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Quantum Engi Q and A_Part_15

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

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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 0 e 2 r/a 0 r 2 sin θ d r d θ d φ or rearranging 1 π integraldisplay r/a 0 =0 e 2 r/a 0 r 2 a 2 0 d r a 0 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 0 ] 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 0 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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Quantum Engi Q and A_Part_15 - 4.2 THE HYDROGEN ATOM 43 The...

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