We study the interaction of an atomic system with an electromagnetic radiation field. The Schrödinger equation for a mass point of charge q in an electromagnetic field described by a vector potential A is ( 10-1 ) i h ∂ ψ ∂ t = [ − ħ 2 2 m ∇ 2 + i q ħ m c A · ∇ + V ] ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493645/749bab20-4a79-4f5a-8c66-d9e9a36a87b9/content/eq876.tif"/>
The interaction of an atomic system with an unquantized electromagnetic field is studied by means of the Heisenberg-operator equations of motion. The electromagnetic fields created by the atom are taken into account by assuming that the charge and probability current densities are the actual charge and current source terms in Maxwell's equation. When Ehrenfest's theorem is written to order (1/c)2, the equations of motion are found to have a constant of motion that can be interpreted as stating that the sum of the atomic energy, energy of interaction, and energy stored in the electromagnetic field is constant. This constant of integration can be expressed as an expectation of a new semiclassical Hamiltonian that now includes the effects of the atomic self-fields to order 1/c2. This Hamiltonian is related to the classical Darwin Lagrangian. Since the new approximate Hamiltonian is the sum of an atomic and a field Hamiltonian, it provides a formulation of semiclassical radiation theory that is formally close to the usual formulation of quantum electrodynamics. A Schr\"odinger equation can be derived by applying the variational principle to the expectation of the new Hamiltonian. The result is a nonlinear integro-differential equation, in \ensuremath{\Psi}, which is somewhat similar to Hartree's self-consistent equation for a multielectron atom.When the Heisenberg equations are written to include the next-higher-order terms, 1/c3, it is found that the total energy of the atom and the electromagnetic field in the vicinity of the atom decreases at a rate that is given by the Larmor power formula (2e2/3c3)(d〈v〉/dt)2, where 〈v〉 is the quantum-mechanical expectation of the electron's velocity operator. This approximate formulation of semiclassical radiation theory is applied to semiclassical calculations of atomic radiative shifts. It is then shown that, if some retardation effects are included to all orders of 1/c in the vector potential, the semiclassical analysis provides a formula for radiative energy-level shifts that are time dependent and that contain coefficients similar to the starting point of the Bethe calculation of the Lamb shift. The incorporation of mass renormalization in semiclassical theory is then discussed.
In this Chapter we discuss the nonrelativistic theory of the elastic scattering of spin 1 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493645/749bab20-4a79-4f5a-8c66-d9e9a36a87b9/content/eq1266.tif"/> particles of mass μ in a spin-dependent potential which contains an ℓ · s term. The Schrödinger equation for the particle is ( 16-1 ) ( − ħ 2 2 μ ∇ 2 + V(r) + 2 ℓ ⋅ s U ( r ) − E ) Ψ = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493645/749bab20-4a79-4f5a-8c66-d9e9a36a87b9/content/eq1267.tif"/>
The Hamiltonian for a system of n particles that are identical, that is, particles that can be substituted for each other with no physical change, must be completely symmetric under any interchange of its arguments. Let Ψ(1, 2, …, n) be any solution of the Schrödinger equation, depending on the coordinates (spatial and spin) of the n identical particles 1 to n, and let P be any permutation of the n numbers 1 to n. Then PΨ(1, 2, …, n) ≡ Ψ(P1, P2, …, Pn) is a function which depends on the coordinates of particle Ψ in the same manner as the original function Ψ depended on the coordinates of particle i. Then using the symmetry of H, we can show that the operator P commutes with the Hamiltonian ( 2-1 ) H ( P Ψ ) = P ( H Ψ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493645/749bab20-4a79-4f5a-8c66-d9e9a36a87b9/content/eq89.tif"/>
Abstract To describe the interaction of a two-level atom with radiation we shall use a semiclassical treatment, i.e. the radiation is treated as a classical electric field but we use quantum mechanics to treat the atom. From the theory of the interaction with radiation, we will be able to find the conditions for which the equations reduce to a set of rate equations that describe the populations of the atomic energy levels (with a steady-state solution). In particular, for an atom illuminated by broad band radiation, this approach allows us to make a connection with Einstein’s treatment of radiation that was presented in Chapter 1; we shall find the Einstein B coefficient in terms of the matrix element for the transition.
