Fund Quantum Mechanics Lect & HW Solutions 61

Fund Quantum - 4.2 THE HYDROGEN ATOM 43 The total probability of finding the particle integrated over all possible positions is using the

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Unformatted text preview: 4.2. THE HYDROGEN ATOM 43 The total probability of finding the particle integrated over all possible positions is, using the techniques of volume integration in spherical coordinates: ∞ π 2π r =0 |ψ100 |2 d3 r = θ=0 φ=0 or rearranging 1 π ∞ −2r/a0 r/a0 =0 e giving r2 r d a2 a0 0 1 −2r/a0 2 r sin θ drdθdφ e πa3 0 π θ=0 sin θ dθ 2π φ=0 1 dφ 11 × × 2 × 2π π4 which is one as required. 4.2.2.2 Solution hydb-b Question: Use the generic expression ψnlm = − 2 n2 (n − l − 1)! [(n + l)!a0 ]3 l 2ρ n +1 L2l+l n 2ρ −ρ/n m e Yl (θ, φ) n with ρ = r/a0 and Ylm from the spherical harmonics table to find the ground state wave function ψ100 . Note: the Laguerre polynomial L1 (x) = 1 − x and for any p, Lp is just its p-th 1 derivative. Answer: You get, substituting n = 1, l = 0, m = 0: ψ100 = − 2 12 0! [1!a0 ]3 0 2ρ 1 L1 1 2ρ −ρ/1 0 e Y0 (θ, φ) 1 where 0! = 1! = √ L1 (x) is the first derivative of L1 (x) = 1 − x with respect to x, which is 1, 1 0 −1, and Y0 = 1/ 4π according to the table. So you get ψ100 = 1 π a3 0 e−r/a0 . as in the previous question. 4.2.2.3 Solution hydb-c Question: Plug numbers into the generic expression for the energy eigenvalues, h2 1 ¯ En = − , 2me a2 n2 0 ...
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This note was uploaded on 01/06/2012 for the course PHY 3604 taught by Professor Dr.danielarenas during the Fall '11 term at UNF.

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