1
vote
1answer
71 views

Inner product of position and momentum eigenkets

Let's define $\hat{q},\ \hat{p}$ the positon and momentum quantum operators, $\hat{a}$ the annihilation operator and $\hat{a}_1,\ \hat{a}_2$ with its real and imaginary part, such that $$ \hat{a} = ...
1
vote
0answers
36 views

Transition Between Position and Momentum Basis

I'm having some trouble following pages 55-56 of Sakurai's Modern Quantum Mechanics. We're trying to transfer from position space into momentum space. Here's a quote: Let us now establish the ...
0
votes
2answers
109 views

Representations in quantum mechanics [closed]

This might be a very simple question. I just want someone to point me the right direction to understand things like this: $$ \langle x|x'\rangle=\delta(x-x') \\ \psi(x)=\langle x|\psi\rangle \\ ...
0
votes
1answer
79 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 ...
4
votes
0answers
137 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( ...
3
votes
0answers
78 views

Non-Hermiticity when Fourier transforming onto a finite lattice

I'm doing numerical simulations. I have the Haldane model in a honeycomb lattice where $$ H = \sum \limits_{<ij>}a^\dagger_i b_j + h.c $$ Where $i$ belongs to sublattice $A$, and $j$ to ...
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 ...
7
votes
2answers
573 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 ...