Coupling to Electromagnetic Fields
Electromagnetism is the most important interaction for the study of atoms, molecules and materials. It determines most of the potentials or perturbation operators V which are studied in practical applications of quantum mechanics, and it also serves as a
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Coupling to Electromagnetic Fields
Electromagnetism is the most important interaction for the study of atoms, molecules and materials. It determines most of the potentials or perturbation operators V which are studied in practical applications of quantum mechanics, and it also serves as a basic example for the implementation of other, more complicated interactions in quantum mechanics. Therefore the primary objective of the current chapter is to understand how electromagnetic fields are introduced in the Schrödinger equation.
15.1 Electromagnetic Couplings The introduction of electromagnetic fields into the Schrödinger equation for a particle of mass m and electric charge q can be inferred from the description of the particle in classical Lagrangian mechanics. The Lagrange function for the particle in electromagnetic fields E(x, t) = − ∇(x, t) −
∂A(x, t) , ∂t
B(x, t) = ∇ × A(x, t)
(15.1)
is L=
m ˙ 2 + q x(t) ˙ · A(x(t), t) − q(x(t), t). x(t) 2
(15.2)
Let us check (or review) that Eq. (15.2) is indeed the correct Lagrange function for the particle. The electromagnetic potentials in the Lagrange function depend on the time t both explicitly and implicitly through the time dependence x(t) of the trajectory of the particle. The time derivative of the conjugate momentum
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_15
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15 Coupling to Electromagnetic Fields
p=
∂L = mx˙ + qA ∂ x˙
(15.3)
is therefore ∂A ∂A dp = mx¨ + q x˙i . +q dt ∂xi ∂t
(15.4)
According to the Euler-Lagrange equations (cf. Appendix A), this must equal ∂L = q x˙i ∇Ai − q∇. ∂x
(15.5)
The property (7.34) of the tensor implies ei x˙j ∂i Aj − x˙j ∂j Ai = ei ij k klm x˙j ∂l Am = x˙ × B,
(15.6)
and therefore the Euler-Lagrange equation yields the Lorentz force law mx¨ = q(E + v × B),
(15.7)
as required. The classical Hamiltonian for the particle follows as H = p · x˙ − L =
m 1 (p − qA)2 + q = x˙ 2 + q. 2m 2
(15.8)
The Hamilton operator of the charged particle therefore becomes H =
1 [p − qA(x, t)]2 + q(x, t), 2m
(15.9)
and the Schrödinger equation in x representation is ih¯
1 ∂ 2 (x, t) = − [h∇ ¯ − iqA(x, t)] (x, t) + q(x, t)(x, t). ∂t 2m
(15.10)
This is the Schrödinger equation for a charged particle in electromagnetic fields. If we write this in the form ih¯
∂ 1 2 − q = (ih∇ ¯ + qA) ∂t 2m
(15.11)
we also recognize that this arises from the free Schrödinger equation through the substitutions ih∇ ¯ → ih∇ ¯ + qA,
ih¯
∂ = ihc∂ ¯ 0 → ihc∂ ¯ 0 − q. ∂t
(15.12)
15.1 Electromagnetic Couplings
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These equations can be combined in 4-vector notation with p0 = −E/c, A0 = −/c, pμ = − ih∂ ¯ μ → pμ − qAμ = − ih∂ ¯ μ − qAμ .
(15.13)
This observation is useful for recognizing a peculiar symmetry property of Eq. (15.10). Classical electromagnetism is invariant under gauge transformations of the electromagnetic potentials (here
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