34.1 The Pauli-Schr¨odinger equation and free spin 1/2 particles in d=3

34.1 The Pauli-Schr¨odinger equation and free spin 1/2 particles in d=3

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We have so far seen two quite diferent quantum systems based on three dimensional space:

• The free particle of chapter 19. This had classical phase space ${\mathbf{R}}^6$ with coordinates $q_1,q_2,q_3,p_1,p_2,p_3$ and Hamiltonian $\frac{1}{2m}|\bar{\mathbf{p}}{|}^2$ . Quantization using the Schr¨odinger representation gave operators $Q_1,Q_2,Q_3,P_1,P_2,P_3$ on the space ${\mathcal{H}}_B=L^2({\mathbf{R}}^3)$ of square-integrable functions of the position coordinates. The Hamiltonian operator is

$$H=\frac{1}{2m}|\mathbf{P}{|}^2=-\frac{1}{2m}\left(\frac{{\partial }^2}{\partial q_1^2}+\frac{{\partial }^2}{\partial q_2^2}+\frac{{\partial }^2}{\partial q_3^2}\right)$$

• The spin $\frac{1}{2}$ quantum system, discussed first in chapter $7$ and later in section 31.1.1. This had a pseudo-classical fermionic phase space ${\mathbf{R}}^3$ with coordinates ${\xi }_1,{\xi }_2,{\xi }_3$ which after quantization became the operators

$$\frac{1}{\sqrt{2}}{\sigma }_1,\frac{1}{\sqrt{2}}{\sigma }_2,\frac{1}{\sqrt{2}}{\sigma }_3$$

on the state space ${\mathcal{H}}_F={\mathbf{C}}^2$ . For this system we considered the Hamiltonian describing its interaction with a constant background magnetic field

$$H=-\frac{1}{2}(B_1{\sigma }_1+B_2{\sigma }_2+B_3{\sigma }_3)\text{\hspace{1em}(34.1)}$$

It turns out to be an experimental fact that fundamental matter particles are described by a quantum system that is the tensor product of these two systems, with state space

$$\mathcal{H}={\mathcal{H}}_B\otimes {\mathcal{H}}_F=L^2({\mathbf{R}}^3)\otimes {\mathbf{C}}^2\text{\hspace{1em}(34.2)}$$

which can be thought of as two-component complex wavefunctions. This system has a pseudo-classical description using a phase space with six conventional coordinates $q_j,p_j$ and three fermionic coordinates ${\xi }_j$ . On functions of these coordinates one has a generalized Poisson bracket $\{\cdot ,\cdot {\}}_{\pm }$ which provides a Lie superalgebra structure on such functions. On generators, the non-zero bracket relations are

$$\{q_j,p_k{\}}_{\pm }={\delta }_{jk},\{{\xi }_j,{\xi }_k{\}}_{\pm }={\delta }_{jk}$$

For now we will take the background magnetic field $\mathbf{B}=0$ . In chapter 45 we will see how to generalize the free particle to the case of a particle in a general background electromagnetic field, and then the Hamiltonian term 34.1 involving the B field will appear. In the absence of electromagnetic fields the classical Hamiltonian function will still be

$$h=\frac{1}{2m}(p_1^2+p_2^2+p_3^2)$$

but now this can be written in the following form (using the Leibniz rule for a Lie superbracket)

$$h=\frac{1}{2m}{\left\{\sum _{j=1}^3{p}_j{\xi }_j,\sum _{k=1}^3{p}_k{\xi }_k\right\}}_{\pm }=\frac{1}{2m}\sum _{j,k=1}^3{p}_j{\left\{{\xi }_j,{\xi }_k\right\}}_{\pm }{p}_k=\frac{1}{2m}\sum _{j=1}^3{p}_j^2$$

Note the appearance of the function $p_1{\xi }_1+p_2{\xi }_2+p_3{\xi }_3$ which now plays a role even more fundamental than that of the Hamiltonian (which can be expressed in terms of it). In this pseudo-classical theory $p_1{\xi }_1+p_2{\xi }_2+p_3{\xi }_3$ is the function generating a “supersymmet ${\mathrm{ry}}^{\mathrm{y}}$ , Poisson commuting with the Hamiltonian, while at the same time playing the role of a sort of “square root” of the Hamiltonian. It provides a new sort of symmetry that can be thought of as a “square root” of an infinitesimal time translation.

