Periodic Boundary Conditions and which Hamiltonian to use

[rwooduk]

Homework Statement

Example Question: an electron with mass m is confined in a thin wire, with periodic boundary conditions applied in the x direction and harmonic potentials in the y and z direction. Write an expression for the wave functions in the ground state. Write down all the energy eigen values.

Homework Equations

Schrodinger Equation
aΨ₀=0

The Attempt at a Solution

I have no trouble deriving the wave function for the periodic boundary conditions.

Question:

1. I've used the same method to derive the harmonic potential wave functions as a simple harmonic oscillator, are they the same?

2. I'm having real trouble with knowing the hamiltonian to use for periodic boundary conditions. I (now) understand that the only thing that changes in the hamiltonian for a system is the potential term (and time depencancy). However what would the potential term be for a periodic boundary condition system?

This is a revision question for exams (not assessed).

Thanks in advance for any guidance on this.

 

[Simon Bridge]

I have no trouble deriving the wave function for the periodic boundary conditions.
...
I'm having real trouble with knowing the hamiltonian to use for periodic boundary conditions

... these two statements seem to contradict each other: how can you "derive" the wavefunction if you don't know the Hamiltonian? Have you tried doing it backwards - work from the wavefunction to the Hamiltonian?

Say the wire is length L, what is the potential inside the length L?
Put V= that potential in the schrodinger equation and you have your Hamiltonian.

I've used the same method to derive the harmonic potential wave functions as a simple harmonic oscillator, are they the same?

... yes.
You should have something about this in your notes - it's just like when you saw a 3D particle in a box, you took infinite square-well wavefunctions, independently, in each direction.

 

[rwooduk]

... these two statements seem to contradict each other: how can you "derive" the wavefunction if you don't know the Hamiltonian? Have you tried doing it backwards - work from the wavefunction to the Hamiltonian?

Hi, I derived the wavefunction for the periodic conditions by starting with the general form X(x) = A sin kx + B cos kx = A exp (ikx) + B exp (-ikx) and then applying the boundary conditions X(L) = X(0) for B=0 etc. I derived the wavefunction for the harmonic oscillator by applying the annihilation operator to an unknown ground state wave function Y(y).

Say the wire is length L, what is the potential inside the length L?
Put V= that potential in the schrodinger equation and you have your Hamiltonian.

This is what I'm struggling with, how would you know that potential? I could derive the potential at a point r in the wire using last years electromagnetism, but I'm sure that would not be the right way to do it. I really have a mental block with this. Also, other questions simply say the system is periodic, is there a general potential for this case?

... yes.
You should have something about this in your notes - it's just like when you saw a 3D particle in a box, you took infinite square-well wavefunctions, independently, in each direction.

Excellent, yes I see now, thanks!

 

[Simon Bridge]

Hi, I derived the wavefunction for the periodic conditions by starting with the general form X(x) = A sin kx + B cos kx = A exp (ikx) + B exp (-ikx) and then applying the boundary conditions X(L) = X(0) for B=0 etc. I derived the wavefunction for the harmonic oscillator by applying the annihilation operator to an unknown ground state wave function Y(y).

... so basically you guessed at the general form of the wavefunction without understanding why it works.
You know what sort of potential gives rise to HO solutions - what sort of potential would normally have the sine/cosine solutions?

This is what I'm struggling with, how would you know that potential? I could derive the potential at a point r in the wire using last years electromagnetism, but I'm sure that would not be the right way to do it. I really have a mental block with this. Also, other questions simply say the system is periodic, is there a general potential for this case?

... in this case you would want to model the wire as a perfect conductor with no internal structure.
Consider: the infinite square well is the model for a block of conducting material where the walls perfectly reflect.
If you took a long thin bit of such a material, then joined the ends together, what would the boundary conditions be like?

 

[rwooduk]

... so basically you guessed at the general form of the wavefunction without understanding why it works.
You know what sort of potential gives rise to HO solutions - what sort of potential would normally have the sine/cosine solutions?

to be honest you are right, I had always assumed that since the particle could be considered a wave then X(x) = A sin kx + B cos kx is used as it's a solution to the wave equation, is that not correct? the solution would also be of the sine/cosine form. A free particle / wave where there was no potential and a confined particle / wave where there was an infinite potential would both give sine/cosine solutions. In fact thinking about it wouldn't all potentials give sine/cosine solutions in one form or another.

