Anti-symmetric Operator - Spin

Spin

First, we will review spin for non-relativistic quantum mechanics. Spin, an intrinsic property similar to angular momentum, is defined by a spin operator S that plays a role on a system similar to the operator L for orbital angular momentum. The operators and whose eigenvalues are and respectively. These formalisms also obey the usual commutation relations for angular momentum, and . The raising and lowering operators, and, are defined as and respectively. These ladder operators act on the state in the following and respectively.

The operators S_x and S_y can be determined using the ladder method. In the case of the spin 1/2 case (fermion), the operator acting on a state produces and . Likewise, the operator acting on a state produces and . The matrix representations of these operators are constructed as follows:

= \begin{bmatrix} \langle+|S_+|+\rangle & \langle+|S_+|-\rangle \\ \langle-|S_+|+\rangle & \langle-|S_+|-\rangle \end{bmatrix} = \hbar \cdot \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}
= \begin{bmatrix} \langle+|S_-|+\rangle & \langle+|S_-|-\rangle \\ \langle-|S_-|+\rangle & \langle-|S_-|-\rangle \end{bmatrix} = \hbar \cdot \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}

Therefore and can be represented by the matrix representations:

= \frac{ \hbar}{2} \cdot \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
= \frac{ \hbar}{2} \cdot \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}

Recalling the generalized uncertainty relation for two operators A and B, \Delta_{\psi} A \, \Delta_{\psi} B \ge \frac{1}{2} \left|\left\langle\left\right\rangle_\psi\right| , we can immediately see that the uncertainty relation of the operators and are as follows:

\Delta_{\psi} S_x \, \Delta_{\psi} S_y \ge \frac{1}{2} \left|\left\langle\left\right\rangle_\psi\right| = \frac{1}{2} (i \hbar S_z) = \frac{ \hbar}{2} S_z

Therefore, like orbital angular momentum, we can only specify one coordinate at a time. We specify the operators and .

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