PHYS 507
Answers to Homework III 1. (a) The Taylor series is 1 V (x) = V (a) + V (a)(x - a) + V (a)(x - a)2 + 2 and the expectation value is 1 V (x) = V (a) + V (a)x2 + 2 Including only the lowest order difference we have V (x) - V (a) V (a)x2 /2 and the
PHYS 507
Homework I (Fall '05) Assigned: September 28, 2005, Wednesday. Due: October 7, 2005, Friday, at 5:00 pm. Notes: (i) You can discuss and solve the problems with your friends. However, you should write your own solution on your homework. Directly c
PHYS 507
Homework I (Fall '06) Assigned: October 20, 2006, Friday. Due: November 1, 2006, Wednesday, at 5:00 pm. Notes: (i) You can discuss and solve the problems with your friends. However, you should write your own solution on your homework. Directly co
1
Hartree-Fock Approximation
N N
We have a system of N interacting fermions with a Hamiltonian H=
i=1
h(i) +
i<j
Vint (i, j)
.
(1)
Here, h(i) is the "single-particle Hamiltonian" for the particle-i and Vint (i, j) is the part of the Hamiltonian describing
PHYS 507 Quantum Mechanics I Final Examination
Assigned: January 2, 2006. Due: January 6, 2006, Friday, 16:00. Rules: As opposed to the homeworks, for the final exam you are not allowed to talk to your friends (or to anybody) about the exam. Solve all pro
We have seen that the wavefunctions of electrons are completely antisymmetric. By this statement we mean that under the exchange of any two particles' coordinates and spins the wavefunction changes sign. Inclusion of the spin degree of freedom complicates
Lecture Notes on Hilbert spaces
State
State of a physical system at a given time is basically all information that identies the particular state the system is in. For example, I am reading these notes is a description of your current state, but not a comp
PHYS 507
Homework II (Fall '06) Assigned: November 17, 2006, Friday. Due: November 26, 2006, Sunday. 1. Consider a state of a particle in 1D having the following position-space wavefunction (x ) = N exp - |x | a ,
where a is a distance and N is an appropr
PHYS 507
Homework V Assigned: December 12, 2003, Friday. Due: December 22, 2003, Monday, at 5:00 pm. 1. You know that a classical magnetic dipole moment in a magnetic field B has energy - B and it feels a torque B. In quantum mechanics we use the same Ham
PHYS 507
Answers to Homework I (Fall '05) 1. (a) We know that u2 has to be orthogonal to u1 , i.e., u1 |u2 = 0. Using this, we can completely determine u2 . Let u2 = Then 1 u1 |u2 = 6 a b a b .
2 1+i
=
2a + (1 + i)b =0 . 6
From here we find a = -(1 + i)b/
PHYS 507
Answers to Homework III (Fall '05) 1. (a) [X, Y ] = (XY - Y X) = Y X - X Y = [Y , X ] = [Y, X] = -[X, Y ]. (b) (A A) = A (A ) = A A. Let | be an eigenket of A A with eigenvalue . Let | be normalized. In that case = |A A| = |A A = A|A = |A|2 , i.e
PHYS 507
Homework III (Fall '05) Assigned: October 17, 2005, Monday. Due: October 26, 2005, Wednesday, at 5:00 pm. 1. (a) Show that if X and Y are hermitian operators, then [X, Y ] is anti-hermitian. (b) Show that for any operator A, the operator A A is h
PHYS 455 Answers to Homework II
1. Consider a system with a 3 dimensional state space which is described as 3 1 column vectors. Let the state of the particle be 1 =N 1+i , 2i and the following observable is measured on the 0 0 A= 0 1 1 0 system, 1 0 0
(a)
PHYS 507
Answers to Homework II 1. (a) i i T (a)|p = exp - pa |p = exp - p a |p h h ~ (p ) = p | = p |T (a)| .
(b) There are different ways of doing this. This is a longer way. Start with .
