In physics, an operator is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Because of this, they are very useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.
In classical mechanics, the movement of a particle (or system of particles) is completely determined by the Lagrangian or equivalently the Hamiltonian , a function of the generalized coordinates q, generalized velocities and its conjugate momenta:
If either L or H is independent of a generalized coordinate q, meaning the L and H do not change when q is changed, which in turn means the dynamics of the particle are still the same even when q changes, the corresponding momenta conjugate to those coordinates will be conserved (this is part of Noether's theorem, and the invariance of motion with respect to the coordinate q is a symmetry). Operators in classical mechanics are related to these symmetries.
More technically, when H is invariant under the action of a certain group of transformations G:
the elements of G are physical operators, which map physical states among themselves.
|Time translation symmetry|
where is the rotation matrix about an axis defined by the unit vector and angle θ.
If the transformation is infinitesimal, the operator action should be of the form
where is the identity operator, is a parameter with a small value, and will depend on the transformation at hand, and is called a generator of the group. Again, as a simple example, we will derive the generator of the space translations on 1D functions.
As it was stated, . If is infinitesimal, then we may write
This formula may be rewritten as
where is the generator of the translation group, which in this case happens to be the derivative operator. Thus, it is said that the generator of translations is the derivative.
The whole group may be recovered, under normal circumstances, from the generators, via the exponential map. In the case of the translations the idea works like this.
The translation for a finite value of may be obtained by repeated application of the infinitesimal translation:
with the standing for the application times. If is large, each of the factors may be considered to be infinitesimal:
But this limit may be rewritten as an exponential:
To be convinced of the validity of this formal expression, we may expand the exponential in a power series:
The right-hand side may be rewritten as
which is just the Taylor expansion of , which was our original value for .
The mathematical properties of physical operators are a topic of great importance in itself. For further information, see C*-algebra and Gelfand-Naimark theorem.
The mathematical formulation of quantum mechanics (QM) is built upon the concept of an operator.
Physical pure states in quantum mechanics are represented as unit-norm vectors (probabilities are normalized to one) in a special complex Hilbert space. Time evolution in this vector space is given by the application of the evolution operator.
Any observable, i.e., any quantity which can be measured in a physical experiment, should be associated with a self-adjoint linear operator. The operators must yield real eigenvalues, since they are values which may come up as the result of the experiment. Mathematically this means the operators must be Hermitian.The probability of each eigenvalue is related to the projection of the physical state on the subspace related to that eigenvalue. See below for mathematical details about Hermitian operators.
In the wave mechanics formulation of QM, the wavefunction varies with space and time, or equivalently momentum and time (see position and momentum space for details), so observables are differential operators.
In the matrix mechanics formulation, the norm of the physical state should stay fixed, so the evolution operator should be unitary, and the operators can be represented as matrices. Any other symmetry, mapping a physical state into another, should keep this restriction.
The wavefunction must be square-integrable (see Lp spaces), meaning:
and normalizable, so that:
Two cases of eigenstates (and eigenvalues) are:
Let ψ be the wavefunction for a quantum system, and be any linear operator for some observable A (such as position, momentum, energy, angular momentum etc.). If ψ is an eigenfunction of the operator , then
where a is the eigenvalue of the operator, corresponding to the measured value of the observable, i.e. observable A has a measured value a.
If ψ is an eigenfunction of a given operator , then a definite quantity (the eigenvalue a) will be observed if a measurement of the observable A is made on the state ψ. Conversely, if ψ is not an eigenfunction of , then it has no eigenvalue for , and the observable does not have a single definite value in that case. Instead, measurements of the observable A will yield each eigenvalue with a certain probability (related to the decomposition of ψ relative to the orthonormal eigenbasis of ).
In bra–ket notation the above can be written;
that are equal if is an eigenvector, or eigenket of the observable A.
Due to linearity, vectors can be defined in any number of dimensions, as each component of the vector acts on the function separately. One mathematical example is the del operator, which is itself a vector (useful in momentum-related quantum operators, in the table below).
An operator in n-dimensional space can be written:
where ej are basis vectors corresponding to each component operator Aj. Each component will yield a corresponding eigenvalue . Acting this on the wave function ψ:
in which we have used
In bra–ket notation:
If two observables A and B have linear operators and , the commutator is defined by,
The commutator is itself a (composite) operator. Acting the commutator on ψ gives:
If ψ is an eigenfunction with eigenvalues a and b for observables A and B respectively, and if the operators commute:
then the observables A and B can be measured simultaneously with infinite precision i.e. uncertainties , simultaneously. ψ is then said to be the simultaneous eigenfunction of A and B. To illustrate this:
It shows that measurement of A and B does not cause any shift of state i.e. initial and final states are same (no disturbance due to measurement). Suppose we measure A to get value a. We then measure B to get the value b. We measure A again. We still get the same value a. Clearly the state (ψ) of the system is not destroyed and so we are able to measure A and B simultaneously with infinite precision.
If the operators do not commute:
they can't be prepared simultaneously to arbitrary precision, and there is an uncertainty relation between the observables,
even if ψ is an eigenfunction the above relation holds.. Notable pairs are position-and-momentum and energy-and-time uncertainty relations, and the angular momenta (spin, orbital and total) about any two orthogonal axes (such as Lx and Ly, or sy and sz etc.).