H a , a =1,2,..., N are identical copies of a one particle Hamiltonian acting in different Hilbert spaces and V ab , a < b are identical copies of the interaction Hamiltonian between two particles We try a wave function | ψ t > = ⊗ k - 1 N | ψ k t > https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1428.tif"/> Substituting this into the Schrodinger equation gives Σ k ψ 1 t ⊗ … ⊗ ( i d ψ k t / d t ) ⊗ … ⊗ ψ n t = = Σ k ψ 1 t ⊗ ... ⊗ H k ψ k t ⊗ ... ⊗ ψ n t + + Σ a < b ψ 1 t ⊗ ... ⊗ V a b ψ a t ... ⊗ ψ b t ⊗ ... ⊗ ψ n t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1429.tif"/> Taking the inner product on both sides with ψ 1 t ⨶.. ψ kt ⨶...⨶ ψ nt where a hat above a symbol means omission of that symbol gives the approximate equations i d | ψ k t > / d t = H k | ψ k t > + Σ m ≠ k < I ⊗ ψ m t | V k m | ψ k t ⊗ ψ m t > https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1430.tif"/> 256 In terms of kernels, we can express this as i d ψ k t ( r ) / d t = H k ψ k t ( r ) + ( Σ m ≠ k ∫ V ( r , r ′ ) | ψ m t ( r ′ ) | 2 d 3 r ′ ) ψ k t ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1431.tif"/> This equation however does not take into account the Pauli exclusion principle. For taking that into account, we must try an antisymmetrized wave function | ψ t > = Σ σ ∈ S n | ψ σ 1 , t > ⊗ ... ⊗ | ψ σ n , t > https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1432.tif"/> Substituting and taking the inner products after assuming orthonormality of the component wave functions gives us i d | ψ k t > / d t = H k | ψ k t > + Σ m ≠ k < I ⊗ ψ m t | V | ψ k t ⊗ ψ m t > − Σ m ≠ k < ψ m t ⊗ I | V | ψ k t ⊗ ψ m t > https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1433.tif"/> which in coordinate form can be expressed as i d ψ k t ( r ) / d t = H k ψ k t ( r ) + Σ m ≠ k ( ∫ V ( r , r ′ ) | ψ m t ( r ′ ) | 2 d 3 r ′ ) ψ k t ( r ) − Σ m ≠ k ( ∫ V ( r , r ′ ) ψ ¯ m t ( r ′ ) ψ k t ( r ′ ) d 3 r ′ ) ψ m t ( r ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1434.tif"/> These equations are special cases of nonlinear Schrodinger equations of the form i d ψ k t ( r ) / d t = ∇ 2 ψ k t ( r ) / 2 m + V 0 ( r ) ψ k t ( r ) + Σ m V k m ( r , ψ 1 t , ... , ψ N t ) ψ m t ( r ) , k = 1 , 2 , ... , N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1435.tif"/> The charge density is ρ ( t , r ) = − e Σ k | ψ k t ( r ) | 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1436.tif"/> and its rate of increase is given by ρ t ( r ) = - e Σ k ( ψ ¯ k t , t ( r ) . ψ k t ( r ) + ψ ¯ k t ( r ) . ψ k t , t ( r ) = ( i e / 2 m ) Σ k [ ψ k t ( r ) ∇ 2 ψ ¯ k t ( r ) - ψ ¯ k t ( r ) ∇ 2 ψ k t ( r ) ] = d i v ( ( i e / 2 m ) Σ k [ ψ k t ( r ) ∇ ψ ¯ k t ( r ) - ψ ¯ k t ( r ) ∇ ψ k t ( r ) ] ) = - d i v J ( t , r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1437.tif"/> 257 where the current density J ( t,r ) is given by J ( t , r ) = ( − i e / 2 m ) Σ k [ ψ k t ( r ) ∇ ψ ¯ k t ( r ) − ψ ¯ k t ( r ) ∇ ψ k t ( r ) ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1438.tif"/> It should be noted that this expression for the current density is the same as that derived using the linear Schrodinger equations but the wave functions ψ kt are calculated by solving the nonlinear Schrodinger equation. It should be noted that in deriving the above formula for the current density, we have made used of the relation V ¯ k m ( r , ψ 1 , … , ψ N ) = V m k ( r , ψ 1 , … , ψ N ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1439.tif"/> which is readily verified.
Abstract The atom-field interaction is treated in semiclassical radiation theory, starting from the transformation from the minimal coupling Hamiltonian to the electric dipole form used extensively in quantum optics. The Heisenberg and density matrix approaches are developed and applied to two-state atoms, Bloch equations, Rabi oscillations, Maxwell-Bloch equations, and transition rates for absorption and emission. Einstein’s theory of blackbody radiation based on momentum fluctuations and dissipation is reviewed. The Einstein fluctuation formula is derived and used to introduce wave-particle duality, Hanbury Brown-Twiss correlations, and photon bunching.