p137bp9sol

# p137bp9sol - Physics 137B Fall 2007 Moore Problem Set 9...

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Physics 137B, Fall 2007, Moore Problem Set 9 Solutions 1. Let a denote the hydrogen 1s state and let b denote the the 2p state with magnetic quantum number m . In the notation | n`m i , we have | a i = | 100 i , and | b i = | 21 m i . The transition rate for spontaneous emission, calculated via the statistical argument due to Einstein, is given by Eq. (11.76): W s ab = ω 3 ba 3 πc 3 ~ ± 0 | D ba | 2 . (1) Here ω ba = E 2 - E 1 ~ = - 3 4 E 1 ~ = 3 μα 2 c 2 8 ~ (2) and D ba = h b | - e r | a i = - e h 21 m | r | 100 i . For a ﬁxed value of the magnetic quantum number m , the dipole moment D ba can be calculated most easily by expressing the vector r in terms of so-called spherical tensor operators . To introduce this notion, ﬁrst deﬁne three new basis vectors e +1 = - 1 2 x - i ˆ y ) , e - 1 = 1 2 x + i ˆ y ) , e 0 = ˆ z . (The fact that we are using complex vectors to span real 3-dimensional space looks ﬁshy, but it turns out to be harmless. Note that these vectors are orthogonal in the complex sense, e * m · e m 0 = δ m,m 0 .) Next we deﬁne new coordinates r m e * m · r for m = 0 , ± 1. Explicitly r +1 = - 1 2 ( x + iy ) , r - 1 = 1 2 ( x - iy ) , r 0 = z . 1

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Using the orthogonality property mentioned above, we can write | D ba | 2 = e 2 ( |h b | r +1 | ψ a i| 2 + |h ψ b | r - 1 | a i| 2 + |h ψ b | r 0 | ψ a i| 2 ) = e 2 X m 0 |h 21 m | r m 0 | 100 i| 2 . The motivation for these deﬁnitions is that they allow us to express r m simply in terms of the spherical harmonic functions that we all know and love. Looking at Table 6.1, for instance, we see that with these deﬁnitions r m = r 4 π 3 rY 1 m ( θ, φ ) , m = 0 , ± 1 . Now the hydrogen atom wavefunctions are ψ n`m ( r, θ, φ ) = R n` Y `m ( θ, φ ), so h 21 m | r m 0 | 100 i = Z d 3 r ψ * 21 m ( r ) r 4 π 3 rY 1 m 0 ( θ, φ ) ψ 100 ( r ) = r 4 π 3 Z 0 r 3 R 21 ( r ) R 10 ( r ) dr Z Y 1 m ( θ, φ ) * Y 1 m 0 ( θ, φ ) Y 00 ( θ, φ ) sin θ dθ dφ = 1 3 δ m,m 0 Z 0 r 3 R 21 ( r ) R 10 ( r ) dr , since Y 00 = 1 4 π and the spherical harmonics have the orthogonality property Z Y `m ( θ, φ ) * Y ` 0
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## This note was uploaded on 08/01/2008 for the course PHYSICS 137B taught by Professor Moore during the Fall '07 term at Berkeley.

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p137bp9sol - Physics 137B Fall 2007 Moore Problem Set 9...

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