Commutator Relations: [x,p]=ih, Proof of p=-iħ∂/∂x+f(x)

alisa
Messages
3
Reaction score
0
given that [x,p]=ih, show that if x=x, p has the representation p=-iħ∂/∂x+f(x) where f(x) is an arbitrary function of x
 
Physics news on Phys.org
alisa, you're supposed to show an attempted solution at the problem. That goes for your other threads as well.

alisa said:
given that [x,p]=ih, show that if x=x,

What do you mean " if x=x". x=x by definition.
 
Tom Mattson said:
What do you mean " if x=x". x=x by definition.

It's the coordinate representation in which the Hilbert space is L^{2}(\mathbb{R},dx). The "x" operator is realized by a multiplication by "x". She's asked to prove that the most general representation of the momentum operator in this Hilbert space is the one written there.
 
OK, so then it should read something like "If \hat{x}|\psi>=x|\psi>...", right?
 
Last edited:
Exactly. That's the spectral equation, but nonetheless, yes.
 

Thread 'Please tell me where I am going wrong in this integral'

##|\Psi|^2=\frac{1}{\sqrt{\pi b^2}}\exp(\frac{-(x-x_0)^2}{b^2}).## ##\braket{x}=\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dx\,x\exp(-\frac{(x-x_0)^2}{b^2}).## ##y=x-x_0 \quad x=y+x_0 \quad dy=dx.## The boundaries remain infinite, I believe. ##\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dy(y+x_0)\exp(\frac{-y^2}{b^2}).## ##\frac{2}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,y\exp(\frac{-y^2}{b^2})+\frac{2x_0}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,\exp(-\frac{y^2}{b^2}).## I then resolved the two...
View full post »

Thread 'Direction of the magnetic field of an oscillating charge'

Hello everyone, I’m considering a point charge q that oscillates harmonically about the origin along the z-axis, e.g. $$z_{q}(t)= A\sin(wt)$$ In a strongly simplified / quasi-instantaneous approximation I ignore retardation and take the electric field at the position ##r=(x,y,z)## simply to be the “Coulomb field at the charge’s instantaneous position”: $$E(r,t)=\frac{q}{4\pi\varepsilon_{0}}\frac{r-r_{q}(t)}{||r-r_{q}(t)||^{3}}$$ with $$r_{q}(t)=(0,0,z_{q}(t))$$ (I’m aware this isn’t...
View full post »

Thread 'Initial conditions for orbits around a wormhole'

Hi, I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem. Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$ Where ##b=1## with an orbit only in the equatorial plane. We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$ Ultimately, I was tasked to find the initial...
View full post »

Similar threads

Eigenfunction of momentum and operators
Replies
14
Views
3K
I Commutators, operators and eigenvalues
Replies
7
Views
1K
Expected Energy Value of a Time Independent Wave
Replies
2
Views
2K
Proving the Commutation Relation for Quantized Boson in a One-Dimensional Box
Replies
1
Views
1K
Solution to Show Relation of X, P with Representation of P=-ih/2π*∂/∂x +f(x)
Replies
3
Views
8K
Time Inversion Symmetry and Angular Momentum
Replies
1
Views
2K
Show that: Commutator relations (QM)
Replies
8
Views
3K
Are These Hermitian Conjugates Also Hermitian Operators?
Replies
4
Views
9K
What Are the Commutation Relations of \( \hat{R}^2 \) with \( \hat{L} \)?
Replies
1
Views
4K
General commutation relations for quantum operators
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top