Position and momentum space

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In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension.

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Position space (also real space or coordinate space) is the set of all position vectors r in space, and has dimensions of length. A position vector defines a point in space. If the position vector of a point particle varies with time it will trace out a path, the trajectory of a particle. Momentum space is the set of all momentum vectors p a physical system can have. The momentum vector of a particle corresponds to its motion, with units of [mass][length][time]−1.

Mathematically, the duality between position and momentum is an example of Pontryagin duality . In particular, if a function is given in position space, f(r), then its Fourier transform obtains the function in momentum space, φ(p). Conversely, the inverse Fourier transform of a momentum space function is a position space function.

These quantities and ideas transcend all of classical and quantum physics, and a physical system can be described using either the positions of the constituent particles, or their momenta, both formulations equivalently provide the same information about the system in consideration. Another quantity is useful to define in the context of waves. The wave vector k (or simply "k-vector") has dimensions of reciprocal length, making it an analogue of angular frequency ω which has dimensions of reciprocal time. The set of all wave vectors is k-space. Usually r is more intuitive and simpler than k, though the converse can also be true, such as in solid-state physics.

Quantum mechanics provides two fundamental examples of the duality between position and momentum, the Heisenberg uncertainty principle ΔxΔpħ/2 stating that position and momentum cannot be simultaneously known to arbitrary precision, and the de Broglie relation p = ħk which states the momentum and wavevector of a free particle are proportional to each other. [1] In this context, when it is unambiguous, the terms "momentum" and "wavevector" are used interchangeably. However, the de Broglie relation is not true in a crystal.

Position and momentum spaces in classical mechanics

Lagrangian mechanics

Most often in Lagrangian mechanics, the Lagrangian L(q, dq/dt, t) is in configuration space, where q = (q1, q2,..., qn) is an n-tuple of the generalized coordinates. The Euler–Lagrange equations of motion are

(One overdot indicates one time derivative). Introducing the definition of canonical momentum for each generalized coordinate

the Euler–Lagrange equations take the form

The Lagrangian can be expressed in momentum space also, [2] L′(p, dp/dt, t), where p = (p1, p2,..., pn) is an n-tuple of the generalized momenta. A Legendre transformation is performed to change the variables in the total differential of the generalized coordinate space Lagrangian;

where the definition of generalized momentum and Euler–Lagrange equations have replaced the partial derivatives of L. The product rule for differentials [nb 1] allows the exchange of differentials in the generalized coordinates and velocities for the differentials in generalized momenta and their time derivatives,

which after substitution simplifies and rearranges to

Now, the total differential of the momentum space Lagrangian L′ is

so by comparison of differentials of the Lagrangians, the momenta, and their time derivatives, the momentum space Lagrangian L′ and the generalized coordinates derived from L′ are respectively

Combining the last two equations gives the momentum space Euler–Lagrange equations

The advantage of the Legendre transformation is that the relation between the new and old functions and their variables are obtained in the process. Both the coordinate and momentum forms of the equation are equivalent and contain the same information about the dynamics of the system. This form may be more useful when momentum or angular momentum enters the Lagrangian.

Hamiltonian mechanics

In Hamiltonian mechanics, unlike Lagrangian mechanics which uses either all the coordinates or the momenta, the Hamiltonian equations of motion place coordinates and momenta on equal footing. For a system with Hamiltonian H(q, p, t), the equations are

Position and momentum spaces in quantum mechanics

In quantum mechanics, a particle is described by a quantum state. This quantum state can be represented as a superposition (i.e. a linear combination as a weighted sum) of basis states. In principle one is free to choose the set of basis states, as long as they span the space. If one chooses the eigenfunctions of the position operator as a set of basis functions, one speaks of a state as a wave function ψ(r) in position space (our ordinary notion of space in terms of length). The familiar Schrödinger equation in terms of the position r is an example of quantum mechanics in the position representation. [3]

By choosing the eigenfunctions of a different operator as a set of basis functions, one can arrive at a number of different representations of the same state. If one picks the eigenfunctions of the momentum operator as a set of basis functions, the resulting wave function is said to be the wave function in momentum space. [3]

A feature of quantum mechanics is that phase spaces can come in different types: discrete-variable, rotor, and continuous-variable. The table below summarizes some relations involved in the three types of phase spaces. [4]

Comparison and summary of relations between conjugate variables in discrete-variable (DV), rotor (ROT), and continuous-variable (CV) phase spaces (taken from arXiv:1709.04460). Most physically relevant phase spaces consist of combinations of these three. Each phase space consists of position and momentum, whose possible values are taken from a locally compact Abelian group and its dual. A quantum mechanical state can be fully represented in terms of either variables, and the transformation used to go between position and momentum spaces is, in each of the three cases, a variant of the Fourier transform. The table uses bra-ket notation as well as mathematical terminology describing Canonical commutation relations (CCR). Phase spaces.png
Comparison and summary of relations between conjugate variables in discrete-variable (DV), rotor (ROT), and continuous-variable (CV) phase spaces (taken from arXiv:1709.04460). Most physically relevant phase spaces consist of combinations of these three. Each phase space consists of position and momentum, whose possible values are taken from a locally compact Abelian group and its dual. A quantum mechanical state can be fully represented in terms of either variables, and the transformation used to go between position and momentum spaces is, in each of the three cases, a variant of the Fourier transform. The table uses bra-ket notation as well as mathematical terminology describing Canonical commutation relations (CCR).

