2001AugQualQUANT - Ben Sauerwine Practice for Qualifying...

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Ben Sauerwine Practice for Qualifying Exams Thanks to Yossef and Shiang Yong for their input in this problem. Problem Source: CMU August 2001 Qualifying Exam Electron in a spherical well An electron is in a spherical well of radius a and depth , i.e., the non-relativistic Hamiltonian is 0 V V m p H + = 2 2 v (1) with m = the mass of an electron and a r V a r V V > = = 0 0 (2) In this problem, we take 1 2 = π h . The solutions of the Schrodinger equation are of the form ( ) ( ) s lm Y r R χ φ θ , , where ( ) r R is the radial wave equation, ( ) φ θ , lm Y is a spherical harmonic, and s χ is a non-relativistic spinor. ( ) r R is a solution to the equation ( ) ( ) 0 1 2 1 2 2 2 = + + R r l l V E m dr dR r dr d r (3) where E is the energy of the state. Note that in spherical coordinates, the operator has the form 2 ( φ θ , 1 1 2 2 2 2 Ω + = r dr dR r dr d r ) (4) where Ω is an operator in the angular variables. The differential equation satisfied by spherical Bessel or Hankel functions is ( ) ( ) ( ) 0 1 1 1 2 2 2 = + + x F x l l dx x dF x dx d x l l (5)
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