Addition of Angular Momenta Spin Angular Momentum Spin Angular Momentum

Spinors

In the basis , a general spin state can be written as

with complex coefficients . Normalisation requires that

The state defined above can also be represented by a two-component column vector called a spinor whose components are given by the projections onto the basis :

The basis spinors are

and the completeness relation in this matrix representation is

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truein Problem :19 An electron is in the spin state

• What is the probability that a measurement of the z-component of the spin will result in ? In ?
• What is the expectation value of ?
• What is the RMS uncertaintity in ?

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Spin is an additional degree of freedom independent of the spatial degrees of freedom. Spin and position (or momentum) can assume precise values simultaneously and independently of one another i.e.

The total quantum state of a particle is constructed from the direct product of the position and spin eigenstates. The states and forms a basis for the Hilbert space (that takes into account the spatial and spin degrees of freedom). A general state in this Hilbert space is then given by

The projection onto position and spin eigenstates are

since and

The quantities express the probability of finding the particle at position with z-component of spin of . The normalisation condition is