Download 5 Problems on Quantum Mechanics II - Assignment 1 | PHYSICS 137B and more Assignments Quantum Mechanics in PDF only on Docsity! Physics 137B, Fall 2007 Quantum Mechanics II Problem set 1: review, esp. angular momentum and spin Assigned Friday, 30 August. Due in box Friday, 7 September, 5 pm. Assume every Friday that there is a new problem set unless I say otherwise in class. Starting next week, problem sets will assume you have a copy of Bransden. They should be downloaded from the course webpage http://socrates.berkeley.edu/˜jemoore/p137b/p137b.html. 1. Show that the commutator [J2, Jz] = 0, where J2 = Jx2 + Jy2 + Jz2. Do not use the orbital angular momentum expression L = r×p. Instead use the general angular momentum commutation relations, which can be written compactly as [Ji, Jj ] = ih̄ijkJk (1) where the “Levi-Civita antisymmetric symbol” is formally defined in Bransden problem 6.4. This symbol gives just the same pattern of signs as in a cross product: (r× p)i = ijkripj . If you don’t have a copy of Bransden yet, the full commutation relations are [Jx, Jy] = ih̄Jz, [Jz, Jx] = ih̄Jy, [Jy, Jz] = ih̄Jx. (2) 2. (based on Bransden 6.13) Consider a free particle of mass µ fixed to move on a ring of radius a. (a) Argue that the Hamiltonian of this system is H = Lz 2 2I (3) where the z-axis is through the centre O of the ring and is perpendicular to the plane, and I is the moment of inertia of the particle with respect to the centre O. (b) Find the energy eigenfunctions for the system. Are these also eigenstates of Lz? Does Lz commute with the Hamiltonian? 3. In units of h̄, what is the classical angular momentum of an electron moving in a circle with radius the Bohr radius a0 = 5.29 × 10−11m at frequency 13.6 eV/h̄? 4. Verify that the following matrix expressions of the operators Sx, Sy, Sz for spin s = 1/2 are Hermitian and satisfy the angular momentum algebra: Sx = h̄ 2 ( 0 1 1 0 ) , Sy = h̄ 2 ( 0 −i i 0 ) , Sz = h̄ 2 ( 1 0 0 −1 ) . (4) 5. Using the representation of spin-half operators S in problem 4, (a) write a spinor χ↑ with 〈Sz〉 = h̄/2. I’ll give the answer: χ↑ = (1, 0), up to overall phase. (b) calculate the expectation value E of Sx in this state: E = 〈χ↑|Sx|χ↑〉. (c) calculate the expectation value of Sx2 in this state. (d) calculate the Sx variance σ2x = 〈(Sx − E)2〉 in this state. (e) Let a general normalized state in this representation be written (z1, z2), where z1 and z2 are complex numbers satisfying |z1|2 + |z2|2 = 1. Calculate the variance σ2x in this state as a function of z1 and z2, using (d) to check your result. (f) Find a normalized state (z1, z2) that is an eigenstate of Sx with eigenvalue h̄/2. Its x variance σ2x should be zero, since it is an eigenstate of Sx.
