- #1

Haorong Wu

- 415

- 90

- Homework Statement
- Given a state of a electron, ##\psi = \begin{pmatrix} \psi_{+} \\ \psi _{-} \end{pmatrix} =R \left ( r \right ) \begin{pmatrix} \sqrt {\frac 3 5} Y_0^0 + \sqrt {\frac 1 {10}} Y_1^1 +\sqrt {\frac 1 {10}} Y_1^-1 \\ \sqrt {\frac 1 5} Y_1^0 \end{pmatrix}##, what is the expectation of ##\left < S_x \right >## ?

- Relevant Equations
- None

I have two different solutions, and I do not know which one is correct and why the other one is wrong.

Solution 1.

In the ##L_z## space, the spin state is ##\begin{pmatrix} \sqrt { \frac 4 5} \\ \sqrt { \frac 1 5} \end{pmatrix}##, and ##S_x=\frac \hbar 2 \begin{pmatrix} 0& 1 \\ 1& 0 \end{pmatrix}##, so ##\left < S_x \right >=\begin{pmatrix} \sqrt { \frac 4 5} & \sqrt { \frac 1 5} \end{pmatrix} \frac \hbar 2 \begin{pmatrix} 0& 1 \\ 1& 0 \end{pmatrix} \begin{pmatrix} \sqrt { \frac 4 5} \\ \sqrt { \frac 1 5} \end{pmatrix}=\frac 2 5 \hbar##.

Solution 2.

##\left < S_x \right >=\int \begin{pmatrix} \psi_{+}^{*} & \psi_{-}^{*} \end{pmatrix} \frac \hbar 2 \begin{pmatrix} 0& 1 \\ 1& 0 \end{pmatrix} \begin{pmatrix} \psi_{+} \\ \psi_{-} \end{pmatrix} dr =0##.

I guess the problem is that I have not make clear whether the spatial functions should be involved in the calculations. Should the orbital momentums affect the calculation of spins?

Thanks!

Solution 1.

In the ##L_z## space, the spin state is ##\begin{pmatrix} \sqrt { \frac 4 5} \\ \sqrt { \frac 1 5} \end{pmatrix}##, and ##S_x=\frac \hbar 2 \begin{pmatrix} 0& 1 \\ 1& 0 \end{pmatrix}##, so ##\left < S_x \right >=\begin{pmatrix} \sqrt { \frac 4 5} & \sqrt { \frac 1 5} \end{pmatrix} \frac \hbar 2 \begin{pmatrix} 0& 1 \\ 1& 0 \end{pmatrix} \begin{pmatrix} \sqrt { \frac 4 5} \\ \sqrt { \frac 1 5} \end{pmatrix}=\frac 2 5 \hbar##.

Solution 2.

##\left < S_x \right >=\int \begin{pmatrix} \psi_{+}^{*} & \psi_{-}^{*} \end{pmatrix} \frac \hbar 2 \begin{pmatrix} 0& 1 \\ 1& 0 \end{pmatrix} \begin{pmatrix} \psi_{+} \\ \psi_{-} \end{pmatrix} dr =0##.

I guess the problem is that I have not make clear whether the spatial functions should be involved in the calculations. Should the orbital momentums affect the calculation of spins?

Thanks!