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Lectures on LightNonlinear and Quantum Optics using the Density Matrix$

Stephen Rand

Print publication date: 2010

Print ISBN-13: 9780199574872

Published to Oxford Scholarship Online: September 2010

DOI: 10.1093/acprof:oso/9780199574872.001.0001

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Appendix C The Classical Hamiltonian of Electromagnetic Interactions

Appendix C The Classical Hamiltonian of Electromagnetic Interactions

Source:
Lectures on Light
Publisher:
Oxford University Press

Appendix C

The Classical Hamiltonian of Electromagnetic Interactions

The nonrelativistic force on a charged particle of velocity is the Lorentz force

F ¯ = q [ E ¯ + ( v ¯ × B ¯ ) ] .
(C.1)

On the basis of the vector identity

¯ · ( ¯ × A ¯ ) = 0 ,
(C.2)

where Ā is an arbitrary vector potential, we conclude from the Maxwell equation ∇̄∙ = 0 that can always be written in the form

B ¯ = ¯ × A ¯ ,
(C.3)

where Ā is a vector potential.

Similarly, applying Eq. (C.3) in the Maxwell equation ¯ × E ¯ = B ¯ t , we see that it can be written as

¯ × E ¯ + t ( ¯ × A ¯ ) = 0 ,
(C.4)

or by reversing the order of derivatives, as

¯ × ( E ¯ + A ¯ t ) = 0.
(C.5)

(p.267) Making use of a second vector identity, namely

¯ × ¯ φ = 0 ,
(C.6)

where φ is a scalar potential, the electric field can clearly always be written as

E ¯ = ¯ φ A ¯ t .
(C.7)

The expression for the Lorentz force becomes

F ¯ = q [ ¯ φ A ¯ t + ( v ¯ × [ ¯ × A ¯ ] ) ] .
(C.8)

Using the replacements d A ¯ d t = A ¯ t + ( v ¯ · ̄ ) A ¯ and × (∇̄ × Ā) = ∇̄ (Ā) − (∙∇̄)Ā, we find

F ¯ = q [ ¯ φ + ¯ ( v ¯ · A ¯ ) A ¯ t ( v ¯ · ¯ ) A ¯ ] = q [ ¯ ( φ v ¯ · A ¯ ) d A ¯ d t ] = q [ ¯ ( φ v ¯ · A ¯ ) d d t ¯ v ( v ¯ · A ¯ ) ] .
(C.9)

Since the scalar potential φ does not depend on velocity, the ith component of the force on a particle may be written as

F i = q [ U x i d d t U v i ] ,
(C.10)

where i = x,y, z and Uq(φ − Ā). Here U is a generalized potential function from which we obtain the Lagrangian in traditional, nonrelativistic mechanics:

L = T U = m v 2 2 q φ + q A ¯ · v ¯ .
(C.11)

The Lagrangian in turn furnishes the Hamiltonian

H = i x ˙ i L x ˙ i L = 1 2 m v 2 + q φ ,
(C.12)

which, when expressed in terms of the canonical momentum p i = L / x ˙ i = m v i + q A i yields

H = 1 2 m ( p ¯ q A ¯ ) · ( p ¯ q A ¯ ) + q φ .
(C.13)

The Hamiltonian in Eq. (C.13) can be converted to a form that is much more convenient for describing the system in terms of the applied fields E and B, to separate out multipole contributions, and for eventual quantization [C.1]. To make the conversion, (p.268) we note that Eqs. (C.3) and (C.7) can be satisfied if φ(,t) and Ā(,t) are expanded in Taylor's series representations as

φ ( r ¯ , t ) = r ¯ . E ¯ ( 0 , t ) 1 2 r ¯ r ¯ : ( ¯ E ¯ ( 0 , t ) ) + ,
(C.14)

and

A ¯ ( r ¯ , t ) = 1 2 B ¯ ( r ) × r ¯ + 1 3 r ¯ · ¯ B ¯ ( 0 , t ) × r ¯ + .
(C.15)

These expansions constitute a specific choice of gauge consistent with Taylor expansions of the fields themselves:

E ¯ ( r ¯ , t ) = E ¯ ( 0 , t ) + r ¯ · ( ¯ E ¯ ( 0 , t ) + ,
(C.16)
B ¯ ( r ¯ , t ) = B ¯ ( 0 , t ) + r ¯ · ( ¯ B ¯ ( 0 , t ) + .
(C.17)

By substituting Eqs. (C.14) and (C.15) into Eq. (C.13), one finds that the interaction Hamiltonian assumes the form

H = μ ¯ ( e ) · E ¯ μ ¯ ( m ) · B ¯
(C.18)

if we retain only the leading electric and magnetic dipole terms, with moments μ̄(e) = er̄ (pointing from the negative to the positive charge) and μ ¯ ( m ) = 1 2 e r ¯ × v ¯ .

References

C.1 L.D. Barron and C.G. Gray, J. Phys. A: Math., Nucl. Gen., 6, 59(1973).