Φ ϕ 1 2 lim s s 1 29 1 2 exp i s ϕ s 1 exp i s 1

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ϕ ϕ + 1 2 lim s →∞ s + 1 ( 29 - 1 2 exp i s ϕ [ ] s + 1 - exp i s + 1 ( 29 ϕ [ ] s - exp - i ϕ [ ] 0 { } [ III-16b ] so that the state ϕ fails to be a strict eigenket of cos Φ by terms that diminish faster than s + 1 ( 29 - 1 2 as s →∞ . Similarly, we can see that diagonal matrix elements of cos Φ and sin Φ are given by ϕ cos Φ ϕ = cos ϕ 1 - lim s →∞ s + 1 ( 29 - 1 { } cos ϕ [ III-17a ] ϕ sin Φ ϕ = sin ϕ 1 - lim s →∞ s + 1 ( 29 - 1 { } sin ϕ [ III-17b ]
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 25 R. Victor Jones, May 2, 2000 25 Reprise of Characteristics and Properties of Phase States: a. The expectation value of the number operator and the fractional uncertainty associated with a state of well-defined phase : ϕ N ϕ = lim s →∞ s + 1 ( 29 - 1 n n = 0 s = lim s →∞ s + 1 ( 29 - 1 s s + 1 ( 29 2 = lim s →∞ s 2 [ III-18a ] fractional uncertainty = ϕ N 2 ϕ - ϕ N ϕ 2 { } ϕ N ϕ = lim s →∞ s + 1 ( 29 - 1 n 2 n = 0 s - lim s →∞ s + 1 ( 29 - 1 n n = 0 s 2 lim s →∞ s + 1 ( 29 - 1 n n = 0 s = lim s →∞ 1 6 2 s 2 + s ( 29 - 1 4 s 2 lim s →∞ s 2 = 1 3 [ III-18b ] b. Expectation value of the fields associated with a single mode : From Equation [ III-3a ] ϕ r E ϕ =- 2 h ϖ 2 ε 0 V ˆ e sin r k r r - ϖ t + ϕ ( 29 lim s →∞ s + 1 ( 29 - 1 n + 1 ( 29 1 2 n = 0 s diverges as s for large s ! [ III-19 ]
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 26 R. Victor Jones, May 2, 2000 26 c. Phase of field associated with single mode : ϕ cos Φ ϕ = cos ϕ ϕ sin Φ ϕ = sin ϕ [ III-20a ] cos Φ = ∆ sin Φ= ϕ cos 2 Φ ϕ - ϕ cos Φ ϕ 2 { } = 0 [ III-20b ] d. Probability of photon number : Finally, we may easily deduce the probability of finding n photons ( i.e. the photon statistics) in a particular state of well defined phase -- viz. P n = n ϕ 2 lim s →∞ s + 1 ( 29 - 1 [ III-50 ] We see that there is a equal, but small probability of any number: this agrees with the intuition that the magnitude of the field is completely undetermined if the phase is precisely known! 3. Coherent Photon States: 16 It would, indeed, be useful to have eigenstates of the destruction operator (electric or magnetic field) -- viz. a r k α r k = α r k α r k [ III-51 ] Reprise of Characteristics and Properties of Coherent States: a. The Fock state representation of the coherent state : 16 The coherent state is a Harvard invention ! See R. J. Glauber, Phys. Rev. 131 , 2766 (1963).
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 27 R. Victor Jones, May 2, 2000 27 Since . a n = n + 1 n + 1 and a a = N + 1 , then n a = n + 1 n + 1 and we are able to write a representative of the sought state in the number state basis -- viz. n a α = n + 1 n + 1 α = α n α [ III-52a ] or n α = α n n - 1 α = α n n ! 0 α [ III-52b ] Using the expansion of the identity operator, the eigenket becomes α = n n α n = 0 α α n n !
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