ls3_unit_3 - THE INTERACTION OF RADIATION AND MATTER...

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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 19 R. Victor Jones, May 2, 2000 19 III. R EPRESENTATIONS OF P HOTON S TATES 1. Fock or “Number” States : . 11 As we have seen, the Fock or number states n r k { } ≡ ∏ s σ n r k s σ [ III-1 ] are complete set eigenstates of an important group of commuting observables -- viz. H rad , N and r M . Reprise of Characteristics and Properties of Fock States: a. The expectation value of the number operator and the fractional uncertainty associated with a single Fock state : n N n = n [ III-2a ] n = "uncertainty" [ ] = n N 2 n - n N n 2 { } = 0 [ III-2b ] b. Expectation value of the fields associated with a single mode : For one mode Equations [ II-24a ] and [ II-24b ] reduce to r E r r , t ( 29 = i ˆ e E a exp i r k r r - i ϖ t [ ] - a exp - i r k r r - i ϖ t [ ] [ III-3a ] r H r r , t ( 29 = i ε 0 μ 0 E ˆ k × ˆ e [ ] a t ( 29 exp i r k r r - i ϖ t [ ] - a t ( 29 exp - i r k r r - i ϖ t [ ] [ III-3b ] 11 In what follows, for simplicity we drop the r k subscripts on the operators and state vectors with the obvious meaning that n r k { } n , a r k a , etc.. .
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 20 R. Victor Jones, May 2, 2000 20 where E= h ϖ 2 ε 0 V n r E n = 0 n r H n = 0 [ III-4a ] E = n r E r E n - n r E n 2 { } = h ϖ ε 0 V n + 1 2 ( 29 1 2 = 2 E n + 1 2 ( 29 1 2 H = n r H r H n - n r H n 2 { } = h ϖ μ 0 V n + 1 2 ( 29 1 2 = ε 0 μ 0 2 E n + 1 2 ( 29 1 2 E H = c h ϖ V n + 1 2 ( 29 = ε 0 μ 0 2 E 2 n + 1 2 ( 29 [ III-4b ] c. Phase of field associated with single mode : To obtain something analogous to the classical theory we would like to separate the creation and destruction operators (and, thus, the electric and magnetic field operators) into a product of amplitude and phase operators. Following Susskind and Glogower, 12 we define a phase operator , Φ such that a N + 1 ( 29 1 2 exp i Φ ( 29 a exp - i Φ ( 29 N + 1 ( 29 1 2 [ III-5 ] Defined in this way, the basic properties of the phase operator may be evaluated from known properties of the creation, destruction and number operators. Inverting, we obtain 12 Susskind, L. and Glogower, J., Physics , 1 , 49 (1964)
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 21 R. Victor Jones, May 2, 2000 21 exp i Φ ( 29 N + 1 ( 29 - 1 2 a exp - i Φ ( 29 a N + 1 ( 29 - 1 2 [ III-6 ] and since aa = N + 1 , it follows that exp i Φ ( 29 exp - i Φ ( 29 = 1 [ III-7 ] but only in this order! Operating on number states with the phase operators, we obtain from Equation [ I-26 ] exp i Φ ( 29 n = N + 1 ( 29 - 1 2 a n = N + 1 ( 29 - 1 2 n ( 29 1 2 n - 1 = n - 1 exp - i Φ ( 29 n = a N + 1 ( 29 - 1 2 n = a n + 1 ( 29 - 1 2 n = n + 1 [ III-8 ] Consequently, the only nonvanishing matrix elements of the phase operator are n - 1 exp i Φ ( 29 n = 1 n + 1 exp - i Φ ( 29 n = 1 [ III-9 ] The phase operators defined by Equation [ III-36 ] do have the felicitous or classically analogous property of revealing magnitude independent information, but unfortunately they are nonHermitian operators -- i.e.
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