# What is Momentum space: Definition and 80 Discussions

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

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7. ### Position to momentum space in three dimensions

Hi! I am trying to change the hydrogen ground state wave funcion from position to momentum space, so i solved the integral Ψ(p)=(2πħ)^(-3/2) (πa^3)^(-1/2)∫∫∫e^(prcosθ/ħ) e^(-r/a) senθ r^2 dΦdθdr and got 4πħ(2πħ)^(-3/2) p^(-1) (πa^3)^(-1/2) I am [(ip/ħ-1/a)^(-2)], which according to the...
8. ### Conservation of Momentum Space Ship Problem

Homework Statement The payload of a spaceship accounts for 20% of its total mass. The ship is traveling in a straight line at 2100km/hr relative to some inertial observer O. When the time is right, the spaceship ejects the payload, which is moving away from the ship at 500km/hr immediately...
9. ### I Calculating the number of energy states using momentum space

A question came up about deducing the number of possible energy states within a certain momentum ##p## using momentum space. To make my question easier to understand, I deliberately chose ##p## and not a particular increment ##dp## and I assume a 2 dimensional momentum space with coordinates...
10. ### I Momentum/Position space wave function

These are from Griffith's: My lecture note says that I am having quite a confusion over here...Does the ##\Psi## in the expression ##\langle f_p|\Psi \rangle## equals to ##\Psi(x,t)##? I understand it as ##\Psi(x,t)## being the component of the position basis to form ##\Psi##, so...
11. ### Probability distribution momentum for particle

Homework Statement A particle with mass m is moving on the x-axis and is described by ## \psi_b = \sqrt{b} \cdot e^{-b |x|}## Find the probability distribution for the particles momentum Homework Equations ## \Phi (p)= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty \Psi(x,0) \cdot e^{-ipx} dx##...
12. M

### Finding the position operator in momentum space

Homework Statement Given ##\hat{x} =i \hbar \partial_p##, find the position operator in the position space. Calculate ##\int_{-\infty}^{\infty} \phi^*(p) \hat{x} \phi(p) dp ## by expanding the momentum wave functions through Fourier transforms. Use ##\delta(z) = \int_{\infty}^{\infty}\exp(izy)...
13. ### A Is the Inverse Momentum Operator an Essential Tool in Quantum Mechanics?

In QM, the inverse distance operator ##\hat{r}^{-1}## appears often because of the association to Coulomb potential. The operator of inverse momentum, ##\frac{1}{\hat{p}}## is a lot more rare. In the book "Exploring Quantum Mechanics: A Collection of 700+ Solved Problems for Students, Lecturers...
14. ### What is the Usefulness of Reciprocal Space in Classical Physics?

how do you describe the speed of an object in momentum space (energy, momentum as the 2 axes) where there is no distance or time? Can you give an example?
15. ### B Spin and polarizations in momentum space

Can momentum space also able to handle spin and polarizations? I'm understanding it that in QM, you have position, momentum, spin, polarization as observables. Position and momentum can be equivalent via Fourier transform. So if you use momentum space instead of position, how do you handle...

36. ### Q:Is it possible to do a coordinate transfomation in momentum space?

Q: How does one do a coordinate transformation in momentum space while insuring conservation of momentum? I have a several particles with momentum components P_x , P_y , P_z . I would like to rotate the x, y, and z axis. By angle θ in the x/y and angle Θ in the y/z . So giving new...
37. ### Talmi-Moshinsky coefficients for momentum space wavefunctions

I've found a number of papers about how to calculate Talmi-Moshinsky coefficients. For example W. Tobocman Nucl. Phys A357 (1981) 293-318 and FORTRAN code base on it Y.-P. Gan et al. Comput. Phys. Commun. 34 (1985) 387. This works well if I want to calculate matrix elements that only depend on...
38. ### Fourier Transform of wavefunction - momentum space

