6.728 APPLIED QUANTUM MECHANICS
Final Exam Topics
1. Fundamentals of Quantum Mechanics:
S-Equation, eigen functions and values, expectation values, Ehrenfests Theorem, prob
ability density and ux.
2. 1D solutions:
Particle in a box, Free particle, Tunneli
WKB Approximation Applied to Tunneling
I
=
eikx + rI eikx
tII (x)e(x) + rII (x)e(x)
2m
k=
E
2
h
x
2m
(x) =
(V (x ) E ) dx
2
h
0
II
=
III
=
AIII eikx
Note, we use t(x) and r(x) in II , but they can only take the WKB AWAY from the classical turning
point.
6.728 APPLIED QUANTUM MECHANICS October 1, 2006
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
Seeing Atoms the Scanning Tunneling Microscope
In 1983, Binnig and Rohrer and their co-workers1 reported a new
Two-Level System with Static and Dynamic Coupling
T.P. Orlando
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
c
2004
Contents
1 Two-State System
1.1 Two-State Hamiltonian . . . . . . . . . . . . . . . . .
Massachusetts Institute of Technology
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6.728 Applied Quantum and Statistical Physics
Problem Set 2
Issued: Sept. 11, 2013
Due: Sept. 18, 2013
Problem 1
(a) A particle is known to be localized at x = 0
Massachusetts Institute of Technology
Electrical Engineering and Computer Science Department
6.728 Applied Quantum and Statistical Physics
Problem Set 3
Issued: Sept. 18, 2013
Due: Sept. 25, 2013
Problem 1
An electron in a semiconductor with a nonlinear b
Massachusetts Institute of Technology
Electrical Engineering and Computer Science Department
6.728 Applied Quantum and Statistical Physics
Problem Set 1
Issued: Sept. 4, 2013
Due: Sept. 11, 2013
Problem 1
What is the largest acceleration possible for a so
Massachusetts Institute of Technology
Electrical Engineering and Computer Science Department
6.728 Applied Quantum and Statistical Physics
Problem Set 4
Issued: Sept. 25, 2013
Due: Oct. 2, 2013
Problem 1
Do Problem 8.2 in HSO.
Problem 2
In this problem we
Massachusetts Institute of Technology
Electrical Engineering and Computer Science Department
6.728 Applied Quantum and Statistical Physics
Problem Set 7
Issued: Oct. 16, 2013
Due: Oct. 23, 2013
Problem 1
We are interested in nding an approximate solution
Massachusetts Institute of Technology
Electrical Engineering and Computer Science Department
6.728 Applied Quantum and Statistical Physics
Problem Set 8
Issued: Oct. 23, 2013
Due: Oct. 30, 2013
Problem 1
Consider a restricted Hamiltonian for an N -level s
Massachusetts Institute of Technology
Electrical Engineering and Computer Science Department
6.728 Applied Quantum and Statistical Physics
Problem Set 9
Issued: Oct. 30, 2013
Due: Nov. 6, 2013
Problem 1
A particle with mass m is incident on a potential st
Massachusetts Institute of Technology
Electrical Engineering and Computer Science Department
6.728 Applied Quantum and Statistical Physics
Problem Set 12
Issued: Nov. 20, 2013
Due: Nov. 27, 2013
Problem 1
Do Problem 27.2 in HSO.
Problem 2
Diamond can be d
Massachusetts Institute of Technology
Electrical Engineering and Computer Science Department
6.728 Applied Quantum and Statistical Physics
Problem Set 11
Issued: Nov. 13, 2013
Due: Nov. 20, 2013
Problem 1
Do Problem 25.2 in HSO.
Problem 2
Do Problem 26.1
Massachusetts Institute of Technology
Electrical Engineering and Computer Science Department
6.728 Applied Quantum and Statistical Physics
Problem Set 13
Issued: Nov. 27, 2013
Due: Dec. 6, 2013
Problem 1
A muon is an elementary particle that has a negativ
6.728 APPLIED QUANTUM AND STATISTICAL PHYSICS
QUIZ 2
November 13, 2006
PLEASE WRITE ALL ANSWERS IN THE BLANK BOOKLET AND NOT ON
THE EXAM ITSELF. HOWEVER, PLEASE RETURN THE EXAM, THE
BOOKLET, AND THE FORMULA SHEETS.
