Linked Questions

2
votes
3answers
1k views

Bra Ket Notation and Derivative [duplicate]

Let $$a$$ be the partial derivative symbol with respect to $x$. What is $$\langle x|a|x \rangle$$ equal to? I think it is 0 but not sure.
3
votes
2answers
117 views

Transformation connecting two representations - Quantum mechanics [duplicate]

I am working on Dirac's paper The Lagrangian in Quantum Mechanics. He looks for the analogy between a classical transformation between two sets of coordinates and momenta $p_r$, $q_r$ and $P_r$, $Q_r$ ...
-3
votes
1answer
253 views

Momentum operator action [duplicate]

I have a density matrix defined as $$\rho(t) = \int_{-\infty}^{+\infty} \sigma(t,x) \otimes |x\rangle\langle x|$$ I want to show that $$\hat{P} \rho(t) \hat{P} - \frac{1}{2} (\hat{P^2} \rho(t) - \...
0
votes
0answers
62 views

Does this notation for the momentum in quantum mechanics make sense? [duplicate]

Let $P$ be the momentum operator. Susskind writes: $$P |\Psi \rangle =-ih \frac d {dx} |\Psi \rangle$$ Then he states that this can be rewritten as $$-ih\frac {d \psi(x)}{dx}$$ Where $\psi(x)$ is the ...
20
votes
2answers
3k 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 ...
5
votes
2answers
420 views

What is $\hat{p}|x\rangle$?

Trying to solve a QM harmonic oscillator problem I found out I need to calculate $\hat{p}|x\rangle$, where $\hat{p}$ is the momentum operator and $|x\rangle$ is an eigenstate of the position operator, ...
6
votes
2answers
519 views

How is an operator applied to a wavefunction in quantum mechanics? [closed]

If you have the Hamiltonian operator written as such: $$\hat H = -\frac{\hbar}{2m}\frac{1}{r}\frac{\partial^2}{\partial r^2}r \tag{1}$$ then to apply the Hamiltonian operator to a wavefunction, do ...
2
votes
1answer
2k views

Matrix elements of the operator $\hat{x} \hat{p}$ in position and momentum basis

I want to calculate the matrix elements of the operator $\hat{x} \hat{p}$ in momentum and position basis, that is the two quantities ($p,q$ - momenta, $x,y$ - positions): $$\langle p|\hat{x} \hat{p}|...
1
vote
3answers
367 views

Momentum operator representation

If $\hat{p}$ acts on position eigenstate, it is $$\tag{1}\hat{p}\left|x\right\rangle=+i\hbar\frac{\partial }{\partial x}\left|x\right\rangle .$$ But in general $$\tag{2}\hat{p} = -i\hbar \frac{\...
2
votes
2answers
409 views

Quantum field theory for the gifted amateur: problem 2.4

I am trying to do problem 2.4 in the book "Quantum field theory for the gifted amateur". I have a math background but little training in physics. I am asked to use the identity $$\langle x \mid p \...
3
votes
1answer
417 views

Momentum operator expression

In the course of calculating $$\langle x|\hat{p}|\psi\rangle$$ I have a step which is: $$ \langle x|\frac{\hbar}{i}\frac{d}{dx}|\psi\rangle=\frac{\hbar}{i}\frac{d}{dx}\langle x|\psi\rangle.$$ What is ...
1
vote
4answers
266 views

Calculating $\langle x | \hat{x} | p \rangle$ in $p$ basis

I am trying to calculate $\langle x\ |\ \hat{x}\ |\ p\rangle$. I can work in the $x$-basis like so: $$\langle x\ |\ \hat{x}\ |\ p\rangle=\int dx'\langle x\ |\ \hat{x}\ |\ x'\rangle\langle x'\ |\ p\...
3
votes
2answers
321 views

Space translation of operators, states, and particle densities

In Sidney Coleman's Lectures he talked about space translations such that $$\tag{1} e^{ia P}\rho(x) e^{-ia P} ~=~ \rho(x-a),$$ but when I expanded the exponentials and used the commutation relation ...
1
vote
2answers
374 views

Understanding momentum space and position space

I'm starting a more 'formal' course on quantum mechanics, and we're going through momentum and position space, which I'm kind of confused on. I would just like clarification on this since I'm not ...
1
vote
3answers
312 views

What's the correct link between Dirac notation and wave mechanics integrals?

In wave mechanics when we compute the expectation value of energy we write the following $$\left<\hat{H}\right>=\int_{-\infty}^\infty\mathrm{d}x\ \psi^*(x)\hat{H}\psi(x)=\int_{-\infty}^\infty\...

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