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PHYSICS 373, SPRING 2013 HOMEWORK 6, Due Friday, 03/08 1. A free particle in one dimension has a Gaussian wave function such that the expectation

problem number 2, 3, and 5 from the attached document.
PHYSICS 373, SPRING 2013 HOMEWORK 6, Due Friday, 03/08 1. A free particle in one dimension has a Gaussian wave function such that the expectation value of x 2 is b 2 / 2 at time t = 0. What are the expectation values of x 2 and p 2 at any later time t ? 2. (a) Show the following relations between the Pauli matrices: σ a σ b = δ ab + abc σ c where a,b,c = x,y,z . (b) Proof the following equality: e -→ σ -→ n 2 = cos θ 2 + i -→ σ -→ n sin θ 2 where -→ n is a unit vector denoting the axis of station and θ is the angle of rotation. (c) Rederive the result of problem 2 in homework 1, using rotations. 3. Derive the commutation relations for orbital angular momentum. 4. The Hamiltonian for a spin 1/2 particle of mass m has the form: H = p 2 2 m + 2 x 2 2 - ωS z where S z is the projection of the spin operator along the z-direction. (a) Find the possible energies of this system and the energy eigenstates. Show that the excited states are doubly degenerate. These levels are related to the so-called ”Landau levels” for an electron in a constant magnetic field. (b) Consider the operator Q , Q = (¯ ) 1 / 2 - , where σ - = σ x - y and a is the annihilation operator encountered in the harmonic oscillator. The fancy name for this operator is ”super-generator”. Show that this operator annihilates the ground state of the system. Show also that Q and Q transform the two members of a degenerate pair of each excited state into each other. This operation is called a super-symmetry transformation. (c) Show that the anti-commutator [ Q,Q ] + H 5. Consider a spin-less free particle of mass m confined to move otherwise freely in a finite pipe of length L 3 with a transverse section that is a two dimensional rectangle with sizes L 1 and L 2 , L 1 < L 2 < L 3
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(a) Find the spectrum of the Hamiltonian. (b) What is the probability to find the particle, when prepared in the ground state, In the region0 < x < L 1 / 2 , 0 < y < L 2 / 2 , 0 < z < L 3 / 2? (c) What is the average value of -→ L 2 and L z where L i are the components of the orbital angular momentum of this particle, when the state of the system is prepared in the ground state? 2
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