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'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...
Quantum Field Theory Purly in Momentum Space?
Hello,
I have a complicated nonlinear-nonlocal-nonrelativistic-effective action in momentum space and would like to do perturbation theory with that. I need to find propagator and Feynman rules. I can not go to x-space and follow the standard...
Hi all,
I understand how to transform between position space and momentum space; it's a Fourier transform:
\varphi|p>=\frac{1}sqrt{2\hbar\pi}\int_{\infty}^{\infty} <x|\varphi> exp(-ipx/\hbar)dx
But I can't figure out how to transform the operators. I know what they transform into (e.g...
here is the notes:
we have the MB statistics or distribution
ns = Agsexp(-es/kT) (1)
This is only for a set of discrete energy levels es. In this section, we shall see how (1) can be applied to a variety of situations. For instance, how can we use this distribution for an ideal gas which...
Hi,
I am having trouble understanding some things about k-space or momentum space in a crystal. The trouble began when I was first introduced to the Bloch theorem, a few weeks back.
It is:
\psi_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r}).
In...
does anyone knw the code for how to produce the d slash notation in the integration measure for momentum space? Where (d slash)^n X=(d^n)X/((2pi)^n).
Basically all i want to do is replace the h:
\hslash
with a d.
does anyone knw the code for how to produce the d slash notation in the integration measure for momentum space? Where (d slash)^n X=(d^n)X/((2pi)^n).
Basically all i want to do is replace the h:
\hslash
with a d.
Problem
Derive the Schroedinger equation (for harmonic oscillator) in momentum space.
The attempt at a solution
We have
ih \frac{\partial}{\partial t} \langle p' | \alpha \rangle = \langle p' | \frac{p^2}{2m} | \alpha \rangle + \langle p' | V(x) | \alpha \rangle
\iff ih...
I have learned that to transform from one space to another, we can use
g(e) = g(p)/f’, where de/dp = f’
Can we use this relation to transform wavefunctions of energy space to momentum space?
If not, why?
If so, that's very strange as E= p^2/2m and dE/dp= p/m and put into...
Homework Statement
given A(k)=N/(k2+a2) calculate psi(x) and show that
(delta k * delta x) > 1
independent of the choice of a
The Attempt at a Solution
I calculated psi(x) to be (N*pi/a)*e-|ax|
Would it be ok to compute <x> and <x2> in coordinate space and <k> and <k2> in...
Hi!
I wonder if the Feynman rules in momentum space can also be applied to disconnected diagrams. So aussume I have a disconnected Feynman diagram 1 , i.e. the product of two connected Feynman diagrams 2 and 3.
I can translate my diagrams with the position space Feynman rules to explicit...
Homework Statement
Check that a given momentum space wave function is normalized. I've done the integral, but the result is not dimensionless. Here is the wave function:
\overline{\phi} = \frac{1}{\pi} ( \frac{2 a_{0}}{\bar{h}})^{3/2} \frac{1}{(1+(a_{0} p / \bar{h})^2)^2}
The units of this...
using the hamiltonian to derive the pathintegral is well known (see schulman), but i have only seen it for diagonal momenta and coupled coordinates:
G(x,t;y) = <x|exp(-itH/hbar)|y> using the trotter formula etc one arrives at:
G(x,t;y) = lim_N->infinity Int...
[b]1. The figure shows the rear view of a space capsule that was left rotating about its axis at 6 rev/min after a collision with another capsule. You are the flight controller and have just moments to tell the crew how to stop this rotation before they become ill from the rotation and the...
Page 152 Robinett:
Consider the (non-normalized) even momentum space wavefunctions for the symmetric well:,
\phi_n^+(p) = 2sin(w-m)/(w-m)+sin(w+m)/(w+m) where
w = sin((n-1/2)pi) and
m = ap/hbar.
Show that
\int_{-\infty}^{\infty}\phi_n^+(p)^*\cdot \phi_n^+(p) dp = \delta_{n,m}
The hint...
Problem:
Find the momentum-space wave function \Phi_n(p,t) for the nth stationary state of the infinite square well.
Equations:
\Psi_n(x,t) = \psi_n(x) \phi_n(t)
\psi_n(x) = \sqrt{\frac{2}{a}}\sin(\frac{n\pi}{a}x)
\phi_n(t) = e^{-iE_n t/\hbar}
\Phi_n(p,t) =...
could somebody please explain to me why position and momentum space are related to one another by a Fourier transform, meaning why do I get momenta when I do a Fourier transform of an expression in position space?
I have this doubt..quantization in momentum space using G(p) as the Fourier transform of the wave function was not common (at least when i studied Q. Physics) my doubt is, if we have that:
x |G(p)>=i \hbar \frac{ \partial G(p)}{\partial p}
But..what would happen if we apply:
\dot...
How exactly does one find a wave function? Specifically, I am asked to find the momentum space wave functoin for the nth stationary state in an infinite square well. Then I am to graph the probability density (phi sqaured) for the first and second energy levels. Lastly, I need to use the...
Hey,
We are given the 1s spatial wave function for the hydrogen atom:
\psi(\vec{r}) = \frac{1}{\sqrt{a_{0}^3r}}e^{-r/a_{0}
We are asked to find the momentum space wave function \phi(\vec{p}). Obviously this is just the Fourier transform of the spatial wave function. In calculating...
I am just starting reading some quantum stuff. There of course be many questions here and there.
One thing comes to bother me now that it seems QM are treating this "state" in either momentum or position representation, if leave alone spin space. It seems to treat "momentum" and "position" as...
Please, I would like to write the time-independent schroedinger equation (describing the motion of a bound electron) in momentum space and in cylindrical coordinates.
Can you help me?
Thank you very much.
Hugues Merlain