Momentum space Definition and 69 Threads

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) [SIZE=16px][FONT=Noto Sans](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)]...

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 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}...

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...

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...

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...

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...

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...

I'm trying to do the following problem: ie. I'm trying to use the Feynman rules for momentum space to write down the mathematical expression that the diagram is supposed to represent. However, I don't feel very confident in what I've managed so far. Now as I understand it, these diagrams...

Hi, Note: I will be sloppy with constant factors in this post. Only the general structure of the equations matters. Consider a particle in a linear potential, \frac{\mathrm d^2}{\mathrm d x^2} \psi(x) + x \psi(x) - E \psi(x) = 0. Mathematically, this is a second-order ODE, and there...

Homework Statement <e^{ip'x}|x^{2}|e^{ipx}> Homework Equations The Attempt at a Solution Its pretty obvious that its difficult to integrate in position-space, so I rewrite x in momentum space (i.e. the second-order differential operator with respect to p). If that is...

Hello, I have a slight problem with Quantumtheory here. Homework Statement I have solved the schrödinger equation in the momentum space for a delta potential and also transferred it into real space. So now I have to find the correlation between the width of the wavefunction in both spaces...

The form of the Fourier transform I love the most (because it is very symmetric) is: f(x) = \int_{-\infty}^\infty g(\xi)e^{2\pi i x \xi}\,d\xi g(\xi) = \int_{-\infty}^\infty f(x)e^{-2\pi i x \xi}\,dx If we take \xi = p then we get: f(x) = \int_{-\infty}^\infty g(p)e^{2\pi i x p}dp =...

Schrodinger Equation in momentum space? ?? I don't know if this makes any sense at all, but I'm studying QM and just trying to generalize some things I'm learning. Please let me know where I go wrong.. Basically by my understanding the most general form of the Schrodinger Equation can be...

Homework Statement Given \Psi(x,0)=\frac{A}{x^{2}+a^{2}}, (-\inf<x<\inf) a) determine A c) find the momentum space wave function \Phi(p,0), and check that it is normalized Homework Equations At t=0, we can find the momentum space wave function by the formula...