Linked Questions

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3answers
6k views

Derivation of momentum operator [duplicate]

From a video lecture on quantum mechanics at MIT OCW I found that you didn't need to know Schrödinger's equation to know the momentum operator which is $\frac{\hbar}{i}\frac{\partial}{\partial x}$. ...
1
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1answer
1k views

Why is quantum mechanical momentum the derivative of the wave function with respect to the position? [duplicate]

In classical mechanics the momentum is defined as mass times the time-derivative of position. In quantum mechanics, however, the time-derivative of the wave function is the hamiltonian, while the ...
0
votes
1answer
47 views

Does the canonical commutation relation give a unique solution for the momentum operator? [duplicate]

So lets say we are in a 1d system and in the position basis just for simplicity. The CCR is: $$ [x,p]=i $$ and the momentum operator is $-i\partial_x$. Is this solution unique or are there other ...
34
votes
6answers
47k views

How to get the position operator in the momentum representation from knowing the momentum operator in the position representation?

I know that $$\tag{1}\hat{p}~=~-i\hbar \frac{\partial}{\partial x}~.$$ How can I get $$\tag{2}\hat{x}~=~i\hbar \frac{\partial}{\partial p}~?$$ I think this simple and I'm just over thinking it, ...
16
votes
7answers
3k views

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. ...
8
votes
3answers
2k views

Derivation of position operator in QM

For the definition of the momentum operator $$\hat{P } = -i \hbar \nabla$$ in quantum mechanics, as I understand you can derive this by either considering a more general definition of momentum, i.e. '...
1
vote
1answer
429 views

Commutator identities and Fourier transform

Is it possible to derive one side of the arrow below from the other by using only the Fourier transform and its reciprocal? $$[\hat{p},f(\hat{x})]=-i\hbar f'(\hat{x}) \leftrightarrow [\hat{x},f(\hat{...
0
votes
2answers
255 views

Determine $p_x$ from $[x,p_x]=i\hbar $ [closed]

With $[x,p_x]=i\hbar $, how to determine the form of the operator $p_x$?
0
votes
0answers
412 views

How to prove $\hat{p}|x\rangle=i\hbar\frac{\partial}{\partial x}|x\rangle$,using $[\hat{x},\hat{p}]=i\hbar$? [duplicate]

How to prove $$\hat{p}|x\rangle=i\hbar\frac{\partial}{\partial x}|x\rangle,$$ using $$[\hat{x},\hat{p}]=i\hbar~?$$ The question seems to be uncomplete because for any $f(x)$ $$[\hat{x},\hat{p}+f(x)]=i\...
4
votes
1answer
150 views

Momentum operator ambiguous?

In nonrelativistic quantum mechanics, are different operators possible as a candidate for the momentum operator, given that one has fixed one position operator and a hilbert space that this position ...
2
votes
1answer
101 views

Common observables and associated operators: operator momentum [duplicate]

Starting from my previous question Commutators in quantum mechanics and considering that the commutator $$\left[i\hbar\frac{\partial}{\partial x},x\right]=i\hbar, \tag{1}$$ the associated linear ...
1
vote
1answer
149 views

Eq. (2.4.7) of Weinberg, “Lectures on QM” from eq. (2.4.6)

Could anyone please tell me the proof of (2.4.7), using (2.4.6)? Substituting (2.4.7) into the right equation of (2.4.4), we get: $$P_1 = -i \hbar \left( \frac{\partial}{\partial x_{1e}} + \frac{\...
0
votes
0answers
72 views

How can $\hat p = - i \hbar \partial_q$ be derived starting from the definitions of $\hat q$ and $\hat p$ in terms of creation/destruction operators? [duplicate]

Consider the position and momentum operators $\hat q$ and $\hat p$, defined respectively in terms of creation and destruction operators in the usual way: $$ \hat q = c (\hat a + \hat a^\dagger), \...