Problem Set 3: due Monday, Sept. 26
JJS 1.21, 1.26, 1.28, 1.29, 1.32, 1.33
Problem Set 4: due Monday, Oct. 3
JJS 2.3, 2.4, 2.6, 2.8, 2.13, 2.16
Hint:
2.6. Recognize the commutator is just the time-derivative of xp. Conside the state as an eigenstate of H.
2.8 d. think of a spin in a magnetic field.
2.13 Virial theorem says the average kinetic energy is equal to average potential energy in an harmonic oscialltor.
2.17 The most general normalized state is
con\theta e^{-\phi}|0> + sin\theta e^{+\phi/2}|1>
find the values of \theta and \phi such that the expectation value of x in this state is the largest
First Mid-Term Exam solution 1 2 3 4
Problem Set 5: due Monday, Oct. 24
Problem set 6: due Monday, Oct. 31
JJS 2.25, 2.34, 2.36, 2.37, 3.2, 3.3
hint:
2.25, follow what I did in the class. Find
vector potential which gives the right magnetic flux.
2.34, (2.6.25) is simple, but (2.6.27) is
somewhat difficult. the key is to work out the Heisenberg equations of motion of
PI.
2.36, the commutator between Pi_x and Pi_y is very
similar to the commutation relation between x and p for one-dimensional harmonic
oscillator!
2.37, using Feynman path integral, one realizes
that the phase difference between the paths is just the integral of the
lagrangian density over the time. The time is the distance divided by velocity,
and the lagrangian density is equal to the minus of the interaction energy
between the magnetic moment and magnetic field.
3.2, using the expression of U that I gave in the
lecture in terms of \theta vector.
3.3 In part b, remember the vector product of the
two spins is the sum of three components.
Problem set 7: due Monday, Nov. 7
JJS 3.4, 3.8, 3.14, 3.16, 3.18, 3.22
hint: 3.4 you can work out the matrices S_z and S_x and calculate directly. or you can argue using some facts.
3.14 we did similar derivation in class.
3.16 use L_x^2 + L_y^2+L_z^2 = L^2
3.18 Use rotation matrix to rotate the state. Find out what is the rotation matrix (one of the m is zero!)
3.22 you can show J_y^3 is a linear combination of 1, J_y and J_y^2.
Problem set 8: will not be collected
Solution for the second midterm: p1, p2, p3, p4, p5
Problem set 9: due Monday, Nov. 21
HINT： 1) This problem is designed to demonstrate that the
spherical tensors are just ordinary vectors in the simplest case. Therefore
the transformation property of a rank-1 tensor must be the same as that of an
ordinary vector.
2) I sort of did that in class. Just use Clebsch-Gordon's
3) Start with some state Y_{lm} \chi_{m_s} and apply the projection operators on
it. Show the result is an eigenstate of total
angular momentum. Then work out the details of the projection operator on Y\chi
state.
4) a) use Wigner Eckart theorem. Consider the ratios of the matrix elements b)
Calculate the matrix element directly and
consider ratios
5) a) start with second-order spherical harmonics and write them in Cartesian cooridnates. Express the quadrupole moment in
terms of spherical tensors and use W-E theorem.
6) Consider the derivatives as constants. use the definition of S_+ and S_- to rewrite the hamiltonian. Work out the hamiltonian
in m-basis (a 4 times 4 matrix), and try to diagonalize it.
Problem set 10: due Friday, Dec. 2
JJS 4.1, 4.2, 4.3, 4.4, 4.5, 4.6
Hint: 1) the particles are distinguishable here. Give them a label like 1, 2, 3,4 and think of the label as physical.
2) consider these transformations physically.
3) Apply AB+BA on the common eigenstate and see what happens.
4) b) write (sigma dot x) as a 2 by 2 matrix in sphereical coordinate system and then use the definition of spherical harmonics.
5) V violates parity. The C-coefficient can be calculated as matrix elements between nljm and n'l'j'm' states. Please consider what selection rule apply to the matrix elements.
6) Solve one-dim schrodinger equation and obtain eigenvalue equations for even and odd states separately.
Problem set 11: due Monday, Dec. 11
JJS 4.7, 4.8, 4.9, 4.10, 4.11, 4.12
Hint: 4.7 a) write down \psi(x,t) explicitly and work out \psi^*(x, -t)
b) An explicit form of \chi(n) can be worked out from (4.4.62)
4.8 a) Any eigenfunction of H can be chosen to be a real function of coordinate times a coordinate-independent phase factor. Thus, at any time t, one can choose the phases of the eigenfuctions to cancel the phases of the coefficients in an expansion of a general state.
b) read comments on the top of page 277.
4.9 study text around eq. (4.4.61)
4.10 a) Apply time-reversal operator on this state. Consider the time-reversal transformation on the operator D(R) and the state |jm> separately. The result can be expressed in terms of Wigner-D function. Use e^{-i\pi J_y}|jm> = (-1)^{j-m}|j-m>, a property of state under rotation in y direction by \pi.
b) Work out D(R)|jm> in terms of Wigner D-matrix first, then apply time reversal on the state. Equate the result to that from a).
c) this is a matter of phase choice. show the result can be obtained by a judicious choice of phase \eta. see equations (4.4.73) and eq.(4.4.78).
4.11 a) calculate the matrix element, and apply time-reversal to the operator and state. b) calculate the matrix element use the expanded state. consider cancellation between positive and negative m.
4.12 Write H as a 3X3 matrix and diagonalize it.
Solution for the sample final: p1, p2, p3
