solq(1) - Physics 742 Graduate Quantum Mechanics 2 Solution...

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Physics 742 – Graduate Quantum Mechanics 2 Solution Set Q 1. [15] A spin ½ particle of mass m lies in a one-dimensional spin-dependant potential 2 22 0 1 P Hm Q P m ω + =+ where 10 00 P + ⎛⎞ = ⎜⎟ ⎝⎠ . In other words, the potential only affects particles in a spin-up state. (a) [4] Find the discrete energy eigenstates for spin-up (they should look something like , i + ) and the continuum energy eigenstates for spin-down (they should look something like , β ). Also, identify their energies. For spin up particles, the Hamiltonian is 2 1 0 2 2 HPmm Q , which is a harmonic oscillator potential. The eigenstates are labeled , n + , where the + denotes spin up, and have energy () 1 2 n + = . The ground state, for example, is given by 1/4 2 0, exp 2 mm x x ωω ψ π + = −+ == For spin down particles, the Hamiltonian is 2 0 2 HPm = , which is just the free particle Hamiltonian. These states can be chosen to be labeled by their wave number, which we could denote by , k . They have energy 2 km = . Properly normalized, they look like , 1 2 ikx k xe =
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(b) [11] At t
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This note was uploaded on 02/15/2011 for the course PHYS 742 taught by Professor Unknow during the Spring '04 term at Harvard.

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solq(1) - Physics 742 Graduate Quantum Mechanics 2 Solution...

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