Questions tagged [bloch-oscillation]
22 questions
25
votes
4 answers
Crystal momentum and the vector potential
I noticed that the Aharonov–Bohm effect describes a phase factor given by $e^{\frac{i}{\hbar}\int_{\partial\gamma}q A_\mu dx^\mu}$. I also recognize that electrons in a periodic potential gain a phase factor given by…
Alex Eftimiades
- 643
- 4
- 11
9
votes
1 answer
What exactly is Crystal Momentum, $\hbar k$?
The title says it all really.
Does this mean that the crystal is moving?
From my notes, I read that
The effect of an external force on an electron in the crystal is to
change the crystal momentum $\hbar k$. In the absence of a force, the crystal…
user267061
7
votes
3 answers
How does Bloch's theorem generalize to a finite sized crystal?
I would be fine with a one dimensional lattice for the purpose of answering this question. I am trying to figure out what more general theorem (if any) gives Bloch's theorem as the number of unit cells-->infinity.
Alex Eftimiades
- 643
- 4
- 11
6
votes
3 answers
Peierls Substitution with Time-Dependent Vector Potential
My question is whether Peierls substitution really holds true for time-dependent electromagnetic (EM) potentials and, if yes, why.
To implement an electromagnetic field in a condensed matter system described by a Bloch Hamiltonian, I have often seen…
Fred
- 195
4
votes
1 answer
Matrix elements $\langle n,k|x|n',k'\rangle$ for Bloch states
I believe this is just elementary QM, but I'm getting awfully confused. The question is drawn from this paper on Wannier-Stark localization (but is self-contained):…
dsfkgjn
- 127
4
votes
1 answer
Advanced atomic physics: From Liouville Equations to the Bloch equations
I'm trying to derive the Bloch equations from the Liouville equation. This should be possible according to this paper, which discusses higher order Bloch equations (second order spherical tensors). I'm trying to derive the same for the simple vector…
The Quantum Physicist
- 3,517
3
votes
2 answers
Two different expressions for Rabi Frequency (on resonance)
I've been searching through several articles and books and I found two different expressions for the Rabi Frequency on the semiclassical rotating-wave approximation, and they are different by a factor of two.
Consider a eletric field…
3
votes
2 answers
Bloch oscillations - Scattering to other bands
In the free electron approximation, a Bloch state $|k\rangle$ is the linear superposition of free plane wave states $\sum_G C_G(k) |k+G\rangle$, where $G$ are the conjugate lattice. Since the coefficients $C_G$ varies with $k$, under an electric…
felix
- 1,776
2
votes
1 answer
Is current density independent of applied fields for Bloch electrons?
Following Ashcroft-Mermin chapter 12 the semiclassical dynamics is governed by
$
\dot{\vec{r}} = \vec{v}_n(\vec{k}) = \frac{1}{\hbar}\frac{\partial \epsilon_n(\vec{k})}{\partial \vec{k}}
$
and
$
\hbar \dot{\vec{k}} = -e\vec{E} -\frac{e}{c} …
Uphyscs
- 49
2
votes
2 answers
In the context of bloch waves, where does the formula $| k \rangle = A\sum_{X} e^{ikX} |X \rangle$ come from?
I am following a course on condensed matter physics. In our lecture notes, the lecturer has postulated the following formula in the context of Bloch waves:
$$| k \rangle = A\sum_{X} e^{ikX} |X \rangle$$
It is either assumed knowledge or bad practice…
Mikkel Rev
- 1,470
2
votes
1 answer
Bloch waves at large momenta
I am trying to come to grip with some solid state theory. Bloch waves, energy eigenstates for hamiltonians with lattice periodic potential in $\mathbb R^d$, are frequently written as
$$\phi_{n,k}(r)=e^{2\pi ik\cdot r}u_{n,k}(r)$$ with $u_{n,k}$…
plm
- 191
2
votes
1 answer
Wannier functions on a ring
Let's say I have a single particle hamiltonian in a periodic potential, for example a 1D lattice such that:
$$H = -\frac{\partial_x^2}{2m} + V(x) $$
with $ V(x+a) = V(x)$ where $a$ is the lattice spacing between the atoms or sites.
It is known by…
Koby Yavilberg
- 117
1
vote
1 answer
Understanding electric conduction in tight binding model
Let's consider a system of free electrons moving in a one dimensional lattice with dispersion $\varepsilon(k) = -2t\cos{ka}$, ($a$ is the lattice spacing and $t$ the hopping amplitude). Let's now superimpose a uniform force $F$ to the system, so…
Matteo
- 3,261
1
vote
2 answers
Why is the relaxation of coherence rate half the spontaneous emission rate?
Consider a two-level atom of which the lower and upper levels are denoted, respectively, a and b. If spontaneous emission from the upper to the lower level is the only source of relaxation, then the rate of change of the diagonal density matrix…
Nicolas Schmid
- 1,150
1
vote
1 answer
Solving Schrodinger Equation for scattering off a periodic potential
I am interesting in solving the time-independent Schrodinger equation (TISE) for the scenario where we have an electron plane wave of fixed energy incident upon a potential that is infinite and periodic only the transverse plane (i.e. a very thin or…
bdforbes
- 83