Quantum Engi Q and A_Part_15

Quantum Engi Q and A_Part_15 - 4.2. THE HYDROGEN ATOM 43...

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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: integraldisplay | ψ 100 | 2 d 3 vectorr = integraldisplay ∞ r =0 integraldisplay π θ =0 integraldisplay 2 π φ =0 1 πa 3 e − 2 r/a r 2 sin θ d r d θ d φ or rearranging 1 π integraldisplay ∞ r/a =0 e − 2 r/a r 2 a 2 d r a integraldisplay π θ =0 sin θ d θ integraldisplay 2 π φ =0 1 d φ giving 1 π × 1 4 × 2 × 2 π which is one as required. 4.2.2.2 Solution hydb-b Question: Use the generic expression ψ nlm = − 2 n 2 radicaltp radicalvertex radicalvertex radicalbt ( n − l − 1)! [( n + l )! a ] 3 parenleftbigg 2 ρ n parenrightbigg l L 2 l +1 n + l parenleftbigg 2 ρ n parenrightbigg e − ρ/n Y m l ( θ, φ ) with ρ = r/a and Y m l from the spherical harmonics table to find the ground state wave function ψ 100 . Note: the Laguerre polynomial L 1 ( x ) = 1 − x and for any p , L p 1 is just its p-th derivative....
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This note was uploaded on 11/13/2011 for the course PHY 4458 taught by Professor Garvin during the Fall '11 term at University of Florida.

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Quantum Engi Q and A_Part_15 - 4.2. THE HYDROGEN ATOM 43...

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