NAME:
Problem Possible Score
1
20
2
20
3
6.728 Applied Quantum and Statistical Physics:
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
Quiz 1
Quiz Out: 10/16/06
Quiz Due: 10/18/06 at the beginning of class
NAME:
Problem
1
2
3
Total
Possible
35
35
T rigonometric arld Hyperbolic Iderltities
1
ci:,sr = - ( ri'
2
+ cCi')
I
cohh s = - (,?I+,:-I) = c oh(ix)
2
= ci:,sh(is)
s i n ( s i 11) = s in r ci:,s y & ci:,s r s in y
h inh(r i y ) = hinh r ci:,sh y i cash s sinh !J
ci:,s(s f y ) = ci:,s r c i : ~
y
6.728 APPLIED QUANTUM and STATISTICAL MECHANICS
TWO-Level Formula
Consider the two level Hamiltonian:
H11 + H22
E=
2
and
=
H22 H11
2
(1)
and H12 = H21 = V with V real. Then the Hamiltonian with basis states u1 and u2 is
H=
E
V
+
E
V
Hij = ui |H |uj
wher
Gaussian Wavepacket supplenlent to 6 .728 notes
S.D. Seilturia
TVe start, 114th a st,ationary (unllorlnalizedj Gaussian navepacket,
for which t,he Fourier transforin is
T his can be ~a-ritten
Recalling from high school algebra the trick of .completing the
6.728 Applied Quantum and Statistical Physics:
Department of Electrical Engineering and Computer Science
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PROBLEM SET 1
Problem Set Out: 9/6/06
Problem Set Due: 9/13/06 at the beginning of class
READINGS
Chapters 13 of
6.728 Applied Quantum and Statistical Physics:
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
PROBLEM SET 2
Problem Set Out: 9/13/06
Problem Set Due: 9/20/06 at the beginning of class
READINGS
Chapters 46 o
6.728 Applied Quantum and Statistical Physics:
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PROBLEM SET 3
Problem Set Out: 9/20/06
Problem Set Due: 9/27/06 at the beginning of class
PROBLEMS
Problem 3.1 S
6.728 Applied Quantum and Statistical Physics:
Department of Electrical Engineering and Computer Science
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PROBLEM SET 4
Problem Set Out: 9/27/06
Problem Set Due: 10/4/06 at the beginning of class
PROBLEMS
Problem 4.1
R
6.728 Applied Quantum and Statistical Physics:
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
PROBLEM SET 5
Problem Set Out: 10/4/06
Problem Set Due: 10/11/06 at the beginning of class
PROBLEMS
Problem 5.1
6.728 Applied Quantum and Statistical Physics:
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
PROBLEM SET 6
Problem Set Out: 10/18/06
Problem Set Due: 10/25/06 at the beginning of class
Problem 6.1 Quantum
6.728 Applied Quantum and Statistical Physics:
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
PROBLEM SET 7
Problem Set Out: 10/25/06
Problem Set Due: 11/01/06 at the beginning of class
PROBLEMS
Problem 7.1
6.728 Applied Quantum and Statistical Physics:
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
PROBLEM SET 8
Problem Set Out: 11/01/06
Problem 8.1
Problem Set Due: 11/8/06 at the beginning of class
A pair of
6.728 Applied Quantum and Statistical Physics:
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
PROBLEM SET 9
Problem Set Out: 11/15/06
Problem Set Due: 11/22/06 at the beginning of class
Problem 9.1 In this
6.728 Applied Quantum and Statistical Physics:
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
PROBLEM SET 10
Problem Set Out: 11/27/06
Problem Set Due: 12/8/06 at the beginning of class
Note that the due da
Massachusetts Institute of Technology
Electrical Engineering and Computer Science Department
6.728 Applied Quantum and Statistical Physics
Problem Set 14
Issued: Dec. 4, 2013
Due:
Problem 1
When an atom undergoes a radiative decay, we expect the atom to