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# Wavefunction of 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 . 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

 (5.15)

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:

 (5.16) (5.17) (5.18)

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 a meaning to the equation

 (5.19)

The multiplication in the previous 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 dimensionality a product space is equal to the product of the dimensionalities of each of the factor spaces. Actually, a general ket in the product space is not of the form (5.19), 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:

 (5.20) (5.21)

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

 (5.22)

Next: Rotation Operators in Spin Up: Spin Angular Momentum Previous: Properties of Spin Angular
Richard Fitzpatrick 2016-01-22