# About Density Matrix of a Particle

The quantum state of a spin- 1/2 particle can be written, in the momentum representation, as a two-component spinor,

$$\textit{Ψ}(\textbf{p})=\left(\begin{matrix}a_{1}(\textbf{p})\\a_{2}(\textbf{p})\end{matrix}\right),$$

With the density matrix corresponding to 2-component spinor above is

$$\textit{ρ}(\textbf{p′,p′′})=\left(\begin{matrix}a_{1}(\textbf{p′})a_{1}(\textbf{p′′})^*&a_{1}(\textbf{p′})a_{2}(\textbf{p′′})^*\\a_{2}(\textbf{p′})a_{1}(\textbf{p′′})^*&a_{2}(\textbf{p′})a_{2}(\textbf{p′′})^*\end{matrix}\right),$$

The reduced density for spin, increase irrespective of momentum, is obtained by setting $$\textbf{p} = \textbf{p′} = \textbf{p′′}$$ and integrating over p, with the bloch vector n is given by

$$n_{z} = \int [|a_{1}(\textbf{p})|^2-|a_{2}(\textbf{p})|^2]d\textbf{p}$$

and

$$n_{x} - in_{y} = \int a_{1}(\textbf{p})a_{2}(\textbf{p})^*d\textbf{p}$$

we can write the reduced density matrix for spin as $$\textit{𝜏}=\left(\begin{matrix}1+n_{z}&n_{x}-in_{y}\\n_{x}+in_{y}&1-n_{z}\end{matrix}\right),$$

i still dont understand how integrating over p can get the 𝜏 form of matrix, can someone help me ?

## 2 Answers

The Hilbert space is $$\mathcal H= L^2(R^3)\times \mathbb C^2$$, to find the reduced spin density matrix you have to trace over $$L^2(R^3)$$. That is why you are integrating $$\int d^3 p \; \rho(p,p)$$.

Remember that your "wavefunctions" are normalized (or, the density matrix should contain a normalization factor) so that $$\int d^3p |a_1(p)|^2 + |a_2(p)|^2=1$$.

Now, take your original $$\rho(p,p')$$, and write down the integral I mentioned above. Then use the normalization condition to rewrite the diagonal components, and you will get the $$\tau$$ expression.

Integrating over p will not get you a matrix but will get you the matrix elements in the matrix.

Your $$\rho$$ is hermitian with trace=1 as per general properties of the density matrix. Since any hermitian matrix with trace=1 can be written in the form $$\textit{𝜏}=\left(\begin{matrix}1+n_{z}&n_{x}-in_{y}\\n_{x}+in_{y}&1-n_{z}\end{matrix}\right),$$ it's really just a matter matching the various matrix elements in terms of spinors with those in terms of the $$n_k$$ parametrization.

If you prefer, $$n_z$$ is defined as the average difference $$n_{z} = \int [|a_{1}(\textbf{p})|^2-|a_{2}(\textbf{p})|^2]d\textbf{p}$$ with the average taken over momenta, etc.