operator

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An operator \mathcal{A}\, is a, \mathcal{A} : \mathcal{H} \to \mathcal{H} \,, from the set of states \mathcal{H}\, 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:

A \left(  a_1 \left|\Psi_1\right\rangle + a_2 \left|\Psi_2\right\rangle \right) = a_1 A\left|\Psi_1\right\rangle + a_2 A \left|\Psi_2\right\rangle\,, \forall a_1, a_2 \in \mathbb{C}\,, and \left|\Psi_1\right\rangle, \left|\Psi_2\right\rangle \in \mathcal{H}\,.

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

A \left(  a_1 \left|\Psi_1\right\rangle + a_2 \left|\Psi_2\right\rangle \right) = a_1^* A\left|\Psi_1\right\rangle + a_2^* A \left|\Psi_2\right\rangle\,, \forall a_1, a_2 \in \mathbb{C}\,, and \left|\Psi_1\right\rangle, \left|\Psi_2\right\rangle \in \mathcal{H}\,.

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

Contents

Mathematical properties

Given a basis \left\{\left|\phi_i\right\rangle\right\}\,, the action of an operator A\, on any state \left|\Psi\right\rangle\, is determined by the action of A\, on the basis:

A \left|\Psi\right\rangle = A \sum_i c_i \left|\phi_i\right\rangle = \sum_i c_i \left(A \left|\phi_i\right\rangle\right)\,.

Let

\left|\Psi'\right\rangle = \sum_i c_i' \left|\phi_i\right\rangle = A \left|\Psi\right\rangle\,.

Then the resulting state is determined through

c_i' = \left\langle \phi_i | \Psi' \right\rangle = \left\langle \phi_i | A |\Psi \right\rangle =  \sum_j c_j \left\langle\phi_i | A | \phi_j\right\rangle,\,.

The values

A_{ij} \equiv \left\langle \phi_i | A | \phi_j \right\rangle\,,

are termed the matrix elements of A\,. The naming is suggestive, and just as we can associate a column or row vector to \left|\Psi\right\rangle\, or \left\langle\Psi\right|\,, we can associate a matrix to each operator, where the elements of the matrix are simply A_{ij}\,. The action of an operator on a state is then equivalent to the action of a matrix on a vector. Thus

\left|\Psi'\right\rangle = A \left|\Psi\right\rangle \sim c_i' = \sum_j A_{ij} c_j\,.

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

\left|\Psi'\right\rangle = A\left|\Psi\right\rangle\,,

and

\left|\Psi''\right\rangle = B\left|\Psi'\right\rangle\,,

then

\left|\Psi''\right\rangle = BA\left|\Psi\right\rangle\,,

where

(BA)_{ij} = \sum_k B_{ik} A_{kj}\,.

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

AB \neq BA\,,

and we say that the operators A\, and B\, do not commute unless the commutator

[A,B] = AB - BA\,

is zero.

Adjoint

The Hermitian adjoint A^{\dagger}\, of an operator A\, is defined to be the operator A^{\dagger}\, that satisfies

\left\langle \Phi | A^{\dagger} | \Psi\right\rangle = \left\langle \Psi | A | \Phi\right\rangle^*\, \forall \left|\Psi\right\rangle, \left|\Phi\right\rangle \in \mathcal{H}\,.

The matrix elements of A^{\dagger}\, are therefore given by A^{\dagger}_{ij} = A^*_{ji}\,. In an expression like \left\langle \Psi | A | \Phi\right\rangle\, we can equally well act A\, on \left\langle \Psi \right|\, on the left, since

\left(\left\langle \Psi | A\right) | \Phi\right\rangle = \left\langle \Phi | A^{\dagger} | \Psi\right\rangle^*\,.

Thus

\left\langle \Psi \right| A = \left(  A^{\dagger} \left| \Psi\right\rangle  \right)^{\dagger}\,.

Unitary operator

A unitary operator satisfies

A^{\dagger} = A^{-1}\,.

Hermitian operator

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

A^{\dagger} = A\,,

or

A_{ij} = A^*_{ji}\,.

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 U\, such that

U A U^\dagger = D\,,

where D\, is diagonal. Since D^{\dagger} = (U AU^{\dagger})^{\dagger} = U A^{\dagger} U^\dagger = U A U^\dagger = D\,, 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 |\phi_1\rangle\, and |\phi_2\rangle\, are eigenvectors of A\, corresponding to eigenvalues \lambda_1\, and \lambda_2\,, then

\langle \phi_1 | A | \phi_2\rangle - \langle \phi_1 | A | \phi_2\rangle\, =\lambda_1^* \langle \phi_1 |\phi_2\rangle  - \lambda_2 \langle \phi_1 | \phi_2\rangle\,,
=(\lambda_1^* - \lambda_2) \langle \phi_1 |\phi_2\rangle\,,
=(\lambda_1 - \lambda_2) \langle \phi_1 |\phi_2\rangle\,,
=0\,,

thus either \lambda_1 = \lambda_2\, or \langle \phi_1 |\phi_2\rangle=0\,.

Simultaneous diagonalizability

If two Hermitian operators A\, and B\, commute, that is

[A,B] = 0\,,

then there exists a unitary operator U\, such that

U A U^\dagger = D_1\,,

and

U B U^\dagger = D_2\,,

where both D_1\, and D_2\, 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 \mathbf{1}\, which maps every state onto itself:

\mathbf{1} \left|\Psi\right\rangle = \left|\Psi\right\rangle\,.

