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## 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

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

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(C.1)

On the basis of the vector identity

$Display mathematics$
(C.2)

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

$Display mathematics$
(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

$Display mathematics$
(C.4)

or by reversing the order of derivatives, as

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(C.5)

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

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(C.6)

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

$Display mathematics$
(C.7)

The expression for the Lorentz force becomes

$Display mathematics$
(C.8)

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

$Display mathematics$
(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

$Display mathematics$
(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:

$Display mathematics$
(C.11)

The Lagrangian in turn furnishes the Hamiltonian

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(C.12)

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

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(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

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(C.14)

and

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(C.15)

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

$Display mathematics$
(C.16)
$Display mathematics$
(C.17)

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

$Display mathematics$
(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).