Tagged Questions
0
votes
0answers
77 views
Prove that the position operator is $\hat{x} = i\hbar \frac{d}{{dp}}$ in the momentum representation [closed]
Proof that: $x = i\hbar \frac{d}{{dp}}$
I did this, could you tell me if I am false or true
$\begin{array}{l}
x{e^{\frac{{ipx}}{\hbar }}} = - i\hbar \frac{{d{e^{\frac{{ipx}}{\hbar }}}}}{{dp}} = ...
0
votes
1answer
151 views
Weird integration of gaussian wave packet
I have been learning Fourier transformation of a gaussian wave packet and i don't know how to calculate this integral:
In the above integral we try to calculate $\varphi(\alpha)$ where $\alpha$ is ...
0
votes
1answer
114 views
Fourier transform between $x$ and $p$
On this page right at the top they mention two sets of fourier transform. First set is connection between $x$ (position) and $k$ (wave vector) space:
$$
\begin{split}
f(x) &= ...
0
votes
1answer
353 views
Gaussian wave packet
At our QM intro our professor said that we derive uncertainty principle using the integral of plane waves $\psi = \psi_0(k) e^{i(kx - \omega t)}$ over wave numbers $k$. We do it at $t=0$ hence $\psi = ...
1
vote
1answer
142 views
What's the average position of oscillating particles in a box with periodic boundary conditions?
Imagine an open box repeating itself in a way that a if a particle crossing one of the box boundary is "teleported" on the opposite boundary (typical periodic boundary position in 3D).
Now put a ...
3
votes
1answer
346 views
Physical Significance of Fourier Transform and Uncertainty Relationships
What is the physical significance of a fourier transform?
I am interested in knowing exactly how it works when crossing over from momentum space to co ordinate space and also how we arrive at the ...
7
votes
2answers
254 views
Was uncertainty principle inferred by Fourier analysis?
I would like to know: did Heisenberg chance upon his Uncertainty Principle by performing Fourier analysis of wavepackets, after assuming that electrons can be treated as wavepackets?
4
votes
3answers
553 views
Canonical Commutation Relations
Is it logically sound to accept the canonical commutation relation (CCR)
$$[x,p]~=~i\hbar$$
as a postulate of quantum mechanics? Or is it more correct to derive it given some form for $p$ in the ...
6
votes
2answers
232 views
Evaluating propagator without the epsilon trick
Consider the Klein–Gordon equation and its propagator:
$$G(x,y) = \frac{1}{(2\pi)^4}\int d^4 p \frac{e^{-i p.(x-y)}}{p^2 - m^2} \; .$$
I'd like to see a method of evaluating explicit form of $G$ ...
6
votes
2answers
332 views
What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases?
I'm curious whether I can find the overlap $\langle q | p \rangle$ knowing only the following:
$|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$.
$|p\rangle$ is an eigenvector of ...
3
votes
4answers
603 views
Intuitive explanation of why momentum is the Fourier transform variable of position?
Does anyone have a (semi-)intuitive explanation of why momentum is the Fourier transform variable of position?
(By semi-intuitive I mean, I already have intuition on Fourier transform between ...
3
votes
1answer
334 views
Finding $\psi(x,t)$ for a free particle starting from a Gaussian wave profile $\psi(x)$
Consider a free-particle with a Gaussian wavefunction,
$$\psi(x)~=~\left(\frac{a}{\pi}\right)^{1/4}e^{-\frac12a x^2},$$
find $\psi(x,t)$.
The wavefunction is already normalized, so the next thing to ...
2
votes
1answer
220 views
Is there a relation between quantum theory and Fourier analysis?
These days I was studying the quantum theory.I found that some theories about that is similar to Fourier Transform theory.For instance, it says "A finite-time light's frequency can't be a certain ...
2
votes
2answers
179 views
Measurement and uncertainty principle in QM
The Wikipedia says on the page for the uncertainty principle:
Mathematically, the uncertainty relation between position and momentum arises because the expressions of the wave function in the two ...
3
votes
1answer
69 views
Describing quantum intereference with only currents and densities
I know about and believe to understand the general wave equation based Kirchhoff diffraction formula, which in the Fraunhofer limit leads to a farfield complex wave function by Fourier transforming ...
4
votes
4answers
502 views
Uncertainty Principle for a Totally Localized Particle
If a particle is totally localized at $x=0$, its wave function $\Psi(x,t)$ should be a Dirac delta function $\delta(x)$. Accordingly, its Fourier transform $\Phi(p,t)$ would be a constant for all $p$, ...
1
vote
2answers
169 views
Expressing a particle's matter wave in terms of its momentum
I'm following Zettili's QM book and on p. 39 the following manipulation is done,
Given a localized wave function (called a wave packet), it can be expressed as $$\psi(x,t) = \frac{1}{\sqrt{ 2 \pi}} ...
1
vote
1answer
377 views
Conjugate Variables and Fourier Transforms in Classical Physics
Let q be a generalized coordinate with a conjugate momentum p and a potential resulting in a periodic motion of q. What is the meaning of the Fourier transform of q(t) over its period? Can this be ...
4
votes
2answers
176 views
Does a finite wave necessarily have to be non-monochromatic in reality?
Does a finite wave necessarily have to be non-monochromatic in reality, or is that implication just a result of the mathematical analysis? I always wonder at these sort of things that come out of a ...
5
votes
1answer
499 views
What is the meaning of the Fourier transform of Feynman propagator?
I know $K(a,b,t)$ is the probability amplitude of find a particle that starts at point a in b in a time t later. There is also an expression that sometimes is called green function:
...
4
votes
3answers
395 views
Very simple example of the way the Fourier transform is used in quantum mechanics?
According to a book I'm reading, the Fourier transform is widely used in quantum mechanics (QM). That came as a huge surprise to me. (Unfortunately, the book doesn't go on to give any simple examples ...
4
votes
3answers
998 views
What is the relation between position and momentum wavefunctions in quantum physics?
I have read in a couple of places that $\psi(p)$ and $\psi(q)$ are Fourier transforms of one another (e.g. Penrose). But isn't a Fourier transform simply a decomposition of a function into a sum or ...
