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The mathematical relationship between the quantum state and the probability distribution is, again, widely accepted among physicists, and has been experimentally confirmed countless times. This section summarizes this relationship, which is stated in terms of the mathematical formulation of quantum mechanics.

Measurable quantities ("observables") as operators

It is a postulate of quantum mechanics that all measurements have an associated operator (called an observable operator, or just an observable), with the following properties:

1. The observable is a Hermitian (self-adjoint) operator mapping a Hilbert space (namely, the state space, which consists of all possible quantum states) into itself.

2. The observable's eigenvalues are real. The possible outcomes of the measurement are precisely the eigenvalues of the given observable.

3. For each eigenvalue there are one or more corresponding eigenvectors (which in this context are called eigenstates), which will make up the state of the system after the measurement.

4. The observable has a set of eigenvectors which span the state space. It follows that each observable generates an orthonormal basis of eigenvectors (called an eigenbasis). Physically, this is the statement that any quantum state can always be represented as a superposition of the eigenstates of an observable.

Important examples of observables are:

• The Hamiltonian operator, representing the total energy of the system; with the special case of the nonrelativistic

Hamiltonian operator: .

• The momentum operator: (in the position basis).

• The position operator: , where (in the momentum basis).

Operators can be noncommuting. Two Hermitian operators commute if (and only if) there is at least one basis of vectors, each of which is an eigenvector of both operators (this is sometimes called a simultaneous eigenbasis).

Noncommuting observables are said to be incompatible and cannot in general be measured simultaneously. In fact, they are related by an uncertainty principle, as a consequence of the Robertson–Schrödinger relation.

Measurement probabilities and wavefunction collapse

There are a few possible ways to mathematically describe the measurement process (both the probability distribution and the collapsed wavefunction). The most convenient description depends on the spectrum (i.e., set of eigenvalues) of the observable.

Discrete, nondegenerate spectrum

Let be an observable, and suppose that it has discrete eigenstates (in bra-ket notation) for and corresponding eigenvalues , no two of which are equal.

Assume the system is prepared in state . Since the eigenstates of an observable form a basis (the eigenbasis), it follows that can be written in terms of the eigenstates as

(where are complex numbers). Then measuring can yield any of the results , with corresponding probabilities given by

Usually is assumed to be normalized, in which case this expression reduces to

If the result of the measurement is , then the system's quantum state after the measurement is

so any repeated measurement of will yield the same result . (This phenomenon is called wavefunction collapse.)

Continuous, nondegenerate spectrum

Let be an observable, and suppose that it has a continuous spectrum of eigenvalues filling the interval (a,b).

Assume further that each eigenvalue x in this range is associated with a unique eigenstate . Assume the system is prepared in state , which can be written in terms of the eigenbasis as

(where is a complex-valued function). Then measuring can yield a result anywhere in the interval (a,b), with probability density function ; i.e., a result between y and z will occur with probability

Again, is often assumed to be normalized, in which case this expression reduces to

If the result of the measurement is x, then the new wave function will be

Alternatively, it is often possible and convenient to analyze a continuous-spectrum measurement by taking it to be the limit of a different measurement with a discrete spectrum. For example, an analysis of scattering involves a continuous spectrum of energies, but by adding a "box" potential (which bounds the volume in which the particle can be found), the spectrum becomes discrete. By considering larger and larger boxes, this approach need not involve any approximation, but rather can be regarded as an equally valid formalism in which this problem can be analyzed.

Degenerate spectra

If there are multiple eigenstates with the same eigenvalue (called degeneracies), the analysis is a bit less simple to state, but not essentially different. In the discrete case, for example, instead of finding a complete eigenbasis, it is a bit more convenient to write the Hilbert space as a direct sum of eigenspaces. The probability of measuring a particular eigenvalue is the squared component of the state vector in the corresponding eigenspace, and the new state after measurement is the projection of the original state vector into the appropriate eigenspace.

Density matrix formulation

Instead of performing quantum-mechanics computations in terms of wavefunctions (kets), it is sometimes necessary to describe a quantum-mechanical system in terms of a density matrix. The analysis in this case is formally slightly different, but the physical content is the same, and indeed this case can be derived from the wavefunction formulation above. The result for the discrete, degenerate case, for example, is as follows:

Let be an observable, and suppose that it has discrete eigenvalues , associated with eigenspaces respectively. Let be the projection operator into the space .

Assume the system is prepared in the state described by the density matrix ρ. Then measuring can yield any of the results , with corresponding probabilities given by

where denotes trace. If the result of the measurement is n, then the new density matrix will be

Alternatively, one can say that the measurement process results in the new density matrix

where the difference is that is the density matrix describing the entire ensemble, whereas is the density matrix describing the sub-ensemble whose measurement result was .

Statistics of measurement

As detailed above, the result of measuring a quantum-mechanical system is described by a probability distribution.

Some properties of this distribution are as follows:

Suppose we take a measurement corresponding to observable , on a state whose quantum state is .

• The mean (average) value of the measurement is (see Expectation value (quantum mechanics)) .

• The variance of the measurement is

• The standard deviation of the measurement is

These are direct consequences of the above formulas for measurement probabilities.

Example

Suppose that we have a particle in a 1-dimensional box, set up initially in the ground state . As can be computed from the time-independent Schrödinger equation, the energy of this state is (where m is the

particle's mass and L is the box length), and the spatial wavefunction is . If the energy is now measured, the result will always certainly be , and this measurement will not affect the wavefunction.

Next suppose that the particle's position is measured. The position x will be measured with probability density

If the measurement result was x=S, then the wavefunction after measurement will be the position eigenstate . If the particle's position is immediately measured again, the same position will be obtained.

The new wavefunction can, like any wavefunction, be written as a superposition of eigenstates of any observable. In particular, using energy eigenstates, , we have

If we now leave this state alone, it will smoothly evolve in time according to the Schrödinger equation. But suppose instead that an energy measurement is immediately taken. Then the possible energy values will be measured with relative probabilities:

and moreover if the measurement result is , then the new state will be the energy eigenstate .

So in this example, due to the process of wavefunction collapse, a particle initially in the ground state can end up in any energy level, after just two subsequent non-commuting measurements are made.