All Questions
40
questions
12
votes
2
answers
7k
views
Proving Gauge invariance of Schrodinger Equation
I am trying to proof explicitly that Schrodinger equation:
$$ i\hbar \partial_t \psi = \left[ -\frac{1}{2m}\left(\frac{\hbar}{i}\nabla-q\vec{A}\right)^2+qV \right]\psi$$
remains the same under the ...
- 980
12
votes
1
answer
3k
views
Lorentz force derivation in quantum mechanics [closed]
In Sakurai and Napolitano, chapter 2, there's a derivation of the QM Lorentz force.
Given $$H=\frac{1}{2m}\left(\mathbf{p}-\frac{e\mathbf{A}}{c}\right)^2+e\phi = \frac{\mathbf{\Pi}^2}{2m}+e\phi$$
...
- 877
8
votes
1
answer
5k
views
The gauge-invariance of the probability current
It is simple to show that under the gauge transformation $$\begin{cases}\vec A\to\vec A+\nabla\chi\\
\phi\to\phi-\frac{\partial \chi}{\partial t}\\
\psi\to \psi \exp\left(\frac{iq\chi}{\hbar}\right)\...
- 81
8
votes
1
answer
406
views
Quantum Cyclotron Frequency - Why is it off by a factor of 2?
Say you have a magnetic field $\vec{B}=(0,0,B_0)$. Then the Schrodinger Equation Hamiltonian for a spin-2 particle of charge $e$ moving in this field is:
$$H = \frac{1}{2m}[\vec{p}-e\vec{A}]^2-\vec{\...
- 1,212
5
votes
1
answer
598
views
Aharonov-Bohm Effect in Torus
I had a very brief introduction to the Aharonov-Bohm effect in class. The lecturer introduced the notion that $H(\Phi=\Phi_0)$ and $H(\Phi=0)$ gives identical energy spectrum and that the Hamiltonians ...
- 87
4
votes
1
answer
2k
views
How to solve the Schrödinger equation with magnetic field?
The Schrödinger equation of electron in a magnetic field is
$$
\frac{1}{2m} \left(-\mathrm{i}\hbar\nabla+\frac{e}{c}\mathbf{A}\right)^2 \psi + V\psi = E\psi
$$
where $V=-e\phi$ and $\phi$ is the ...
- 369
4
votes
0
answers
1k
views
Landau quantization: degeneracy of first level
In some books the degeneracy of one Landau level in a two-dimensional gas of free electrons is calculated in the following way:
Note: The electron spin is not considered.
Number of states of a free ...
- 171
3
votes
1
answer
5k
views
How do we derive the minimal coupling Hamiltonian?
Is there a way to rigorously derive the minimal coupling hamiltonian for a system interacting with electromagnetic radiation. How de we arrive at the expression:
$$\hat{H} = \frac{1}{2m}(p-\frac{q}{c}...
- 352
3
votes
1
answer
833
views
Symmetry in Landau's problem
Suppose a particle is forced to move in the $x-y$ plane, under a constant magnetic field $\vec{B} = B\hat{z}$. The Hamiltonian can the be written as
$$
H = \frac{\Pi^2}{2m}
$$
where $\vec{\Pi} = \vec{...
- 980
3
votes
1
answer
583
views
Obtaining quantum Hamiltonian for charged particle from path integral formulation
I was working on Shankar 8.6.4, which is about obtaining in one dimension the Hamiltonian operator of a charged particle from the path integral formulation.
First, I get the propagator over a time ...
- 557
2
votes
2
answers
121
views
Wave equation solution satisfied? [closed]
I'm having a problem trying to show that this solution satisfies the wave equation. I discovered this solution given by $$\psi(x,t)=e^{-(ax-bt)^2}$$ but I'm stuck trying to prove that solution ...
- 23
2
votes
1
answer
134
views
Problem in deriving Pauli equation
I’m trying to derive the equation from the Hamiltonian
$$H=\frac{1}{m}[\vec{\sigma} \cdot (\vec p-q\vec A)]^2+ q \phi$$
and
$$[\vec{\sigma} \cdot (\vec p-q\vec A)]^2= (\vec p-q\vec A)^2-iq \vec{\...
- 483
2
votes
1
answer
88
views
Migdal's problem about rotating a particle in a magnetic field [closed]
I was given a problem by my professor, which belongs to Migdal. The problem is as follows:
If a particle is rotated by 2$\pi$ in a magnetic field its
wave-function $\psi$ transforms into $\exp(i\...
- 389
2
votes
1
answer
3k
views
Derive probability current density - factors of 2 discrepancy [closed]
To derive the probability current density for a particle
in an electromagnetic field, we calculate
$\dfrac{\partial \rho}{\partial t}
=
\dfrac{\partial}{\partial t} (\Psi^* \Psi)
=
\dfrac{\partial \...
- 307
2
votes
1
answer
448
views
Landau levels in rotationally invariant gauge
I try to find wave-function of electron in external constant magnetic field in gauge
$$A=\frac{B}{2}(-y,x,0).$$
I substitute anzats, $\psi=e^{-i\omega t}e^{ip_zz}F(x,y)$. Then, I rewrite equation in ...
- 2,664