In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. Mathematically, they are written:
where
r
{\displaystyle \mathbf {r} }
is position,
ψ
{\displaystyle \psi }
is the wave function,
u
{\displaystyle u}
is a periodic function with the same periodicity as the crystal, the wave vector
k
{\displaystyle \mathbf {k} }
is the crystal momentum vector,
e
{\displaystyle \mathrm {e} }
is Euler's number, and
i
{\displaystyle \mathrm {i} }
is the imaginary unit.
Functions of this form are known as Bloch functions or Bloch states, and serve as a suitable basis for the wave functions or states of electrons in crystalline solids.
Named after Swiss physicist Felix Bloch, the description of electrons in terms of Bloch functions, termed Bloch electrons (or less often Bloch Waves), underlies the concept of electronic band structures.
These eigenstates are written with subscripts as
ψ
n
k
{\displaystyle \psi _{n\mathbf {k} }}
, where
n
{\displaystyle n}
is a discrete index, called the band index, which is present because there are many different wave functions with the same
k
{\displaystyle \mathbf {k} }
(each has a different periodic component
u
{\displaystyle u}
). Within a band (i.e., for fixed
n
{\displaystyle n}
),
ψ
n
k
{\displaystyle \psi _{n\mathbf {k} }}
varies continuously with
k
{\displaystyle \mathbf {k} }
, as does its energy. Also,
ψ
n
k
{\displaystyle \psi _{n\mathbf {k} }}
is unique only up to a constant reciprocal lattice vector
I was thinking about a problem I had considered a long time ago in some thread, finding an example of a wave function ##\displaystyle \psi (x) =e^{iax}\phi (x)## with ##\displaystyle\phi (x)## being periodic with period ##\displaystyle L## and the corresponding Schrödinger equation...
So I thought I understood something well, and then I went to explain it to someone and it turns out I'm missing something, and I'd appreciate any insight you might have.
If I think about Bloch's theorem, it states that
ψk(r)=eik⋅ruk(r) where uk has the periodicity of the lattice. If u is...
Hi,
to describe electronic transport and for example bloch oscillations, one uses a wave-packet build of bloch waves (with a band index n and an effective mass m*).
Do these wave-packets of blochwaves also spread (disperse) over time?
Hello!
The time reversal operator, ##\hat{\Theta}## transforms a Bloch state as follows:
##\hat{\Theta} \psi_{nk}=\psi^*_{nk}##.
How does one proceed to prove the condition that ##\psi_{nk}## and ##\psi^*_{nk}## must satisfy in order for our system to be time reversal invariant?
Thanks in advance!
Homework Statement
I did not manage to get the final form of the equation. My prefactor in the final form always remain quadratic, whereas the solution shows that it is linear,
Homework Equations
w refers to wannier function, which relates to the Bloch function
##\mathbf{R}## is this case...
Homework Statement
Find band gaps for Dirac Comb potential
$$V = \sum_n aV_0(x-na) $$
Homework Equations
Bloch Theorem
$$\psi(x+a) = e^{ika}\psi(x)$$
The Attempt at a Solution
I can solve exactly up to
$$\cos(k a) = \cos(\kappa a) + \frac{2ma^2V_0}{\hbar^2}\frac{\sin(\kappa a)}{\kappa a} =...
Homework Statement
This is just a problem to help me understand. Determine the dispersion relations for the three lowest electron bands for a 1-D potential of the form
##U(x) = 2A\cos(\frac{2\pi}{a} x)##
Homework Equations
I will notate ##G, \,G'## as reciprocal lattice vectors.
$$\psi_{nk}(x)...
A Bloch wave has the following form..
## \Psi_{nk}(r)=e^{ik\cdot r}u_{nk}(r)##
The ##u_{nk}## part is said to be periodic in real space. But what about reciprocal space? I've had a hard time finding a direct answer to this question, but I seem to remember reading somewhere that the entire...
How to understand that Bloch wave solutions can be completely characterized
by their behaviour in a single Brillouin zone? Given Bloch wave:
\begin{equation*}
\psi_{\mathbf{k}}(\mathbf{r}) = u_{\mathbf{k}}(\mathbf{r}) \exp (i\mathbf{k}\mathbf{r})
\end{equation*}
I can write wavefunction for...
Homework Statement
(a) Find energies of states at ##(\frac{\pi}{a},0)##.
(b) Find secular equation
Homework EquationsThe Attempt at a Solution
Part(a)[/B]
In 1D, the secular equation for energy is:
E = \epsilon_0 \pm \left| V(x,y) \right|
When represented in complex notation, the potential...
Background information:
The wave function for an electron in a crystal lattice is modeled by a Bloch wave. A Bloch wave is a function with the periodicity of the lattice multiplied times a complex exponential function. This exponential function has a wave vector k, called the crystal momentum...