# Momentum space

The momentum space is a three-dimensional coordinate system , with each base vector corresponding to an impulse of the corresponding spatial direction. The momentum space is a subspace of the phase space and must therefore be distinguished from the spatial space . By assigning appropriate coordinates, a phase space can be spanned with the momentum space.

## Quantum mechanics

In quantum mechanics , the states of a system are described by wave functions that can be represented in space or momentum space. Depending on the problem, the bill in one of the two rooms can be cheaper. The two types of representation are related to the Fourier transformation :

• Position space wave function ${\ displaystyle \ psi (x) = {\ frac {1} {\ sqrt {2 \ pi \ hbar}}} \, \ int _ {- \ infty} ^ {+ \ infty} \ phi (p) \, e ^ {\ frac {ipx} {\ hbar}} \, \ mathrm {d} p \ qquad \ Leftrightarrow}$
• Momentum space wave function ${\ displaystyle \ phi (p) = {\ frac {1} {\ sqrt {2 \ pi \ hbar}}} \, \ int _ {- \ infty} ^ {+ \ infty} \ psi (x) \, e ^ {- {\ frac {ipx} {\ hbar}}} \, \ mathrm {d} x}$

With

• the reduced Planck quantum of action ${\ displaystyle \ hbar}$
• the impulse ${\ displaystyle p}$
• the imaginary unit ${\ displaystyle i.}$

## Solid state physics

In solid-state physics and crystallography , the terms reciprocal space and reciprocal lattice , which are analogous to momentum space, are used. The wavenumbers or spatial frequencies customary there correspond to the crystal geometry, i.e. a Fourier transformation of the crystal structure . The difference by the factor results from the non-uniform definition of the Fourier transform . The discrete crystal lattice is just as discrete in reciprocal space and clearly a three-dimensional crystal has a three-dimensional reciprocal lattice and a two-dimensional crystal has a two-dimensional reciprocal lattice. Through the spatial frequencies associated with impulses, such as by phonons (see. But also the Mossbauer effect ) together and thus allow for calculations. B. using the Ewald sphere . ${\ displaystyle k = 2 \ pi / \ lambda}$${\ displaystyle {\ hat {k}} = 1 / \ lambda}$${\ displaystyle 2 \ pi}$${\ displaystyle p = \ hbar k}$