Periodic potential - energy bands

[Konte]

Hello everybody,

I have some questions about treatment of Schrodinger equation where ## \hat{V}(\theta)##, the potential energy part of Hamiltonian ##\hat{H}=\hat{T}(\theta)+\hat{V}(\theta)## is a trigonometric function like:
##\hat{V}(\theta) = a sin(\theta)##
or
##\hat{V}(\theta) = a cos(\theta)+ b sin(c\theta)## where ##\theta## is an angular variable.
I read something in solid-state physics that a system which evolve inside a periodical potential ends up with energy bands as eigenvalues solutions.

Do I have the same case here, with those two examples of potential energy?
In other words, will I obtain energy band too, even here I have nothing to do with lattice nor crystals?

Thank you everybody.

Konte

 

[Jilang]

I wouldn't think so. The potential is just a step function, the height of which depends on the direction.

 

[Spinnor]

Interesting potential. For a=b and c = 3.4 Wolfram Alpha will plot your function,

http://www.wolframalpha.com/input/?i=plot+y+=+cos(theta)+++sin(3.4xtheta)

Change c.

 

[houlahound]

The potential would be easier to follow with vectors. Is that from a point or repeats or?

Sorry I need more spatial information. What am I missing?

 

[stevendaryl]

Hello everybody,

I have some questions about treatment of Schrodinger equation where ## \hat{V}(\theta)##, the potential energy part of Hamiltonian ##\hat{H}=\hat{T}(\theta)+\hat{V}(\theta)## is a trigonometric function like:
##\hat{V}(\theta) = a sin(\theta)##
or
##\hat{V}(\theta) = a cos(\theta)+ b sin(c\theta)## where ##\theta## is an angular variable.

That's not a periodic potential. When people call a potential periodic, what they mean is that there is some vector [itex]\vec{\delta r}[/itex] such that [itex]V(\vec{r} + \vec{\delta r}) = V(\vec{r})[/itex]. For example, in one dimension, if the potential looks like this: [itex]V(x) = a cos(kx)[/itex], then [itex]V(x+\frac{2\pi}{k}) = V(x)[/itex]

The potential [itex]V = a sin(\theta)[/itex] isn't periodic in this sense. Yes, [itex]V(\theta + 2 \pi) = V(\theta)[/itex], but the angle [itex]\theta + 2\pi[/itex] is the same angle as [itex]\theta[/itex].

 

[houlahound]

Check out this baby;

http://demonstrations.wolfram.com/SchroedingerWavefunctionsInAContinuouslyVaryingPotential/

Or

http://demonstrations.wolfram.com/BandSpectrumInAPeriodicPotential/

 

[Konte]

Thanks everybody for all of your answers.

The potential would be easier to follow with vectors. Is that from a point or repeats or?

Sorry I need more spatial information. What am I missing?

To be clear, it's about an angular (thus periodic) potential that can hinder a quantum rotor.

As an example, I show here the case of ## \hat{V}(\theta)\,=\, 3\,-\,sin(7\,\theta)##.

That's not a periodic potential. When people call a potential periodic, what they mean is that there is some vector [itex]\vec{\delta r}[/itex] such that [itex]V(\vec{r} + \vec{\delta r}) = V(\vec{r})[/itex]. For example, in one dimension, if the potential looks like this: [itex]V(x) = a cos(kx)[/itex], then [itex]V(x+\frac{2\pi}{k}) = V(x)[/itex]

The potential [itex]V = a sin(\theta)[/itex] isn't periodic in this sense. Yes, [itex]V(\theta + 2 \pi) = V(\theta)[/itex], but the angle [itex]\theta + 2\pi[/itex] is the same angle as [itex]\theta[/itex].

So the word "periodic" has two different senses, that I want to understand. However, it stays really blurred for me. In looking at the case of one dimension as you shown, mathematically, we cannot make a difference between:

## V(x) = a cos(kx) ## and ## V(\theta) = a cos(n \theta)##, since ##\theta## and ##x## are dummies variables.

So I am troubled when the first gives band structure and the second doesn't.

Thank you much.

Konte

 

[stevendaryl]

## V(x) = a cos(kx) ## and ## V(\theta) = a cos(n \theta)##, since ##\theta## and ##x## are dummies variables.

[itex]x[/itex] and [itex]\theta[/itex] are not interchangeable, though, so they aren't completely "dummy variables". In 1-D quantum mechanics, it is assumed that space is described by a coordinate [itex]x[/itex] that runs from [itex]-\infty[/itex] to [itex]+\infty[/itex]. In contrast, the angular variable [itex]\theta[/itex] runs from [itex]-\pi[/itex] to [itex]+\pi[/itex], and the point [itex]\theta = -\pi[/itex] is identified with the point [itex]\theta = +\pi[/itex]. These differences lead to different constraints on the wave function. In terms of [itex]x[/itex], it must be the case that [itex]lim_{x \rightarrow \pm \infty} \psi(x) = 0[/itex]. In terms of [itex]\theta[/itex], it must be the case that [itex]\psi(\theta) = \psi(\theta + 2\pi)[/itex]. So they are very different types of variables, and they have different consequences.

On the other hand, if you demand that EVERYTHING be periodic in [itex]x[/itex] with period [itex]L[/itex]---the wave function, the potential, everything---then the distinction disappears; In that case you can identify [itex]x[/itex] with the angular variable [itex]\theta = \frac{2\pi x}{L}[/itex].

 

[Konte]

[itex]x[/itex] and [itex]\theta[/itex] are not interchangeable, though, so they aren't completely "dummy variables". In 1-D quantum mechanics, it is assumed that space is described by a coordinate [itex]x[/itex] that runs from [itex]-\infty[/itex] to [itex]+\infty[/itex]. In contrast, the angular variable [itex]\theta[/itex] runs from [itex]-\pi[/itex] to [itex]+\pi[/itex], and the point [itex]\theta = -\pi[/itex] is identified with the point [itex]\theta = +\pi[/itex]. These differences lead to different constraints on the wave function. In terms of [itex]x[/itex], it must be the case that [itex]lim_{x \rightarrow \pm \infty} \psi(x) = 0[/itex]. In terms of [itex]\theta[/itex], it must be the case that [itex]\psi(\theta) = \psi(\theta + 2\pi)[/itex]. So they are very different types of variables, and they have different consequences.

On the other hand, if you demand that EVERYTHING be periodic in [itex]x[/itex] with period [itex]L[/itex]---the wave function, the potential, everything---then the distinction disappears; In that case you can identify [itex]x[/itex] with the angular variable [itex]\theta = \frac{2\pi x}{L}[/itex].

Ok, thanks a lot.