Introduction to Quantum and Statistical Mechanics
Homework 1
Prob. 1.1. To establish the intuitive link between the Hamiltonian mechanics and the wave equation, we will use the example of the longitudinal vibrations in classical mechanics written as the w
Introduction to Quantum and Statistical Mechanics
Homework 1 Solution
Prob. 1.1.
(a)
(10 pts)
m d 2z j dt 2 =k ( z j +1 2 z j + z j 1 ), j =1,.,n
From Newtons second law we have:
F =ma =m d 2z j dt 2 ,
The total force experienced by particle j according t
Introduction to Quantum and Statistical Mechanics
Homework 2
Prob. 2.1. We will observe the variance of an operator through the wave functions expanded by the eigenfunctions of the operator. The mathematical form is very helpful when we choose only a few
Introduction to Quantum and Statistical Mechanics
Homework 3
Prob. 3.1. Find the following commutators. Remember that when the commutator of two operators does not vanish, it implies that the two operators cannot be determined with uncertainty simultaneou
Introduction to Quantum and Statistical Mechanics
Homework 3 Solution
Prob. 3.1. (a) x and x 2 commutate, so [ x, V ( x )] = 0 (4 pts) 2 2 (b) [ p,V ( x)] = i m0 x = m0 xp , where V is the same as in (a). (4 pts) x (c) [ p, p 2 ] = 0 (4 pts)
(d)
[ xp, p
Introduction to Quantum and Statistical Mechanics
Homework 6
Prob. 6.1 For an infinite-wall with a small step potential defined as: 0 V ( x) = V0 x0 0 xd dxL xL
We would like to look at the formalism to find the ground-state energy E. Assume that E > V0.
ECE 3060 Fall 2008 Prelim Exam 1 Solution Rules of the Exam (Please read carefully before start) 1. This is an open-book, open-note exam. You are allowed to use your computer as a browser for downloaded course files, but you are NOT allowed to connect to
ECE 4060: Introduction to Quantum and Statistical Mechanics
Quiz 1 Solution
Prob. 1. Given a wave function of a free particle at t = 0 as: 1 x 1 A exp( ik 0 x ) ( x,0 ) = otherwise 0 (a) Find A. (5 pts) 1 * By normalization requirement of dx = 1 , A = . 2
ECE 4060, 2011, Take Home Final Examination.
Due:Dec.15,2011,5pm.Undermydoorthoughmypreferenceandsendmeascannedpdfdocument.
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Introduction to Quantum and Statistical Mechanics
Homework 1
Prob. 1.1. To establish the intuitive link between the Hamiltonian mechanics and the wave equation, we will use the example of the longitudinal vibrations in classical mechanics written as the w
ECE 4060: Introduction to Quantum and Statistical Mechanics
Quiz 1 Solution
Prob. 1. Given a wave function of a free particle at t = 0 as: 1 x 1 A exp(ik 0 x ) ( x,0) = otherwise 0 (a) Find A. (5 pts) 1 . By normalization requirement of *dx = 1 , A = 2
(b
Introduction to Quantum and Statistical Mechanics
Homework 1 Solution
Prob. 1.1. (a) (10 pts)
m d 2z j dt 2 =k ( z j +1 2 z j + z j 1 ), j =1,.,n
From Newtons second law we have:
F =ma =m d 2z j dt 2 ,
The total force experienced by particle j according t
Introduction to Quantum and Statistical Mechanics
Homework 2
Prob. 2.1. We will observe the variance of an operator through the wave functions expanded by the eigenfunctions of the operator. The mathematical form is very helpful when we choose only a few
Introduction to Quantum and Statistical Mechanics
Homework 2 Solution
Prob. 2.1. We will observe the variance of an operator through the wave functions expanded by the eigenfunctions of the operator. The mathematical form is very helpful when we choose on
Introduction to Quantum and Statistical Mechanics
Homework 3
Prob. 3.1. Find the following commutators. Remember that when the commutator of two operators does not vanish, it implies that the two operators cannot be determined with uncertainty simultaneou
Introduction to Quantum and Statistical Mechanics
Homework 3 Solution
Prob. 3.1. (a) x and x 2 commutate, so [ x, V ( x )] = 0 (4 pts) 2 2 (b) [ p,V ( x)] = ihm0 x = m0 xp , where V is the same as in (a). (4 pts) x (c) [ p, p 2 ] = 0 (4 pts)
(d)
[ xp, px
Introduction to Quantum and Statistical Mechanics
Homework 4
a b Prob. 4.1 For a 2 2 Hermitian matrix c d , (a) Prove that a and d are real, and b = c* (5 pts) (b) Find the eigenvalues and prove that they are real. (5 pts) (c) Find the corresponding eigen
Introduction to Quantum and Statistical Mechanics
Homework 4 Solution
Prob. 4.1
(a)
(5 pts)
a b a * c * T A= c d = ( A )* = b * d * , i.e., a = a* and d = d* where a and d are real, and b = c*.
(b)
(5 pts) b 2 = 0 , ( a )( b ) c = 0 d , and we can see th
Introduction to Quantum and Statistical Mechanics
Homework 5
Prob. 5.1 A quantum well laser is operated by electrons relaxing from the first excited state E2 to the ground state E1 in a finite-potential well, which is formed by a superlattice of semicondu
Introduction to Quantum and Statistical Mechanics
Homework 5 Solution
Prob. 5.1 A quantum well laser is operated by electrons relaxing from the first excited state E2 to the ground state E1 in a finite-potential well, which is formed by a superlattice of
Introduction to Quantum and Statistical Mechanics
Homework 6
Prob. 6.1 For an infinite-wall with a small step potential defined as: 0 V (x ) = V0 x0 0 xd d xL xL
We would like to look at the formalism to find the ground-state energy E. Assume that E > V0.
Introduction to Quantum and Statistical Mechanics
Homework 6 Solution
Prob. 6.1 For an infinite-wall with a small step potential defined as: 0 V ( x) = V0 x0 0 xd dxL xL
We would like to look at the formalism to find the ground-state energy E. Assume that