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 ord
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 shou
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 shoul
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 an
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 you
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 si
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
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 e
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 fee
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
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 nor
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.
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 observa
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 resul
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 w
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
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
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. Calcul
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
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
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
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
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 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 che
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 p
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 part
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