3
votes
0answers
99 views

“Derivation” of the Heisenberg Uncertainty Principle

Ok, so I posted this in the mathematics StackExchange, but got no response. The question I outline below is my textbook's "derivation" of the Heisenberg Uncertainty Principle. The "derivation" my ...
0
votes
1answer
73 views

How to find different operator representations in QM?

I read that any observable operator may be represented as: $$\Omega = \sum_n \omega _n | \omega _n \rangle \langle \omega_n |$$ Where the little omegas are the eigenvectors/eigenvalues of the ...
3
votes
1answer
116 views

confusion in discrete transform to solve kronig penney matrix equation in fourier space

I have a periodic potential $$V(x) =\sum_{K}e^{iKx}V_{K} =\sum_{n}e^{\iota2\pi nx/a}V_{n} $$ where $K =\frac{2\pi n}a$ is the reciprocal lattice vector and $a$ is the lattice constant and $n =\pm ...
2
votes
1answer
75 views

Three dimensional wave packets in momentum space

I am given the 3D wave packet: $$\psi(x,y,z)=N\,\exp\left(\frac{-(x^2+y^2+2z^2)}{2a^2}\right).$$ I was asked to find N (easy enough). Then I was asked the probability that we measure $z$ greater than ...
2
votes
0answers
72 views

Momentum representation of a state

I am trying to figure out the momentum representation of the state which has the properties $$\langle \psi |\hat q |\psi \rangle=-q_0,$$ $$\langle\psi|\hat p|\psi \rangle=p_0, $$$$\Delta q\Delta ...
2
votes
1answer
47 views

Transition from coordinate space to momentum space for SHO

I am given that the ground state of the SHO in position space is given as $$\langle q|\psi_0\rangle=\frac{1}{a^{\frac12}\pi^{\frac14}}e^{-q/4a^2}$$ Where a is a constant with units of length. I am ...
4
votes
0answers
105 views

Action of Parity operator on Impulse representation

Is my derivation of the action of the parity operator $\mathbb{P}$ on the $|p\rangle$ representation correct? $$\left( \mathbb{P}\tilde\psi \right)(p)= - \tilde\psi (p).$$ Obtained from $$\left( ...
0
votes
1answer
356 views

Wave packets and the derivation of Schrodinger's equation

I studied in my class, that a plane progressive wave cannot be used to represent the wave nature of a particle as it is not square integrable. Also, the phase velocity can get above the value of $c$, ...
0
votes
1answer
82 views

Momentum representation of a function with discontinuous derivative [closed]

Consider the following wave packet $$\psi = Ce^{2\pi i p_0x/h}e^{-|x|/(2\Delta x)}$$ where $h$ is the Planck's constant and $C$ is the normalization constant. The derivative of this function is ...
9
votes
3answers
256 views

Reconstruction of “wavefunction” phases from $|\psi(x)|$ and $|\tilde \psi(p)|$

Consider a "wavefunction" $\psi(x)$, which has a Fourier transform $\tilde \psi(p)$ Suppose that we know, for each $x$, $|\psi(x)|^2$, and that we know, for each $p$, $|\tilde \psi(p)|^2$. Have we ...
3
votes
2answers
454 views

Particle in a 1D box (momentum representation)

I have this problem. I want to find the wave function in the momentum space for the particle in a 1D box. We know that the wave function in the position space is: $$Y_n(x) = A\sin{(n\pi x/L)}$$ ...
4
votes
3answers
152 views

No well-defined frequency for a wave packet?

There are similar questions to mine on this site, but not quite what I am asking (I think). The de Broglie relations for energy and momentum $$ \lambda = \frac{h}{p}, \\ \nu = E/h .$$ equate a ...
2
votes
0answers
107 views

Number theoretical function applied in physics? [closed]

I have a series of number theoretic phenomena (mathematics) that I can describe exactly by the superpositions or linear combination of the below function (I know it is an inverse Fourier type). Does ...
1
vote
2answers
1k views

From position space to momentum space

Lets say I have a state vector $\left|\Psi(t)\right\rangle$ in a position space with an orthonormal position basis. If I now use an operator $\hat{p}$ on this basis I will get basis which corresponds ...
0
votes
1answer
256 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
349 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
2k 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
256 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
845 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
426 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
1k 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 ...
10
votes
3answers
720 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 $G$ ...
7
votes
2answers
531 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
1k 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
863 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
357 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
194 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
76 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
708 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
216 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
558 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
186 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
669 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
435 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 ...
5
votes
3answers
1k 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 ...