)
)
= ( x + i ) dx;
(x)
1 2
x + i x + c;
) log (x) =
2
1 2
)
(x) = C exp
x +i x :
2
)
where c; C; are constants.
b) We
Ph 12b Quantum Physics
26 January 2010
Decoherence
Part of our motivation for developing the density matrix formalism is to prepare the way for a discussion of
decoherence. Decoherence provides the explanation for why, although the fundamental laws of phy
PH12b 2010 Solutions HW#1
1.
a) hi = 0 = hi imply that ()2 = 2 2 + h i = 2 and ()2 = 2 Using this we get () 1 1 2 2 = + 2 ()2 2 2 2 1 + 2 ()2 2 8 ()
2 2
Now, from the uncertainty relation we know that 2, therefore h i b) 2
solving for we get
2 2 h i = +
1
Ph 12b
Homework Assignment No. 6
Due: 5pm, Thursday, 25 February 2010
1. Geometric phase (15 points).
Recall that the coherent state | of a harmonic oscillator, where is
an arbitrary complex number, can be obtained by applying a unitary
displacement ope
Ph 12b Quantum Physics
9 February 2010
The free particle
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Fourier series and the Fourier transform:
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The free-particle "propagator":
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Ph 12b
Homework Assignment No. 7
Due: 5pm, Thursday, 4 March 2010
1. Damped harmonic oscillator (15 points).
Lets suppose the oscillations of a quantum harmonic oscillator with
circular frequency are damped because the oscillator can emit photons with e
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Ph 12b Quantum Physics
19 January 2010
Quantum evolution
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Homework 2 Solutions
Ph 12b Winter 2010
January 30, 2010
1. Quantized Rotor
a. We know that L is the conjugate momentum of . From equation 1.14 in Libo we can evaluate the
equations of motion to see that
L
H
= = .
L
I
Since is missing from the Hamiltonian
1
Ph 12b
Homework Assignment No. 8
Due: 5pm, Thursday, 11 March 2010
1. A barrier in a well (10 points).
A free quantum-mechanical particle with mass m moves inside a onedimensional box with impenetrable walls located at x = a. Furthermore, a repulsive -f
Homework 8 Solutions
Ph 12b Winter 2010
March 13, 2010
1.
A Barrier in a Well
a. For an even energy eigenstate , (x) = (x) and (x) = (x). functions, From the previous
problem set, we know
lim (x ) (x + ) (0 ) (0+ ) = 0 (0 ) = (0+ ) (0 ), (0+ ) (0)
0
The d
PH12b 2010 Solutions HW#1
1.
a) hi = 0 = hi imply that ()2 = 2 2 + h i = 2 and ()2 = 2 Using this we get () 1 1 2 2 = + 2 ()2 2 2 2 1 + 2 ()2 2 8 ()
2 2
Now, from the uncertainty relation we know that 2, therefore h i b) 2
solving for we get
2 2 h i = +
1
Ph 12b
Homework Assignment No. 3 Due: 5pm, Thursday, 28 January 2010 1. A watched quantum state never moves. Consider a simple model of an atom with two energy levels the ground state |g has energy Eg and the excited state |e has energy Ee > Eg , where
Ph 12b Quantum Physics
9 February 2010
The free particle
Fourier series and the Fourier transform:
The free-particle "propagator":
"Quantum diffusion":
Example: Propagation of a Gaussian wave packet.
Ph 12b Quantum Physics
28 January 2010
Measurement
Our study of decoherence has indicated how a quantum system can exhibit classical behavior. Macroscopic
quantum systems become correlated with the environment. If we observe only the system, and do not mo
Ph 12b Quantum Physics
26 January 2010
Decoherence
Part of our motivation for developing the density matrix formalism is to prepare the way for a discussion of
decoherence. Decoherence provides the explanation for why, although the fundamental laws of phy
1
Ph 12b
Homework Assignment No. 5 Due: 5pm, Thursday, 18 February 2010 1. Minimal uncertainty I: particle in one dimension (10 points). If we measure the Hermitian operator A in the state vector | , the variance of the measurement outcomes is (A)2 = | A