The expectation value (equivalently the average or mean value) is the average measurement of an observable, for particle in region R. The expectation value of the operator is calculated from:
This can be generalized to any function F of an operator:
An example of F is the 2-fold action of A on ψ, i.e. squaring an operator or doing it twice:
The definition of a Hermitian operator is:
Following from this, in bra–ket notation:
Important properties of Hermitian operators include:
An operator can be written in matrix form to map one basis vector to another. Since the operators are linear, the matrix is a linear transformation (aka transition matrix) between bases. Each basis element can be connected to another, by the expression:
which is a matrix element:
A further property of a Hermitian operator is that eigenfunctions corresponding to different eigenvalues are orthogonal.In matrix form, operators allow real eigenvalues to be found, corresponding to measurements. Orthogonality allows a suitable basis set of vectors to represent the state of the quantum system. The eigenvalues of the operator are also evaluated in the same way as for the square matrix, by solving the characteristic polynomial:
where I is the n × n identity matrix, as an operator it corresponds to the identity operator. For a discrete basis:
while for a continuous basis:
A non-singular operator has an inverse defined by:
If an operator has no inverse, it is a singular operator. In a finite-dimensional space, an operator is non-singular if and only if its determinant is nonzero:
and hence the determinant is zero for a singular operator.
The operators used in quantum mechanics are collected in the table below (see for example,). The bold-face vectors with circumflexes are not unit vectors, they are 3-vector operators; all three spatial components taken together.
|Operator (common name/s)||Cartesian component||General definition||SI unit||Dimension|
|Momentum||General ||General ||J s m−1 = N s||[M] [L] [T]−1|
|Electromagnetic field ||Electromagnetic field (uses kinetic momentum; A, vector potential) ||J s m−1 = N s||[M] [L] [T]−1|
|Kinetic energy||Translation ||J||[M] [L]2 [T]−2|
|Electromagnetic field ||Electromagnetic field (A, vector potential) ||J||[M] [L]2 [T]−2|
|Rotation (I, moment of inertia) ||Rotation |
[ citation needed ]
|J||[M] [L]2 [T]−2|
|Potential energy||N/A||J||[M] [L]2 [T]−2|
|Total energy||N/A||Time-dependent potential:|
|J||[M] [L]2 [T]−2|
|Hamiltonian||J||[M] [L]2 [T]−2|
|Angular momentum operator||J s = N s m||[M] [L]2 [T]−1|
|Spin angular momentum|
where σ is the vector whose components are the Pauli matrices.
|J s = N s m||[M] [L]2 [T]−1|
|Total angular momentum||J s = N s m||[M] [L]2 [T]−1|
|Transition dipole moment (electric)||C m||[I] [T] [L]|
The procedure for extracting information from a wave function is as follows. Consider the momentum p of a particle as an example. The momentum operator in position basis in one dimension is:
Letting this act on ψ we obtain:
if ψ is an eigenfunction of , then the momentum eigenvalue p is the value of the particle's momentum, found by:
For three dimensions the momentum operator uses the nabla operator to become:
In Cartesian coordinates (using the standard Cartesian basis vectors ex, ey, ez) this can be written;
The process of finding eigenvalues is the same. Since this is a vector and operator equation, if ψ is an eigenfunction, then each component of the momentum operator will have an eigenvalue corresponding to that component of momentum. Acting on ψ obtains:
In quantum mechanics, bra–ket notation, or Dirac notation, is ubiquitous. The notation uses the angle brackets, "" and "", and a vertical bar "", to construct "bras" and "kets".
In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's energy spectrum or its set of energy eigenvalues, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.
In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, x, and momentum, p, can be predicted from initial conditions.
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.
The Klein–Gordon equation is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. It is a quantized version of the relativistic energy–momentum relation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation. Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pions are unstable and also experience the strong interaction the practical utility is limited.
In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities. For example,
In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory, to the greatest extent possible.
A rotational transition is an abrupt change in angular momentum in quantum physics. Like all other properties of a quantum particle, angular momentum is quantized, meaning it can only equal certain discrete values, which correspond to different rotational energy states. When a particle loses angular momentum, it is said to have transitioned to a lower rotational energy state. Likewise, when a particle gains angular momentum, a positive rotational transition is said to have occurred.
In quantum mechanics, a two-state system is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.
In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimension, the definition is:
The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators x and p to the expectation value of the force on a massive particle moving in a scalar potential ,
In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum problems involving rotational symmetry. Such an operator is applied to a mathematical representation of the physical state of a system and yields an angular momentum value if the state has a definite value for it. In both classical and quantum mechanical systems, angular momentum is one of the three fundamental properties of motion.
In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle.
In quantum mechanics, Landau quantization refers to the quantization of the cyclotron orbits of charged particles in a uniform magnetic field. As a result, the charged particles can only occupy orbits with discrete, equidistant energy values, called Landau levels. These levels are degenerate, with the number of electrons per level directly proportional to the strength of the applied magnetic field. It is named after the Soviet physicist Lev Landau.
Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.
The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.
This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.
In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator.
In quantum mechanics, a translation operator is defined as an operator which shifts particles and fields by a certain amount in a certain direction.
Different subfields of physics have different programs for determining the state of a physical system.