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# 3.2 Representing a general spin state

A ket can also be considered as a vector. The spin-up state is often represented by the symbol and the spin-down state by the symbol . The spin-up state obeys

Equation label:(7)

and the spin-down state obeys

Equation label:(8)

The spin vectors are normalised and orthogonal to one another, as represented by the following relations:

where these equations use additional notation. is known as a bra and the combination of the bra and ket together is an inner product, .

A general spin state can be written as a linear combination of and , thus

Equation label:(9)

where and are complex numbers referred to as probability amplitudes.

In other words, and provide an orthonormal basis for spin space. The vectors and are called basis vectors. Because any spin state can be written as a linear combination of just two basis vectors, the spin space of a spin-½ particle is two-dimensional.

For an atom in any spin state, , as given in Equation 9 the probability of the outcome of a measurement indicating spin-up is and for spin-down is . Since these are the only possible outcomes, the corresponding probabilities must sum to one, therefore

Matrices can be used as an alternative representation of spin states to simplify calculations. and are represented by the following column vectors:

These matrices have two elements because spin space is two-dimensional. Spin states that do not have definite values of are expressed as linear combinations of and .

Any vector in spin space may be written as a linear combination of and . This means that , as defined in Equation 9 becomes:

In this way any spin state of a spin-½ particle can be represented as a two-element matrix, which is called a spinor.

The inner product in spin space of two vectors in matrix form is written

which is consistent with the matrix multiplication:

and thus the inner product of two spin vectors can be written in the matrix form:

We can therefore identify the separate bra and ket vectors as follows:

A ket spin vector is represented by a column spinor, and a bra spin vector is represented by a row spinor. To convert a column spinor into the corresponding row spinor, the rule is to turn the column into a row and take the complex conjugate of all elements. So

When using a matrix representation, the spin operators are also represented as matrices, for example,

Rewriting Equation 7 using matrices gives,

where you can see that carrying out the matrix multiplication gives the expected result.

The operators for a spin-½ particle are each represented by matrices. Along the three axes these are:

It is common to define the so called Pauli operators, , as such that we have the following Pauli operator matrices:

These will be useful in the context of quantum computing where they can be used to represent the action of quantum gates.

## Exercise 8

Show that the spin vectors

are normalized and orthogonal, i.e. and .

### Answer

Using matrix multiplication

The first two equations show that the vectors and are normalised; the third that they are orthogonal to each other. Hence, they are orthonormal.

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