# operator

### From Physics wiki

An **operator** is a, , from the set of states of a system onto itself.
A **linear operator** is the most common type of operator encountered in quantum mechanics, and has the fundamental property that:

, , and .

An **antilinear operator** is less frequently encountered and has the following action:

, , and .

Unless otherwise noted, an operator is always assumed to be linear.

## Contents |

## Mathematical properties

Given a basis , the action of an operator on any state is determined by the action of on the basis:

- .

Let

- .

Then the resulting state is determined through

- .

The values

,

are termed the **matrix elements** of . The naming is suggestive, and just as we can associate a column or row vector to or , we can associate a matrix to each **operator**, where the elements of the matrix are simply . The action of an operator on a state is then equivalent to the action of a matrix on a vector. Thus

- .

Since the action of an operator on a state yields another state, we can compose two operations into a single operator: If

- ,

and

- ,

then

- ,

where

- .

Note, however, that matrix multiplication is not commutative. In general,

- ,

and we say that the operators and **do not commute** unless the commutator

is zero.

### Adjoint

The Hermitian adjoint of an operator is defined to be the operator that satisfies

- .

The matrix elements of are therefore given by . In an expression like we can equally well act on on the left, since

- .

Thus

.

### Unitary operator

A **unitary operator** satisfies

- .

### Hermitian operator

A **Hermitian operator**, also called a self-adjoint operator**, is one that is its own adjoint:**

- ,

or

- .

### Diagonalizability

According to the spectral theorem of linear algebra, any Hermitian matrix can be diagonalized by some unitary matrix. In other words, there exists a unitary matrix such that

- ,

where is diagonal. Since , the resulting diagonal matrix is real, and so the eigenvalues of a Hermitian matrix are real. Furthermore, the eigenvectors corresponding to distinct eigenvalues are orthogonal. For if and are eigenvectors of corresponding to eigenvalues and , then

, | |

, | |

, | |

, |

thus either or .

### Simultaneous diagonalizability

If two Hermitian operators and commute, that is

- ,

then there exists a unitary operator such that

- ,

and

- ,

where both and are diagonal. In other words, the two matrices are simultaneously diagonalizable. The converse is trivially true.

## Examples

### Identity operator

The simplest type of operator is the **identity operator** which maps every state onto itself:

.

### Projection operator

A **projection operator** is an operator that projects onto some subspace of . For example, the operator

,

where , and , yields zero when acting on states orthogonal to and yields the original state when acting on states that are proportional to . For example, . A property of all projection operators is idempotence:

.

The **identity operator** is a type of projection operator that projects onto the entire vector space . Given a basis , the operator

,

is a resolution of the identity matrix by the completeness relation.

## Observables and expectation values

Returning to the example of the Stern-Gerlach experiment, suppose we repeatedly prepare a state and subsequently measure the observable . Such an experiment will yield an average value for , denoted , called the expectation value of , equal to

- .

Using projection operators, note that

, | |

, | |

. |

Thus we see that the *operator*
has the property that

- ,

for any state . In general,

Given a state and an operator corresponding to some observable, the **expectation value**of that observable is given by- .

Again, in our example we see that

- ,

so we say that is an **eigenstate** of the *operator* with **eigenvalue** , while is an eigenstate of with eigenvalue . In general, we will find that

To each observable corresponds an operator with eigenvalues corresponding to the possible values a measurement of that observable may yield.

For historical reasons, the allowable eigenvalues that a measurement may yield are called **quantum numbers**.

In a given basis, the eigenvalues and eigenstates of an operator correspond to the eigenvalues and eigenvectors of the matrix . The eigenvalues are independent of our choice of basis, while the eigenstates are not. The set of eigenvalues of is known as the spectrum of . Since measurements invariably involve real quantities, operators corresponding to observables must be Hermitian.

In the Stern-Gerlach experiment, the value of the intrinsic angular momentum along some axis corresponds to an operator . A state prepared in such a way that a measurement of would yield with probability is therefore an eigenstate of with eigenvalue . Furthermore, measuring does not disturb the state if the state is an eigenstate of .

### Simultaneous eigenstates

Suppose the system is in an eigenstate of some observable operator , such that

- ,

and we wish to make a measurement of another observable represented by an operator . For the system not to be disturbed, must therefore also be an eigenstate of . This demands, in particular, that , i.e., that the matrices commute within the 1-dimensional subspace spanned by . More generally, we say that two observables corresponding to operators and can **simultaneously be known** if , and are said to be **compatible observables**. Correspondingly, if , we say that the two corresponding observables **cannot simultaneously be known** (in general) ^{[1]}, and are likewise said to be **incompatible observables**. The uncertainty that is intrinsic in measuring incompatible observables is quantified by the Heisenberg uncertainty principle.

Returning to the Stern-Gerlach experiment, it is reasonable to conclude that the operators and do not commute, since their corresponding observables cannot simultaneously be known. This is in fact the case, and when discussing angular momentum and spin, we will find that .

It is sometimes the case that we can choose a maximal set of compatible observables to simultaneously diagonalize. If the eigenvalues corresponding to the observables completely (and therefore uniquely) specify the state of a system, then the smallest such set is called a **complete set of commuting observables**.

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## References

- ↑ It may be possible that a subset of states are eigenstates of both and . One could, by a suitable choice of basis, cast and in block diagonal form. The block matrices corresponding to the subspace spanned by must then necessarily commute. Equivalently, .