operator

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.

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.

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

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

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