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
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 455 Homework IV
Assigned: November 29, 2011, Tuesday. Due: December 7, 2011, Wednesday. 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 probabi
PHYS 455 Homework III
Assigned: November 22, 2011, Tuesday. Due: November 30, 2011, Wednesday. 1. Quantum Zeno Effect is the quantum analog of the Zeno paradox. One of many alternative versions of the paradox is summarized by the statement "A watched pot
PHYS 455 Homework II
Assigned: October 25, 2011, Tuesday. Due: November 2, 2011, Wednesday. 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 ob
PHYS 455 Homework I
Assigned: October 4, 2011, Tuesday. Due: October 12, 2011, Wednesday. Notes: (i) You can discuss and solve the problems together with your friends. However, you should write your own solution on your homework. Directly copying other pe
PHYS 507
Answers to Homework VIII (Fall '05) 1. (a) We have |j1 - j2 | j j1 + j2 which looks like the triangle inequality, but these are the quantum numbers, not the actual lengths of our vectors. In the actual triangle, the side lengths are a = j1 (j1 +
PHYS 507
Homework VIII (Fall '05) Assigned: December 14, 2005, Wednesday. Due: December 21, 2005, Wednesday, at 5:00 pm. 1. Let J = J1 + J2 be the total angular momentum of two independent systems. Suppose 2 2 the quantum numbers for J1 and J2 are j1 and
PHYS 507
Answers to Homework VII (Fall '05) 1. (a) We have Wj = [Ji , Wj ] =
n n jn
Vn U and therefore
jn
[Ji , Vn U ] =
n jn ink Vk U
jn
[Ji , Vn ]U +
n jn i m V n Um
jn
Vn [Ji , U ]
= i h
nk
+ i h
nm
= i h
nk
jn kin Vk U
+ i h
nm
jn
mi
V n Um (jm ni - j
PHYS 507
Homework VII (Fall '05) Assigned: November 30, 2005, Wednesday. Due: December 7, 2005, Wednesday, at 5:00 pm. 1. We have said in class that for any vector operator V , the commutation relations [Ji , Vj ] = i k ijk Vk are satisfied where J is the
PHYS 507
Answers to Homework VI (Fall '05) 1. (a,b) The gauge transformation is ~ A = A+ , 1 ~ = - , c t
~ and we want = 0. Therefore, = From here we find ~ A(r, t) = (c) =- Ze 1 = r c t - = Zect r .
Zect ^ . r r2 (r , t) .
(-e) Ze2 t ~ (r , t) = exp i (r
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, )
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
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 455 Answers to Homework III
1. Quantum Zeno Effect is the quantum analog of the Zeno paradox. One of many alternative versions of the paradox is summarized by the statement "A watched pot never boils". It is possible to claim that the quantum analog
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