As a result, we first need to calculate p |T (a). The dual comple
PHYS 507
Homework II Assigned: October 15, 2003, Wednesday. Due: October 24, 2003, Friday, at 5:00 pm. 1. Let T (a) be the translation operator corresponding to displacement a. (a) Calculate T (a)|p where |p is a momentum eigenket. (b) Let | be the state
PHYS 507
Homework II (Fall '05) Assigned: October 10, 2005, Monday. Due: October 19, 2005, Wednesday, at 5:00 pm. 1. Consider a three dimensional Hilbert space. Let cfw_|1 , |2 , |3 be an orthonormal basis for this space. Let | and | be two particular ke
PHYS 507
Answers to Homework I 1. (a) u1 |u1 = u u1 1 = N N 1+i 2 1-i 2
= |N |2 (1 + i)(1 - i) + 4) = 6|N |2 = 1 . 1 |N | = 6 This is the only thing we can say about the value of N , i.e., we can only determine its modulus, |N |, but not its phase. Luckil
PHYS 507
Homework IV Assigned: December 3, 2003, Wednesday. Due: December 12, 2003, Friday, at 5:00 pm. 1. Let |j, m be common eigenstates of J 2 and Jz where J is an angular momentum operator. Calculate the following expectation values in this state.
2 2
PHYS 507
Homework I Assigned: October 1, 2003, Wednesday. Due: October 10, 2003, Friday, at 5:00 pm. Notes: (i) You can discuss and solve the problems with your friends. However, you should write your own solution on your homework. Directly copying other
PHYS 507
Answers to Homework V 1. The assumption = k S is not necessary. We only need to use the commutation relation [Si , j ] = i h , ijk k
k
which expresses the fact that is a vector operator. d Si dt
t
i [H, Si ] t h i i Bj [j , Si ] t = = - h j h = =
PHYS 507
Answers to Homework V (Fall '05) 1. (a) n |[H, A]|n = n |HA|n - n |AH|n = En n |A|n - n |A|n En = (En - En )Ann = 0 .
(b) [T, A] = 1 2 1 [p , xp + px] = [p2 , x]p + p[p2 , x] 2m 2m 1 h h p2 h = (2p)p + p(2p) = 4 =4 T . 2m i i 2m i [V, A] = [V (x)
PHYS 507
Homework V (Fall '05) Assigned: November 7, 2005, Monday. Due: November 16, 2005, Wednesday, at 5:00 pm. Note: First Midterm exam is on November 19, Saturday at 14:00 somewhere in the Physics Dept. 1. The Virial Theorem: Consider the Hamiltonian
PHYS 455 Answers to Homework IV
1. Consider an ensemble for a spin-1/2 particle where the particle is in "spin-up along z" state with probability p1 = 1/6, "spin-down along z" state with probability p2 = 1/3, and "spin-up along x" state with probability p
PHYS 507
Answers to Homework IV (Fall '05) 1. U (t, 0) = (a) U (t, 0)U (t, 0) = cos t - sin t sin t cos t cos t sin t - sin t cos t = 1 0 0 1 . cos t - sin t sin t cos t
.
Normally, we should also check if U (t, 0) U (t, 0) = I is also satisfied, but this
PHYS 507
Homework IV (Fall '05) Assigned: October 24, 2005, Monday. Due: November 2, 2005, Wednesday, at 5:00 pm. 1. A material is called optically active when the polarization direction of linearly polarized photons rotates as they move inside the materi
PHYS 507
Homework III Assigned: October 30, 2003, Thursday. Due: November 10, 2003, Monday, at 5:00 pm. 1. We have said in class that the "equation of motion" for the average position, x t , of a particle with the Hamiltonian H = p2 /2m + V (x) is d2 x t
PHYS 507
Homework VI (Fall '05) Assigned: November 23, 2005, Monday. Due: November 30, 2005, Wednesday, at 5:00 pm. 1. In the Hydrogen atom problem H = p2 /2m - Ze2 /r, the particular gauge chosen is ~ ~ ~ A = 0 and = Ze/r. In another possible gauge (A, )