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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 hydbb 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 pth
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 hydbc 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.
 Fall '11
 Dr.DanielArenas
 mechanics

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