Wave-function of a spin one-half particle

The state of a spin one-half particle is represented as a vector in ket space. Let us suppose that this space is spanned by the basis kets $\vert x', y', z', \pm\rangle$. Here, $\vert x', y', z', \pm\rangle$ denotes a simultaneous eigenstate of the position operators $x$, $y$, $z$, and the spin operator $S_z$, corresponding to the eigenvalues $x'$, $y'$, $z'$, and $\pm \hbar/2$, respectively. The basis kets are assumed to satisfy the completeness relation

$$\int\!\int\!\int \left( \vert x',y',z',+\rangle\langle x',... ...',-\rangle\langle x', y', z',-\vert \right) dx'dy'dz' = 1.$$ (435)

It is helpful to think of the ket $\vert x', y', z', +\rangle$ as the product of two kets--a position space ket $\vert x', y', z'\rangle$, and a spin space ket $\vert+\rangle$. We assume that such a product obeys the commutative and distributive axioms of multiplication:

$$\vert x', y', z'\rangle \vert+\rangle = \vert+\rangle \vert x', y', z'\rangle,$$ (436)

$$\left(c' \vert x', y', z'\rangle + c''\vert x'', y'', z''\rangle\right) \vert+\rangle = c' \vert x', y', z'\rangle \vert+\rangle$$

$$+ c'' \vert x'', y'', z''\rangle \vert+\rangle$$ (437)

$$\vert x', y', z'\rangle\left(c_+ \vert+\rangle + c_- \vert-\rangle\right) = c_+ \vert x', y', z'\rangle\vert+\rangle$$

$$+ c_-\vert x', y', z'\rangle\vert-\rangle,$$ (438)

where the $c$'s are numbers. We can give meaning to any position space operator (such as $L_z$) acting on the product $\vert x', y', z'\rangle \vert+\rangle$ by assuming that it operates only on the $\vert x', y', z'\rangle$ factor, and commutes with the $\vert+\rangle$ factor. Similarly, we can give a meaning to any spin operator (such as $S_z$) acting on $\vert x', y', z'\rangle \vert+\rangle$ by assuming that it operates only on $\vert+\rangle$, and commutes with $\vert x', y', z'\rangle$. This implies that every position space operator commutes with every spin operator. In this manner, we can give meaning to the equation

$$\vert x', y', z', \pm\rangle = \vert x', y', z'\rangle\vert \pm\rangle = \vert \pm\rangle \vert x', y', z'\rangle.$$ (439)

The multiplication in the above equation is of quite a different type to any which we have encountered previously. The ket vectors $\vert x', y', z'\rangle$ and $\vert\pm \rangle$ are in two quite separate vector spaces, and their product $\vert x',y', z'\rangle\vert\pm\rangle$ is in a third vector space. In mathematics, the latter space is termed the product space of the former spaces, which are termed factor spaces. The number of dimensions of a product space is equal to the product of the number of dimensions of each of the factor spaces. A general ket of the product space is not of the form (439), but is instead a sum or integral of kets of this form.

A general state $A$ of a spin one-half particle is represented as a ket $\vert\vert A\rangle\rangle$ in the product of the spin and position spaces. This state can be completely specified by two wavefunctions:

$$\psi_+(x', y', z') = \langle x', y', z' \vert\langle +\vert\vert A\rangle\rangle,$$ (440)

$$\psi_-(x', y', z') = \langle x', y', z' \vert\langle -\vert\vert A\rangle\rangle.$$ (441)

The probability of observing the particle in the region $x'$ to $x'+dx'$, $y'$ to $y'+dy'$, and $z'$ to $z'+dz'$, with $s_z = +1/2$ is $\vert\psi_+ (x', y', z')\vert^2 dx' dy' dz'$. Likewise, the probability of observing the particle in the region $x'$ to $x'+dx'$, $y'$ to $y'+dy'$, and $z'$ to $z'+dz'$, with $s_z = -1/2$ is $\vert\psi_- (x', y', z')\vert^2 dx' dy' dz'$. The normalization condition for the wavefunctions is

$$\int\!\int\!\int \left( \vert\psi_+\vert^2 + \vert\psi_-\vert^2 \right) dx'dy'dz' = 1.$$ (442)