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 . Here, denotes a simultaneous eigenstate of the position operators , , , and the spin operator , corresponding to the eigenvalues , , , and , respectively. The basis kets are assumed to satisfy the completeness relation
It is helpful to think of the ket as the product of two kets--a position space ket , and a spin space ket . We assume that such a product obeys the commutative and distributive axioms of multiplication:
where the 's are numbers. We can give meaning to any position space operator (such as ) acting on the product by assuming that it operates only on the factor, and commutes with the factor. Similarly, we can give a meaning to any spin operator (such as ) acting on by assuming that it operates only on , and commutes with . This implies that every position space operator commutes with every spin operator. In this manner, we can give meaning to the equation
The multiplication in the above equation is of a quite different type to any that we have encountered previously. The ket vectors and lie in two completely separate vector spaces, and their product lies 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 (435), but is instead a sum or integral of kets of this form.
A general state of a spin one-half particle is represented as a ket in the product of the spin and position spaces. This state can be completely specified by two wavefunctions:
The probability of observing the particle in the region to , to , and to , with is . Likewise, the probability of observing the particle in the region to , to , and to , with is . The normalization condition for the wavefunctions is
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)