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.
I am meeting the momentum space wave function ##\Phi\left(p,t\right)## in chapter 3 of Griffiths & Schroeter. I have an integral which I can integrate by parts:
$$\int_{-\infty}^{\infty}{\frac{\partial}{\partial p}\left(e^\frac{ipx}{\hbar}\right)\Phi d...
so from Fourier transform we know that
Ψ(r)=1/2πℏ∫φ(p)exp(ipr/ℏ)dp
I proved that <p>= ∫φ(p)*pφ(p)dp from <p>=∫Ψ(r)*pΨ(r)dr
so will the same hold any operator??
I am having difficulty writing out
##\bra{p',\lambda}\psi^{\dagger}(-\frac{z^-}{2})\gamma^0\gamma^+\psi\frac{z^-}{2})\ket{p,\lambda}## in momentum space.
Here, I am working in light-cone coordinates, where I am defining ##z^-=z^0-z^3##, ##r'=r=(0,z^{-},z^1,z^2)##.
My attempt at this would be...
Since it asks for the time evolution of the wavefunction in the momentum space, I write : ##\tilde{\Psi}(k,t) = < p|U(t,t_{0})|\Psi> = < U^\dagger(t,t_{0})p|\Psi>##
Since ##U(t,t_{0})^\dagger = e^{\frac{i}{\hbar}\frac{\hat{p^2}t}{2m}}##, the above equation becomes
##\tilde{\Psi}(k,t) =...
I am not sure how does the integral was did here. More preciselly, How to go from the first line to the second line? Shouldn't it be $$\frac{4 \pi}{(2 \pi)^3} \int _{0} ^{\infty} p^2 e^{ip*r}/(2 E_p)$$ ? (x-y is purelly spatial)
This seems rather straight forward, but I can't figure out the details... Generally speaking and ignoring prefactors, the Fourier transformation of a (nicely behaved) function ##f## is given by
$$f(x)= \int_{\mathbb{R}^{d+1}} d^{d+1}p\, \hat{f}(p) e^{ip\cdot x} \quad\Longleftrightarrow \quad...
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...
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...
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...
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...
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##...
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)...
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...
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?
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...
Homework Statement
Derive, using the canonical commutation relation of the position space representation of the fields φ(x) and π(y), the corresponding commutation relation in momentum space.Homework Equations
[φ(x), π(y)] = iδ3(x-y)
My Fourier transforms are defined by: $$ φ^*(\vec p)=\int...
In non-relativistic QM, when one speaks of a "wave function" it is understood that one is referring to the position-space version of the wave function. Even if the observable being measured is other than position, like momentum or energy, the associated eigenfunctions are always from the...
Homework Statement
Hi,
Question attached:
inserting ##\phi (x)= \int \frac{d^4k}{(2\pi)^2}\phi(x)e^{-i k_u x^u}##
and similar for ##J(x) ## / ##J(k)## into the action and then integrating over ##k## gives:
Solution attached:
I AM STUCK on this part, completing the square ; so I see...
Homework Statement
So, I'm doing this problem from Townsend's QM book
6.2[/B]
Show that <p|\hat{x}|\psi> = i\hbar
\frac{\partial}{\partial p}<p|\psi>
Homework Equations
|\psi(p)> = \int_\infty^{-\infty} dp |p><p|\psi>
The Attempt at a Solution
So,
<p|\hat{x}|\psi>
= <p|\hat{x}...
What is the relation between the correlators ##\langle 0 | T\phi(x_{1})\phi(x_{2}) | 0 \rangle## and ##\langle 0 | T\phi(p)\phi(x=0) | 0 \rangle##?
I can derive the momentum space Feynman rules for ##\langle 0 | T\phi(x_{1})\phi(x_{2}) | 0 \rangle##. Are the momentum space Feynman rules for...
Homework Statement
This integral has to do with the probability amplitude that a free particle at position x0 is found at x at some time t. With H = p2/(2m), this involves evaluating the integral
1/(2π)3∫d3p e-i(p2/(2m))t eip(x-x0)
The answer is
(m/(2πit))3/2e(im(x-x0)2)/(2t)
2...
Specifically, i do not know hot to express the potential in momentum space. If someone would provide me with a link of source that has the proof in it, it would be appreciated.
Let's say we have a plasma of protons or deuterons and electrons; how thick is the 'shell' that the nuclei form in momentum space compared to the 'shell' formed by D2 or H2 in a gas, at room temperature? Will the distribution of momentum be more spread out or more compact, so to speak, in a...
Homework Statement
How do you get from (3.171) to (3.172)? In particular, why is
##\int e^{-ip.r/{\hbar}}\frac{p_{op}^2}{2m}\Psi(r,t)\,dr=\int\frac{p_{op}^2}{2m}[e^{-ip.r/{\hbar}}\Psi(r,t)]\,dr##? ##\,\,\,\,\,##-- (1)
Homework Equations
The Attempt at a Solution
For (1) to be true, it...
Homework Statement
Using the following expression for the Dirac delta function: $$\delta(k-k')=\frac{1}{2\pi} \int_{-\infty}^{\infty}e^{i(k-k')x} \mathrm{d}x$$
Show that if a position space wave function $$\Psi(x,t)$$ is normalized at time t=0, then it is also true that the corresponding...
Hi, I wish to get position operator in momentum space using Fourier transformation, if I simply start from here,
$$ <x>=\int_{-\infty}^{\infty} dx \Psi^* x \Psi $$
I could do the same with the momentum operator, because I had a derivative acting on |psi there, but in this case, How may I get...
How are protons in an ion source distributed in momentum space? Consider an ion source fed with H2 at low pressure. As soon as the protons are free protons they are accelerated by the extraction voltage of perhaps 10 kV. In momentum space the protons are initially a "shell" with a certain...
So I've read you can get the corresponding wave function of a quantum harmonic oscillator in momentum space from position space by making the substitution ##x \to k## and ##m \omega \to 1/m \omega##.
However in deriving the TISE for momentum space, I seem to be making a mistake. In momentum...
Hey everyone!
It's my first semester with quantum mechanics and I'm uncertain if my solution of this problem is correct, would be nice if someone could check and let me know :smile:
1. Homework Statement
I have to calculate the representation of the state:
|\alpha \rangle \equiv exp[-i...
So I was asked to compute loop contributions to the Higgs and compute the integrals in spherical coordinates, I gave a look to Halzen book but did not found anything. Why, when and how to make that change?
Hey!
1. Homework Statement
I've been given the time dependent schrödinger equation in momentum space and have to calculate the force, as a function of the position, acting on a particle with mass m.
\vec{F}(\vec{r})=-\nabla{V(\vec{r})}...
I didn't put this in the Education forum because I feel the level is probably too high.
I have a physics degree and an education degree, but due to my inability to tolerate students who simply don't care about learning, I became an electrician instead.
My colleagues and I were having a...
I'm working through an article called "Cosmic abundances of stable particles -- improved analysis" (link -- viewable only in Firefox afaik), the result of which, equation (3.8), is cited quite a lot. I'm more interested in how they arrived there.
Particularly, how come momentum space measure...
Might be simple but I couldn't see. We can easily derive momentum operator for position space by differentiating the plane wave solution. Analogously I want to derive the position operator for momentum space, however I am getting additional minus sign.
By replacing $$k=\frac{p}{\hbar}$$ and...
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...
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...
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
=...
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...
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}...
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...
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}...
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...
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) =...
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...
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)...
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...
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...
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...
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...