ls3_unit_1 - THE INTERACTION OF RADIATION AND MATTER...

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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY I. Q UANTUM M ECHANICS OF H ARMONIC O SCILLATORS : 1 To prepare the ground for the quantization of the electromagnetic field, let us revisit the classical treatment of a simple harmonic oscillator with one degree of freedom. We start, with the assertion that the Hamiltonian of such an oscillator must have the form . H = 1 2 m p 2 + m 2 ω 2 q 2 [ ] . [ I-1 ] where p and q are the cannonically conjugate variables. Consequently, the accordant Hamilton equations yield a pair of coupled, inhomogeneous, first-order equations -- viz. dq dt = H p = p m [ I-2a ] d p dt = H q = m ω 2 q . [ I-2b ] Combining these two equations, we see that formal classical analysis of the harmonic oscillator leads to a second-order homogeneous equation of motion d 2 q d t 2 = −ω 2 q 2 [ I-2c ] with the evident complete solution q t ( ) = q 0 ( ) cos ω t + p 0 ( ) m ω sin ω t p t ( ) = m ω q 0 ( ) sin ω t + p 0 ( ) cos ω t [ I-3 ] 1 William Louisell’s Radiation and Noise in Quantum Electronics, McGraw-Hill (1964) has an excellent set of pre-quantum optics reviews.
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 2 R. Victor Jones, April 11, 2000 Equations [ I-2a ] and [ I-2b ] are decoupled by transforming to a new set of dynamic variables -- viz. 2 a = m 2 ω ω q + i p m [ ] a = m 2 ω ω q i p m [ ] [ I-4 ] so that q = 1 2 m ω a + a [ ] p = i m ω 2 a a [ ] [ I-5 ] In terms of these new variables the Hamiltonian becomes H = ω a a [ I-6 ] Thus, Equations [ I-2a ] and [ I-2b ] are transformed to the decoupled normal mode form -- i.e. da dt = i ω a i da dt = H a = ω a da dt = i ω a i da dt = H a = −ω a [ I-7 ] With this background, let us now turn to the quantum treatment of the harmonic oscillator. From the general theory discussed earlier, we make use of the classical analogy and postulate that the dynamic variables q and p become the Hermitian operators q and p which satisfy the commutation relations 2 In classical theory, the a and a variables are frequently referred to as normal mode amplitudes.
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 3 R. Victor Jones, April 11, 2000 q , q [ ] = 0 p , p [ ] = 0 q , p [ ] =
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