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.

1 Mathematical properties
2 Examples
- 2.1 Identity operator
- 2.2 Projection operator
3 Observables and expectation values
- 3.1 Simultaneous eigenstates
4 References

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

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:

$A^{\dagger} = A\,$ ,

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

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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on to density matrix

References

↑ 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, $\left\langle \phi_i | [A,B] | \phi_j\right\rangle = 0\,$ .

Retrieved from "http://www.physics.thetangentbundle.net/wiki/Quantum_mechanics/operator"

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

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