Spin State of an Electron

The goal of this article is to present some information that is useful for computations involving the state of spin of an electron (or any spin-1/2 particle). The spin state can be parameterized by the projective space \(P\mathcal{H}\) of the 2D complex Hilbert space \(\mathcal{H}\). We'll denote vectors by their coordinates in a basis \(\{\uparrow ,\downarrow\}\), i.e. \( (1,0) \) denotes \(\uparrow\).

Experiments show that, if the spin of an electron is measured about any axis, it will be either \(\hbar/2\) or \(-\hbar/2\). We'll call the specific direction in which \((1,0)\) is guaranteed to give (\(+\hbar/2\)) the \(Z\) direction. Since the operator corresponding to this measurement must map \((1,0)\) to \((\hbar/2,0)\) and \((0,1)\) to \((0,-\hbar/2)\) (eigenvector with eigenvalue \(\hbar/2\)) it must be represented by the matrix below (we'll also use \(Z\) to denote the operator and matrix associated with measuring the spin about the \(Z\) direction).

$$Z = \frac{\hbar}{2}\left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right)$$

From experiments we know that once the electron is in an eigenstate of any direction, the expected value of the spin about any orthogonal direction is 0 (50% \(+\hbar/2\) and 50% \(-\hbar/2\)). We can use this fact to find the operator corresponding to measurement about a direction \(X\) orthogonal to \(Z\). Specifically, we look for Hermetian operators in which \(\uparrow^*X\uparrow=0\) and \(\downarrow^*X\downarrow=0\). With a little arithmetic those identities imply the diagonal entries in the associated matrix must be 0. One such operator is represented by the matrix below.

$$X = \frac{\hbar}{2}\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$$ The eigenstates of \(X\) with eigenvalues \(\hbar/2\) and \(-\hbar/2\) will be denoted as \(\rightarrow\) and \(\leftarrow\) respectively. Using the matrix \(X\) above it's easy to check that the coordinates of these states are \((1,1)\) and \((-1,1)\) respectively.

There is now a unique direction \(Y\) left which is orthogonal to both \(X\) and \(Z\) in ordinary space. Its matrix must still have 0 on the diagonal, but it must also satisfy \(\rightarrow^*X\rightarrow=0\) and \(\leftarrow^*X\leftarrow=0\). With a little arithmetic these equations require that the two nonzero entries sum to 0. Combining the fact that the two entries must be conjugates of one another (Hermetian operator) we have the matrix below. $$Y = \frac{\hbar}{2}\left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right)$$ The eigenstates of \(Y\) with eigenvalues \(+\hbar/2\) and \(-\hbar/2\) will be denoted as \(\circlearrowright\) and \(\circlearrowleft\) respectively. Using the matrix \(Y\) above it's easy to check that the coordinates of these states are \((1,i)\) and \((i,1)\) respectively.