Position and momentum spaces
In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension.
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.
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.
Quantum mechanics provides two fundamental examples of the duality between position and momentum, the Heisenberg uncertainty principle ΔxΔp ≥ ħ/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][2] 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.
Introducing the definition of canonical momentum for each generalized coordinate
The Lagrangian can be expressed in momentum space also,[4] 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,
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
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.
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.
If one chooses the (generalized) 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.
[5] 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.
[6] The momentum representation of a wave function and the de Broglie relation are closely related to the Fourier inversion theorem and the concept of frequency domain.
, describing the particle as a sum of frequency components is equivalent to describing it as the Fourier transform of a "sufficiently nice" wave function in momentum space.
holds all the information necessary to reconstruct ψ(r) and is therefore an alternative description for the state
(see matrix calculus for the denominator notation) with appropriate domain.
[8] Conversely, a three-dimensional wave function in momentum space
can be expressed as a weighted sum of orthogonal basis functions
can be made in terms of the eigenfunctions of this operator, which turns out to be the inverse Fourier transform,[8]
The position and momentum 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.
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).
Therefore, k and p are not simply proportional but play different roles.
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.
