# What is the Laplacian in the Pauli Equation?

The Pauli Equation is given by

$$\left[\frac{1}{2m}\left[({\bf\hat{p}}-q {\bf A})^2-q\hbar{\bf\sigma}\cdot {\bf B}]+q\phi\right]\right]|\psi\rangle=i\hbar\frac{\partial}{\partial t}|\psi\rangle.$$

This contains a component $${\bf\hat{p}}^2 |\psi\rangle$$. However, according to Wikipedia:

The state of the system, |ψ⟩ (written in Dirac notation), can be considered as a two-component spinor wavefunction

So $$|\psi\rangle$$ is a vector with two elements, as a function of time. Therefore, I think that $$\bf{\hat{p}}$$ can be viewed as a $$2\times2$$ matrix. What are the matrix elements?

If it is just $$-i\hbar$$ times the identity matrix, why is the equation not simplified to

$$\left[\frac{1}{2m}\left[(-i\hbar\bf{I}-q\bf{A})^2-q\bar{h}\bf{\sigma}\cdot \bf{B}]+q\phi\right]\right]|\psi\rangle=i\hbar\frac{\delta}{\delta t}|\psi\rangle,$$ with $$\bf{I}$$ the identity?

If it is $$-i\hbar\nabla$$ times the identity matrix, then how can we take the laplacian of an element of the two-spinor? After all, each element is just a complex number.

• As usual, if $\psi\in L^2\otimes \mathbb C^2$, for example, then if something like $p^2$ is written, people usually mean $p^2\otimes \mathbb I$, where $\mathbb I$ is the identity matrix on $\mathbb C^2$ Jan 14, 2023 at 19:02
• So we can replace the momentum operator by $-i\bar{h}$? Jan 14, 2023 at 19:04
• Because Wikipedia says: The state of the system, $|\psi\rangle$ (written in Dirac notation), can be considered as a two-component spinor wavefunction Jan 14, 2023 at 19:06
• Indeed, the momentum may be represented by $-i\hbar \nabla$ when acting on any component of the 2-spinor: there is an implicit 2-d identity matrix in it, when acting on spinor space. Is this your question? Jan 14, 2023 at 19:11
• Yes that is my question Jan 14, 2023 at 19:16

As mentioned in the comments, it is understood that $$p^2$$ stands for $$p^2 \otimes \mathbb{I}$$. Since $$p^2 = - \hbar^2 \nabla^2$$, this means you just apply $$p^2$$ to each component of the spinor separately. In other words, the Laplacian is just applied to each entry separately.

Furthermore, don't forget that each component of the spinor is a function. The state of the system is a two-component spinor wavefunction. This means it is composed of two wavefunctions stacked on a spinor.