Linked Questions
10 questions linked to/from Matrix elements of momentum operator in position representation
43
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3
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7k
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What's wrong with this derivation that $i\hbar = 0$?
Let $\hat{x} = x$ and $\hat{p} = -i \hbar \frac {\partial} {\partial x}$ be the position and momentum operators, respectively, and $|\psi_p\rangle$ be the eigenfunction of $\hat{p}$ and therefore $$\...
- 1,270
26
votes
7
answers
6k
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Is the Momentum Operator a Postulate?
I've been studying the postulates of QM and seeing how to derive important ideas from them. One thing that I haven't been able to derive from them, however, is the identity of the momentum operator.
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- 3,012
11
votes
4
answers
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Is the momentum operator diagonal in position representation?
The matrix elements of the momentum operator in position representation are:
$$\langle x | \hat{p} | x' \rangle = -i \hbar \frac{\partial \delta(x-x')}{\partial x}$$
Does this imply that $\langle x |...
- 3,802
12
votes
2
answers
2k
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Is every observable a function of position and momentum?
In the first answer of this question it is said that every quantum observable, let's say $\hat{A}$, can be represented as a function of position and momentum observables. In other words, as I ...
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8
votes
2
answers
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What is the most general expression for the coordinate representation of momentum operator?
I have a question about deriving the coordinate representation of momentum operator from the canonical commutation relation, $$[x,p]= i.$$
One derivation (ref W. Greiner's Quantum Mechanics: An ...
- 6,451
4
votes
1
answer
3k
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Matrix elements of the operator $\hat{x} \hat{p}$ in position and momentum basis
I want to calculate the matrix elements of the operator $\hat{x} \hat{p}$ in momentum and position basis, that is the two quantities ($p,q$ - momenta, $x,y$ - positions):
$$\langle p|\hat{x} \hat{p}|...
- 2,875
3
votes
1
answer
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Commutation relation of position and momentum using Dirac notation
This is likely a very trivial/silly question, but in following a derivation of the position and momentum commutation relation using the dirac notation, I am having trouble justifying a certain step.
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- 127
1
vote
3
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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
$$\left<\hat{H}\right>=\int_{-\infty}^\infty\mathrm{d}x\ \psi^*(x)\hat{H}\psi(x)=\int_{-\infty}^\infty\...
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-1
votes
1
answer
818
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Off-diagonal elements of momentum operator in position representation
In another Phys.SE question, I've proposed the next-cited proof of this statement:
the momentum matrix elements in position representation, $\langle x'|\hat{p}|x\rangle$, are all not null
I'm ...
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3
votes
2
answers
311
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Integrating $\int \frac{e^{ikx}}{x} dk$ by parts to get delta function derivative, how to handle undefined boundary terms?
I'm going through Sergio Dutra's Cavity Electrodynamics: The Strange Theory of Light in a Box. In equation (2.31) he computes:
$$\begin{aligned}\langle x|\hat{p}|x'\rangle&=i\hbar\int\frac{dk}{2\...
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