Homework Statement Find possible momentum, and their probabilities. Find possible energies, and their probabilities. Homework Equations The Attempt at a Solution First, we need to Fourier transform it into momentum space: \psi_k = \frac{1}{\sqrt{2\pi}} \int \psi_x e^{-ikx} dx =...
39. ### Beam of particles in momentum space

I'm mostly concerned with whether or not I understand this problem intuitively in order to answer the final part of this problem. Homework Statement Discuss the implications of Liouville's theorem on the focusing of beams of charged particles by considering the following case. An electron...
40. ### Momentum space particle in a box

I am trying to formulate a solution to the particle in a box energy eigenvalue problem, without solving a differential equation, instead using eigenvectors of p^2. My idea is to do this. Within the box (let's say it is defined between [-a,a] and within this region the hamiltonian is H={p^2}/{2m}...
41. ### 3D Quantum HO in momentum space

Homework Statement I'm trying to prove that the Harmonic oscillator wave function doesn't change (except a phase factor) when I convert from position to momentum space. \Phi_{nlm}(\vec p)=(-i)^{2n+l}\Psi_{nlm}(\vec p) Homework Equations \Phi_{nlm}(\vec p)=\frac{1}{(2\pi)^{3/2}}\int d^3r...
42. ### Density matrix in Momentum Space

I have an one-body density matrix in a Sine wave basis set (Thus psi = psi*). Unfortunately, these are not the natural orbitals (I have correlated particles), so I have off-diagonal elements. I believe I know how to extract the charge density from this density matrix \rho(x;x') = \sum_{ij}...
43. ### Time Dependent expectation value in momentum space

Homework Statement A particle of mass m in the one-dimensional harmonic oscillator is in a state for which a measurement of the energy yields the values hω/2 or 3hω/2, each with a probability of one-half. The average values of the momentum <p> at time t = 0 is √mωh/2. This information...
44. ### Ladder operater for momentum space wavefunction (harmonic oscillator)

Homework Statement I need to find the momentum space wavefuntion Phi(p,t) for a particle in the first excited state of the harmonic oscillator using a raising operator. Homework Equations Phi_1(p,t)= "raising operator" * Phi_0 (p,t)The Attempt at a Solution In position space, psi_1 (x) =...
45. ### Quantum Mechanics: Finding wavefunction in momentum space.

Basically, the problem gives me a wave function and asks me to find the wave function in momentum space. It then asks me to find the expected value. Namely <p> and <p^2>. The problem is, when I try to calculate <p> it blows up to infinity. What am I doing wrong? Here is my work...
46. ### Eigenstates of Dirac Potential in Momentum Space

Homework Statement Consider a particle moving in one dimension and bound to an attractive Dirac δ-function potential located at the origin. Work in units such that m=\hbar=1. The Hamiltonian is given, in real (x) space, by: H=-\frac{1}{2}\frac{d^2}{dx^2}-\delta (x) The (non normalized)...
47. ### Commpute probability in momentum space of particle in box (after walls are removed)

Homework Statement A particle is initially in the nth eigenstate of a box of length 2a. Suddenly the walls of the box are completely removed. Calculate the probability to find that the particle has momentum between p and p + dp. Is energy conserved? Homework Equations solution...
48. ### Momentum space representation for finite lattices - continued

I have been banned, maybe my nickname was not so kind. I let the topic continue here. I report my last comment: "Ok, I got the point. thanks for replying! It's just a change of basis that under boundary condition diagonalize the Hamiltonian. But then a subtle point: In order for...
49. ### Momentum space representation for finite lattices

Hi all, I have a question. For sure the momentum representation used in solid state physics works for infinite lattices or periodic ones. But when it comes to finite lattice, i.e. 100 sites, can the momentum representation be used? What are the errors? Where does this fail? Thanks for...
50. ### Dimensionality of total angular momentum space

Homework Statement There are 2 electrons, one with n=1, l=0 and the other with n=2, l=1. The question asks what is the dimensionality of total angular momentum space. Homework Equations (2j_{1}+1)(2j_{2}+1)The Attempt at a Solution I know for 2 electrons (spin 1/2 each) the possible values of...