Quantization takes

$$p_1{\xi }_1+p_2{\xi }_2+p_3{\xi }_3\to \frac{1}{\sqrt{2}}\mathbf{\sigma }\cdot \mathbf{P}$$

and the Hamiltonian operator can now be written as an anticommutator or a square

$$H=\frac{1}{2m}[\frac{1}{\sqrt{2}}\sigma \cdot \mathbf{P},\frac{1}{\sqrt{2}}\sigma \cdot \mathbf{P}{]}_{+}=\frac{1}{2m}(\sigma \cdot \mathbf{P}{)}^2=\frac{1}{2m}(P_1^2+P_2^2+P_3^2)$$

(using the fact that the ${\sigma }_j$ satisfy the Cliford algebra relations for $\mathrm{Cliff}(3,0,\mathbf{R}))$ .

We will define the three dimensional Dirac operator as

$$\partial /={\sigma }_1\frac{\partial }{\partial q_1}+{\sigma }_2\frac{\partial }{\partial q_2}+{\sigma }_3\frac{\partial }{\partial q_3}=\mathbf{\sigma }\cdot \nabla$$

It operates on two-component wavefunctions

$$\psi (\mathbf{q})=\left(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q})}{{\psi }_2(\mathbf{q})}\right)$$

Using this Dirac operator (often called in this context the “Pauli operator”) we can write a two-component version of the Schr¨odinger equation (often called the “Pauli equation” or “Pauli-Schr¨odinger equation”)

$$\begin{array}{rl}i\frac{\partial }{\partial t}\left(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q})}{{\psi }_2(\mathbf{q})}\right)& =-\frac{1}{2m}{\left({\sigma }_1\frac{\partial }{\partial q_1}+{\sigma }_2\frac{\partial }{\partial q_2}+{\sigma }_3\frac{\partial }{\partial q_3}\right)}^2\left(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q})}{{\psi }_2(\mathbf{q})}\right)\\ & =-\frac{1}{2m}\left(\frac{{\partial }^2}{\partial q_1^2}+\frac{{\partial }^2}{\partial q_2^2}+\frac{{\partial }^2}{\partial q_3^2}\right)\left(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q})}{{\psi }_2(\mathbf{q})}\right)\end{array}\text{\hspace{1em}(34.3)}$$

This equation is two copies of the standard free particle Schr¨odinger equation, so physically corresponds to a quantum theory of two types of free particles of mass m. It becomes much more non-trivial when a coupling to an electromagnetic field is introduced, as will be seen in chapter 45.

The equation for the energy eigenfunctions of energy eigenvalue E will be

$$\frac{1}{2m}(\mathbf{\sigma }\cdot \mathbf{P}{)}^2(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q})}{{\psi }_2(\mathbf{q})})=E(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q})}{{\psi }_2(\mathbf{q})})$$

In terms of the inverse Fourier transform

$${\psi }_{1,2}(\mathbf{q})=\frac{1}{(2\pi {)}^{\frac{3}{2}}}{\int }_{{\mathbf{R}}^3}{e}^{i\mathbf{p}\cdot \mathbf{q}}{\tilde{\psi }}_{1,2}(\mathbf{p})d^3\mathbf{p}$$

this equation becomes

$$((\mathbf{\sigma }\cdot \mathbf{p}{)}^2-2mE)(\genfrac{}{}{0ex}{}{{\tilde{\psi }}_1(\mathbf{p})}{{\tilde{\psi }}_2(\mathbf{p})})=(|\mathbf{p}{|}^2-2mE)\left(\genfrac{}{}{0ex}{}{{\tilde{\psi }}_1(\mathbf{p})}{{\tilde{\psi }}_2(\mathbf{p})}\right)=0\text{\hspace{1em}(34.4)}$$

and as in chapter 19 our solution space is given by distributions supported on the sphere of radius $\sqrt{2mE}=\left|\mathbf{p}\right|$ in momentum space which we will write as

$$\left(\genfrac{}{}{0ex}{}{{\tilde{\psi }}_1(\mathbf{p})}{{\tilde{\psi }}_2(\mathbf{p})}\right)=\delta (|\mathbf{p}{|}^2-2mE)(\genfrac{}{}{0ex}{}{{\tilde{\psi }}_{E,1}(\mathbf{p})}{{\tilde{\psi }}_{E,2}(\mathbf{p})})\text{\hspace{1em}(34.5)}$$

where ${\tilde{\psi }}_{E,1}(\mathbf{p})$ and ${\tilde{\psi }}_{E,2}(\mathbf{p})$ are functions on the sphere $|\mathbf{p}{|}^2=2mE$

Chapter contents · Next section: 34.2 Solutions of the Pauli equation and representations of E(3) · 中文版本

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来源版本：2025-10-20

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