Quantization of the radiation field

Now we enter the quantization of the radiation field. First, we summarize the Hamilton operator, H, and the Schrodinger equation with total wave function, W , and total energy, E , for the radiation field as an ensemble of harmonic oscillators. 

The strategy of quantization of radiation is based on the second quantization of vector potential, A(r, t), given in Eq. (1.76). So arbitrarily introduced expansion coefficients, bk and bkv, in Eq. (1.76) should be related to creation and annihilation operators. If we use the relationships of Eqs. (1.80) and (1.81) between the expansion coefficients and the operators, a and a, and introduce these relationships into Eq. (1.79), we get Eq. (1.102) as the expression for the quantized Hamiltonian, H, of the radiation field, which corresponds to Eq. (1.82) supporting these relationships. The operator for the total momentum, P, is given by Eq. (1.103). 

Let us consider the eigenvalues of operators H and P. As Eqs. (1.102) and (1.103) include creation and annihilation operators, we can assume, intuitively, that their eigenvalue becomes a number. Calculations by using Eq. (1.99) result in  

As the wave function of the harmonic oscillator includes a Hermit polynomial, the eigenvalue of the operator, = a v a v, is nkv as shown in Eq. (1.94). So, the total energy, E, 

The new operator, Nkv, is related to the number of photons specified by the condition (k, v). For example, going from the vacuum without any photons to a state with n photons with a (k, v) condition is described by using the creation operator as  

A key quantity is the vector potential A f,t) = A f), which satisfies a wave equation analogously to the wave equation for the electron field operators ip ). It is chosen divergence free V A r) = 0. Invoking zero scalar potential and this choice is usually referred to as the Coulomb gauge. 

The vector potential is expanded in a complete set of plane waves orthonor-malized over a finite volume V  

Analogously to the situation for electrons, the operator b k) applied to the vacuum state produces a state vector of one photon with propagation vector k and polarization vector n k). The expression 

When the source charges and currents of the photon field are separate (distant) from the electron system under study, the hamiltonian for the interacting electrons and the photon field can be expressed as 

The propagation of photons through matter is governed by Eq. (6.26) and the corresponding ones for electron operators. Define the photon propagator Dx k, ic E) = i(bx k ) b k)))E, which satisfies the equation of motion 

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A crucial step motivated by experimental observations and theoretical considerations is the quantization of the radiation field, whereupon the electric and magnetic fields assume operator character. We first notice that the functions [Pg.115]

The material system is promoted by the pump pulse to the excited electronic state from which it can (spontaneously) emit photons. Since the SE is a quantum process, it is described in the formalism of the quantization of the radiation field [1, 20]. The pump process, on the other hand, can be considered semiclassically. We thus start with the master equation (9.10) with N= 1,... 

Bucci and his coworkers (65,66) treated double resonance by means of a second quantization of the rf fields to avoid the use of the rotating frame. The total Hamiltonian 2 - is then given by equation (1) in which is the normal Hamiltonian of the isolated spin system in the absence of radiation but in the magnetic field, is the radiation Hamiltonian, and represents the interaction between the spins and the radiation field(s). 

The first two terms are the molecular Hamiltonian and the radiation field Hamiltonian. The molecular Schrodinger equation for the first term in (5.2) is assumed solved, with known eigenvalues and eigenfunctions. Solutions for the second term in (3.4) in vacuo are taken in second-quantized form. Hint can be taken in minimal-coupling form (5.3) allowing for the variation of the radiation field over the extent of the molecule,... 

This function is essentially different from Eq. (1.76). The A in Eq. (1.76) is a classical function defined at every point of time and space, while the A in Eq. (1.108) is an operator functioning on various state vectors in the space occupied by photons. This operator is called an operator of quantized field. The image of the quantized radiation field is depicted in Table 1.4. An ensemble of photons with the momentum of hk makes up the substance of the radiation field. 

So far we have treated absorption and stimulated emission of radiation. However, it is well known that an atom can emit radiation even when it is not externally perturbed, i.e. spontaneous emission. It is not possible to treat this process fully here, since consideration of the quantization of the electromagnetic field as described by Quantum ElectroDynamics (QED) is necessary. According to QED a coupling between the atom and the "vacuum state" of the field is responsible for the emission. 

Kramers and Heisenberg [2], who predicted the phenomenon of Raman scattering several years before Raman discovered it experimentally, advanced a semiclas-sical theory in which they treated the scattering molecule quantum mechanically and the radiation field classically. Dirac [3] soon extended the theory to include quantization of the radiatiOTi field, and Placzec, Albrecht and others explored the selection rules for molecules with various symmetries [4, 5]. A theory of the resonance Raman effect based on vibratiOTial wavepackets was developed by Heller, Mathies, Meyers and their colleagues [6-11]. Mukamel [1, 12] presented a comprehensive theory that considered the nonlinear response functions for pathways in LiouvUle space. Having briefly described the pertinent pathways in Liouville space above, we will first develop the Kramers-Heisenberg-Dirac theory by a second-order perturbation approach, and then turn to the wavepacket picture. 

The present derivation of the scattering cross-section is based on a non-relativistic quantum electrodynamic approach. In this picture, the modes of the radiation field are quantized and the electric field is treated as a quantum-mechanical operator that annihilates or creates photons populating the various modes. The field operator is given by... 

We want to learn how to quantize the radiation field. As a first step, consider a continuous elastic system. Any classical continuous elastic system in one dimension can be treated by a normal-mode analysis. Consider an elastic string of length a [m], tied at both ends to some fixed objects, with density per unit length p [kg m ], and tension, or Hooke s law force constant kH [N m-1]. The transverse displacements of the string along the x axis can be described by a transverse stretch y(x, t) at any point x along the string and at a time t. One can describe the y(x, t) as a Fourier sine series in x ... 

The following picture emerges The radiation field, represented classically by a vector potential—that is, by a superposition of plane waves, as before, with transverse electrical and magnetic fields—is now, in quantum electrodynamics, a system with quantized energies. The general eigenfunction is then a product eigenfunction of the type... 

This, plus the quantization of the normal modes of vibration of the electromagnetic radiation field (just demonstrated), form, together, the quantum-mechanical basis for the wave-particle duality A wave can become a particle, and vice versa, but you can never make a simultaneous experiment to test both the wave and the particle nature of the same system. 

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