N n iii 53 to normalize the eigenket write α α α

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n n . [ III-53 ] To normalize the eigenket write α α = α 0 0 α α n α n n ! n = α 0 0 α exp α 2 [ ] = 1 [ III-54 ] so that α 0 = 0 α = exp - 1 2 α 2 . Finally, we see that α = exp - 1 2 α 2 α n n ! n n [ III-55 ] is a normalize representation of the eigenkets of the destruction operator.
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 28 R. Victor Jones, May 2, 2000 28 b. The expectation value of the number operator and the fractional uncertainty associated with a coherent state : α N α = α 2 [ III-56a ] fractional uncertainty = α N 2 α - α N α 2 { } α N α = 1 α 2 exp 2 ( 29 α 2 n n ! n 2 - α 4 = 1 α 2 exp 2 ( 29 α 2 n n ! n n - 1 ( 29 + n [ ] - α 4 = α - 1 [ III-56b ] Thus, we see that the fractional uncertainty diminishes with mean photon number! c. Expectation value of the electric field associated with a single mode : From Equation [ III-3a ] α r E α = - 2 h ϖ 2 ε 0 V ˆ e α sin r k r r t ( 29 [ III-57a ] where α= α exp i ϑ ( 29 . E = α r E r E α - α r E α 2 { } = h ϖ 2 ε 0 V 17 [ III-57b ] 17 Similarly H = 1 c μ 0 h ϖ 2 ε 0 V for the coherent state, so that E H = c h ϖ 2 V .
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 29 R. Victor Jones, May 2, 2000 29 d. Probability of photon number : From the representation of the coherent state given in Equation [ III-55 ] we may easily deduce the probability of finding n photons (the photon statistics) in a particular coherent state is given by a Poisson distribution characterized by the mean value n = α 2 . -- viz. P n = n α 2 = exp - α 2 [ ] α 2 n n ! [ III-58 ] S AMPLE P OISSON D ISTRIBUTIONS - C OHERENT S TATE P HOTON S TATISTICS
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 30 R. Victor Jones, May 2, 2000 30 e. Phase of field associated with single mode : α cos Φ α = 1 2 exp - 1 2 α 2 n α n n ! N + 1 ( 29 - 1 2 a + a N + 1 ( 29 - 1 2 [ ] α n n ! n n n = 1 2 exp - 1 2 α 2 α n + 1 α n + α n α n + 1 n + 1 ( 29 ! n ! { } n = α cos ϑ exp - 1 2 α 2 α 2 n n ! n + 1 ( 29 n [ III-59a] Unfortunately, it is not possible to evaluate this summation analytically. However, Carruthers 18 has given an asymptotic expansion which is valid for a large mean number of photons -- viz. α cos Φ α = cos ϑ 1 - 1 8 α 2 + K α 2 1 [ III-59b] f. Coherent states as a basis: As we will see presently, the coherent states are very useful in describing the quantized electromagnetic field, but, alas, there is a complication -- the coherent states are not truly orthogonal! From Equation [ III-6 ] we see that β α = exp - 1 2 α 2 - 1 2 β 2 β n α n n ! n = exp - 1 2 α 2 - 1 2 β 2 +αβ [ III-60 ] so that 18 Carruthers, P. and Nieto, M. M., Phys. Rev. Lett. 14 , 387 (1965)
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 31 R. Victor Jones, May 2, 2000 31 αβ β α = exp 2 - β 2 +αβ β ( 29 = exp - α-β ( 29 α -β ( 29 ( 29 = exp - α-β 2 ( 29 [ III-61 ] That is, t
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