Projection operator

A projection operator is an operator that projects onto some subspace of \mathcal{H}\,. For example, the operator

P_{\left|\Psi\right\rangle} \equiv \left|\Psi \right\rangle\left\langle \Psi \right|\,,

where P_{\left|\Psi\right\rangle} : \left|\Phi\right\rangle \to \left|\Psi \right\rangle\left\langle \Psi |\Phi\right\rangle\,, and \left\Vert \left|\Psi\right\rangle\right\Vert = 1\,, yields zero when acting on states orthogonal to \left|\Psi\right\rangle\, and yields the original state when acting on states that are proportional to \left|\Psi\right\rangle\,. For example, P_{\left|\Psi\right\rangle} \left|\Psi\right\rangle = \left|\Psi\right\rangle\,. A property of all projection operators is idempotence:

P_{\left|\Psi\right\rangle} ^2 \left| \Phi \right\rangle = P_{\left|\Psi\right\rangle} \left| \Phi \right\rangle\, \forall \left|\Phi\right\rangle\,.

The identity operator is a type of projection operator that projects onto the entire vector space \mathcal{H}\,. Given a basis \left\{\left|\phi_i\right\rangle\right\}\,, the operator

\mathbf{1} \equiv \sum_i \left| \phi_i \right\rangle\left\langle \phi_i \right|\,,

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 \left|\psi \right\rangle = c_\uparrow \left|\uparrow\right\rangle + c_\downarrow \left|\downarrow\right\rangle\, and subsequently measure the observable S_z\,. Such an experiment will yield an average value for S_z\,, denoted \left\langle S_z \right\rangle\,, called the expectation value of S_z\,, equal to

\left\langle S_z \right\rangle = P_\uparrow \left(\frac{\hbar}{2}\right) + P_\downarrow \left(-\frac{\hbar}{2}\right) = |c_\uparrow|^2 \left(\frac{\hbar}{2}\right) + |c_\downarrow|^2 \left(-\frac{\hbar}{2}\right)\,.

Using projection operators, note that

\left\langle\psi \right| \left(  \frac{\hbar}{2} \left|\uparrow\right\rangle\left\langle \uparrow\right| - \frac{\hbar}{2} \left|\downarrow\right\rangle\left\langle \downarrow\right| \right) \left|\psi\right\rangle\, =\left\langle\psi \right| \left(  \frac{\hbar}{2} \left|\uparrow\right\rangle c_{\uparrow} - \frac{\hbar}{2} \left|\downarrow\right\rangle c_{\downarrow} \right)\,,
= \frac{\hbar}{2}  c^*_{\uparrow}  c_{\uparrow} - \frac{\hbar}{2}  c^*_{\downarrow}  c_{\downarrow} \,,
=\left\langle S_z \right\rangle\,.

Thus we see that the operator S_z \equiv \frac{\hbar}{2} \left|\uparrow\right\rangle\left\langle \uparrow\right| - \frac{\hbar}{2} \left|\downarrow\right\rangle\left\langle \downarrow\right|\, has the property that

\left\langle \Psi | S_z | \Psi \right\rangle = \left\langle S_z \right\rangle\,,

for any state \left|\Psi\right\rangle\,. In general,

Given a state \left|\Psi\right\rangle\, and an operator A\, corresponding to some observable, the expectation value of that observable is given by
\left\langle A \right\rangle = \left\langle \Psi | A | \Psi\right\rangle\,.

Again, in our example we see that

S_z \left|\uparrow\right\rangle = \frac{\hbar}{2} \left|\uparrow\right\rangle\,,

so we say that \left|\uparrow\right\rangle\, is an eigenstate of the operator S_z\, with eigenvalue \tfrac{\hbar}{2}\,, while \left|\downarrow\right\rangle\, is an eigenstate of S_z\, with eigenvalue -\tfrac{\hbar}{2}\,. 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 A\, correspond to the eigenvalues and eigenvectors of the matrix A_{mn}\,. The eigenvalues are independent of our choice of basis, while the eigenstates are not. The set of eigenvalues of A\, is known as the spectrum of A\,. 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 \mathbf{\hat{n}}\, corresponds to an operator S_{\hat{n}}\,. A state prepared in such a way that a measurement of S_{\hat{n}}\, would yield \tfrac{\hbar}{2}\, with probability 1\, is therefore an eigenstate of S_{\hat{n}}\, with eigenvalue \tfrac{\hbar}{2}\,. Furthermore, measuring S_{\hat{n}}\, does not disturb the state if the state is an eigenstate of S_{\hat{n}}\,.

Simultaneous eigenstates

Suppose the system is in an eigenstate \left|\Psi\right\rangle\, of some observable operator A\,, such that

A \left|\Psi\right\rangle = \lambda \left|\Psi\right\rangle\,,

and we wish to make a measurement of another observable represented by an operator B\,. For the system not to be disturbed, \left|\Psi\right\rangle\, must therefore also be an eigenstate of B\,. This demands, in particular, that \left\langle \Psi | A B | \Psi \right\rangle = \left\langle \Psi | B A | \Psi \right\rangle\,, i.e., that the matrices commute within the 1-dimensional subspace spanned by \left| \Psi \right\rangle\,. More generally, we say that two observables corresponding to operators A\, and B\, can simultaneously be known if [A, B] = 0\,, and are said to be compatible observables. Correspondingly, if [A, B] \neq 0\,, 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 S_z\, and S_y\, 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 [S_z, S_x] = i\hbar S_y\,.

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

  1. It may be possible that a subset \left\{\, \left|\phi_i\right\rangle \,\right\}\, of states are eigenstates of both A\, and B\,. One could, by a suitable choice of basis, cast A\, and B\, in block diagonal form. The block matrices corresponding to the subspace spanned by \left\{\, \left|\phi_i\right\rangle \,\right\}\, must then necessarily commute. Equivalently, \left\langle \phi_i | [A,B] | \phi_j\right\rangle = 0\,.
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