Relation between space and reciprocal space

The momentum representation of a wave function is very closely related to the Fourier transform and the concept of frequency domain. Since a quantum mechanical particle has a frequency proportional to the momentum (de Broglie's equation given above), describing the particle as a sum of its momentum components is equivalent to describing it as a sum of frequency components (i.e. a Fourier transform). [5] This becomes clear when we ask ourselves how we can transform from one representation to another.

Functions and operators in position space

Suppose we have a three-dimensional wave function in position space ψ(r), then we can write this functions as a weighted sum of orthogonal basis functions ψj(r):

or, in the continuous case, as an integral

It is clear that if we specify the set of functions , say as the set of eigenfunctions of the momentum operator, the function holds all the information necessary to reconstruct ψ(r) and is therefore an alternative description for the state .

In quantum mechanics, the momentum operator is given by

(see matrix calculus for the denominator notation) with appropriate domain. The eigenfunctions are

and eigenvalues ħk. So

and we see that the momentum representation is related to the position representation by a Fourier transform. [6]

Functions and operators in momentum space

Conversely, a three-dimensional wave function in momentum space can be expressed as a weighted sum of orthogonal basis functions ,

or as an integral,

The position operator is given by

with eigenfunctions

and eigenvalues r. So a similar decomposition of can be made in terms of the eigenfunctions of this operator, which turns out to be the inverse Fourier transform, [6]

Unitary equivalence between position and momentum operator

The r and p operators are unitarily equivalent, with the unitary operator being given explicitly by the Fourier transform, namely a quarter-cycle rotation in phase space, generated by the oscillator Hamiltonian. Thus, they have the same spectrum. In physical language, p acting on momentum space wave functions is the same as r acting on position space wave functions (under the image of the Fourier transform).

Reciprocal space and crystals

For an electron (or other particle) in a crystal, its value of k relates almost always to its crystal momentum, not its normal momentum. Therefore, k and p are not simply proportional but play different roles. See k·p perturbation theory for an example. Crystal momentum is like a wave envelope that describes how the wave varies from one unit cell to the next, but does not give any information about how the wave varies within each unit cell.

When k relates to crystal momentum instead of true momentum, the concept of k-space is still meaningful and extremely useful, but it differs in several ways from the non-crystal k-space discussed above. For example, in a crystal's k-space, there is an infinite set of points called the reciprocal lattice which are "equivalent" to k = 0 (this is analogous to aliasing). Likewise, the "first Brillouin zone" is a finite volume of k-space, such that every possible k is "equivalent" to exactly one point in this region.

For more details see reciprocal lattice.

See also

Footnotes

  1. For two functions u and v, the differential of the product is d(uv) = udv + vdu.

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Schrödinger equation Linear partial differential equation whose solution describes the quantum-mechanical system.

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.

Wave function Mathematical description of the quantum state of a system

A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters ψ and Ψ.

Noethers theorem Statement relating differentiable symmetries to conserved quantities

Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918, after a special case was proven by E. Cosserat and F. Cosserat in 1909. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space.

Hamiltonian mechanics Formulation of classical mechanics using momenta

Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena.

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.

Eigenfunction Mathematical function of a linear operator

In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as

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. 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.

Probability amplitude Complex number whose squared absolute value is a probability

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Path integral formulation Formulation of quantum mechanics

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In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the 18th century and onward, after Newtonian mechanics. Since Newtonian mechanics considers vector quantities of motion, particularly accelerations, momenta, forces, of the constituents of the system, an alternative name for the mechanics governed by Newton's laws and Euler's laws is vectorial mechanics.

Canonical quantization Process of converting a classical physical theory into one compatible with quantum mechanics

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 physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. In quantum mechanics, it means the particle is in a region of uniform potential, usually set to zero in the region of interest since the potential can be arbitrarily set to zero at any point in space.

Routhian mechanics Formulation of classical mechanics

In classical mechanics, Routh's procedure or Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the Lagrangian and Hamiltonian functions. As with the rest of analytical mechanics, Routhian mechanics is completely equivalent to Newtonian mechanics, all other formulations of classical mechanics, and introduces no new physics. It offers an alternative way to solve mechanical problems.

In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle.

In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a mathematical constraint, the natural consequence of the equations of motion, rather than a physical constraint. Common examples include energy, linear momentum, angular momentum and the Laplace–Runge–Lenz vector.

In quantum mechanics and quantum field theory, a Schrödinger field, named after Erwin Schrödinger, is a quantum field which obeys the Schrödinger equation. While any situation described by a Schrödinger field can also be described by a many-body Schrödinger equation for identical particles, the field theory is more suitable for situations where the particle number changes.

Lagrangian mechanics Formulation of classical mechanics

Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788 from his work Mécanique analytique, Lagrangian mechanics is a formulation of classical mechanics and is founded on the stationary action principle.

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

References

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