2004AugQualQUANT - Ben Sauerwine Practice for Qualifying...

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Ben Sauerwine Practice for Qualifying Exams Problem Source: CMU August 2004 Qualifying Exam (a) Solve the energy eigenvalue equation for a particle of mass m in the square- well potential that vanishes for 0 < x < a and equals ( ) x V + elsewhere. Show that the energy eigenvalues are given by: ,... 3 , 2 , 1 , 2 2 2 2 2 = = n ma n E n π h with the corresponding normalized energy eigenfunctions ( ) = a x n a x n π ψ sin 2 Explicitly discuss the boundary conditions. The boundary conditions are: ( ) ( ) ( ) ( ) ( ) x E x x V x m a ψ ψ ψ ψ = + = = 2 2 2 2 0 0 h Here boundary conditions on the derivative are waived in the case of an infinite-potential barrier. Showing that these solutions are valid by substitution, I take: ( ) ( ) x E a x n a ma n a x n a n a m a x n a x m x H n n n ψ π π π π π ψ = = = = sin 2 2 sin 2 2 sin 2 2 2 2 2 2 2 2 2 2 2 h h h Showing that these solutions are normal by substitution, I take ( ) normalized a a boundary a x n n a a a dx a x n a dx a x n a dx x a a a n = = = = = 1 2 1 2 2 sin 2 2 1 2 1 2 2 cos 2 1 2 1 2 sin 2 0 0 2 0 2 π π π π ψ (b) Suppose we place two electrons (spin ½) in the infinite square well potential above. Determine the two-particle wave function, whose spatial part is ( 2 1 , , 2 1 x x n n ) ψ
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