45.3 Field equations with background electromagnetic fields

The minimal coupling method described above can be used to write down field equations for our free particle theories, now coupled to electromagnetic fields. They are:

• The Schr¨odinger equation for a non-relativistic particle coupled to a background electromagnetic field is

i (\frac{\partial}{\partial t} - i e A_{0}) ψ = - \frac{1}{2 m} j = 1 \sum 3 (\frac{\partial}{\partial x _{j}} - i e A_{j})^{2} ψ

A special case of this is the Coulomb potential problem discussed in chapter 21, which corresponds to the choice of background field

A_{0} = \frac{1}{r}, A = 0

Another exactly solvable special case is that of a constant magnetic field $B = (0, 0, B)$ , for which one possible choice of vector potential is

A_{0} = 0, A = (- B y, 0, 0)

• The Pauli-Schr¨odinger equation (34.3) describes a free $spin \frac{1}{2}$ non-relativistic quantum particle. Replacing derivatives by covariant derivatives one gets

i (\frac{\partial}{\partial t} - i e A_{0}) (ψ _{2} ( x ) ψ _{1} ( x )) = - \frac{1}{2 m} (σ \cdot (\nabla - i e A))^{2} (ψ _{2} ( x ) ψ _{1} ( x ))

Using the anticommutation

σ_{j} σ_{k} + σ_{k} σ_{j} = 2 δ_{j k}

and commutation

σ_{j} σ_{k} - σ_{k} σ_{j} = i ϵ_{j k l} σ_{l}

relations one finds

σ_{j} σ_{k} = δ_{j k} + \frac{1}{2} i ϵ_{j k l} σ_{l}

This implies that

(σ \cdot (\nabla - i e A))^{2} = \sum_{j = 1}^{3} (\frac{\partial}{\partial x _{j}} - i e A_{j})^{2} + \sum_{j, k = 1}^{3} (\frac{\partial}{\partial x _{j}} - i e A_{j}) (\frac{\partial}{\partial x _{k}} - i e A_{k}) \frac{i}{2} ϵ_{j k l} σ_{l} = \sum_{j = 1}^{3} (\frac{\partial}{\partial x _{j}} - i e A_{j})^{2} + e σ \cdot B

and the Pauli-Schr¨odinger equation can be written

i (\frac{\partial}{\partial t} - i e A_{0}) (ψ _{2} ( x ) ψ _{1} ( x )) = - \frac{1}{2 m} (j = 1 \sum 3 (\frac{\partial}{\partial x _{j}} - i e A_{j})^{2} + e σ \cdot B) (ψ _{2} ( x ) ψ _{1} ( x ))

This two-component equation is just two copies of the standard Schr¨odinger equation and an added term coupling the spin and magnetic field which is exactly the one studied in chapter 7. Comparing to the discussion there, we see that the minimal coupling prescription here is equivalent to a choice of gyromagnetic ratio $g = 1$

• With minimal coupling to the electromagnetic field, the Klein-Gordon equation becomes

(- (\frac{\partial}{\partial t} - i e A_{0})^{2} + j = 1 \sum 3 (\frac{\partial}{\partial x _{j}} - i e A_{j})^{2} - m^{2}) ϕ = 0

The first two equations are for non-relativistic theories, and one can interpret these equations as describing a single quantum particle (with spin $\frac{1}{2}$ in the second case) moving in a background electromagnetic field. In the relativistic Klein-Gordon case, here we are in the case of a complex Klein-Gordon field, as discussed in section 44.1.2. In all three cases, in principle a quantum field theory can be defined by taking the space of solutions of the equation as phase space, and applying the Bargmann-Fock quantization method (in practice this is dificult, since in general there is no translation invariance and no plane-wave basis of solutions).

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正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 484、485、486、487、488、489、490、491、492、493

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

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45.3 Field equations with background electromagnetic fields

45.3 Field equations with background electromagnetic fields

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