p40_051 - 51. The proposed wave function is = 1 er/a a3/2...

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51. The proposed wave function is ψ = 1 πa 3 / 2 e r/a where a is the Bohr radius. Substituting this into the right side of Schr¨ odinger’s equation, our goal is to show that the result is zero. The derivative is dr = 1 πa 5 / 2 e r/a so r 2 dr = r 2 πa 5 / 2 e r/a and 1 r 2 d dr µ r 2 dr = 1 πa 5 / 2 · 2 r + 1 a ¸ e r/a = 1 a · 2 r + 1 a ¸ ψ. The energy of the ground state is given by E = me 4 / 8 ε 2 0 h 2 , and the Bohr radius is given by a = h 2 ε 0 /πme 2 ,so E = e 2 / 8 πε 0 a . The potential energy is given by
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This note was uploaded on 11/12/2011 for the course PHYS 2001 taught by Professor Sprunger during the Fall '08 term at LSU.

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