2002AugQualQUANT2 - Ben Sauerwine Practice for Qualifying...

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Ben Sauerwine Practice for Qualifying Exams Problem Source: CMU Qualifying Exam August 2002 This problem considers electrons confined to a box of length a with a square well potential ( ) { } = otherwise a x x V 0 0 In addition to the confining potential, a strong magnetic field ensures that all electrons have parallel spins. Neglect interactions between electrons. (a) Write down the complete set of single-particle wave functions ( ) x n ψ , properly normalized, and give their energies. These are the well-known solutions to the particle in the box problem: ( ) 2 2 2 2 2 sin 2 a n m E x a n a x n n π π ψ h = = (b) Write down the properly normalized two-electron ground state wave function ( 2 1 0 , x x ) ψ in terms of the single-electron wave functions ( ) x n ψ , and give its energy. The antisymmetric combination of these two wave functions (assuming both electrons are forced to be in the down state) is ( ) ( ) ( ) ( ) ( ) [ ] = 1 2 2 1 2 2 1 1 2 1 0 2 1 , x x x x x x ψ ψ ψ ψ ψ with energy 2 2 2 2 1 2 5
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