... in this case you would want to model the wire as a perfect conductor with no internal structure.
Consider: the infinite square well is the model for a block of conducting material where the walls perfectly reflect.
If you took a long thin bit of such a material, then joined the ends together, what would the boundary conditions be like?

the solution would take integer values i.e. a certain number of wavelenths would "fit" into the wire, not sure how you would describe this as a boundary condition.

thanks for the help

 

[Simon Bridge]

to be honest you are right, I had always assumed that since the particle could be considered a wave then X(x) = A sin kx + B cos kx is used as it's a solution to the wave equation, is that not correct?

Not in general, no.
1. It is not useful at this stage to think of the particle as a wave.
2. X (x) given above is not the solution to the HO potential.

the solution would also be of the sine/cosine form. A free particle / wave where there was no potential and a confined particle / wave where there was an infinite potential would both give sine/cosine solutions. In fact thinking about it wouldn't all potentials give sine/cosine solutions in one form or another.

The free particle wave function is $\psi (x) = 1$ ... a beam of particles would normally be modeled by plane-waves.

the solution would take integer values i.e. a certain number of wavelenths would "fit" into the wire, not sure how you would describe this as a boundary condition.

You need to get used to writing a physical situation in maths. You can probably sketch it using your intuition for standing waves... its the same as the harmonics on the rim of a bell. When you do the sketch, you start out assuming some value at some point, then follow that assuption to its logical conclusion. Same with the maths.

A boundary condition can relate values in one part of the wire to another.
If x is a distance around the wire from some reference point, then x=L is the same place as x=0.
How would you write the relationship between the wavefunction at these two coordinates?

 

[rwooduk]

Not in general, no.
1. It is not useful at this stage to think of the particle as a wave.
2. X (x) given above is not the solution to the HO potential.


The free particle wave function is $\psi (x) = 1$ ... a beam of particles would normally be modeled by plane-waves.


You need to get used to writing a physical situation in maths. You can probably sketch it using your intuition for standing waves... its the same as the harmonics on the rim of a bell. When you do the sketch, you start out assuming some value at some point, then follow that assuption to its logical conclusion. Same with the maths.

A boundary condition can relate values in one part of the wire to another.
If x is a distance around the wire from some reference point, then x=L is the same place as x=0.
How would you write the relationship between the wavefunction at these two coordinates?

Great, many thanks for the insight! To answer your question it would be Ψ(0) = Ψ(L) but can't see how that would help me determine the potential for the wire?

thanks again

 

[Simon Bridge]

Great, many thanks for the insight! To answer your question it would be Ψ(0) = Ψ(L) but can't see how that would help me determine the potential for the wire?

... also $\frac{\partial}{\partial x}\psi(0) = \frac{\partial}{\partial x}\psi(L)$ : the amplitude and gradient have to be the same.
It does not help you find the potential - sorry: I was expecting you to recognize the periodic boundary conditions ;)
What tells you the potential inside the wire is that it is a (classically ideal) conductor.
Think back to classical electrostatics: what is the electric potential inside any conductor?

 

[rwooduk]

... also $\frac{\partial}{\partial x}\psi(0) = \frac{\partial}{\partial x}\psi(L)$ : the amplitude and gradient have to be the same.
It does not help you find the potential - sorry: I was expecting you to recognize the periodic boundary conditions ;)
What tells you the potential inside the wire is that it is a (classically ideal) conductor.
Think back to classical electrostatics: what is the electric potential inside any conductor?

ahh ok thanks! and classically? it's constant, just kQ/r would this fit to the potential term in the Hamiltonian?

just to recap (for myself and anyone reading) I'm trying to find the potential term for the Hamiltonian for a wire, so that it may be applied to the wave function already derived from periodic boundary conditions.

 

[rwooduk]

Does anyone else have any ideas of what form of Hamiltonian you would use given the wave function of a periodic system, such as a wire?

the closest thing I've found is this:

http://rci.rutgers.edu/~ke116/Teaching_files/bloch.pdf [Broken]

but I'm pretty sure the question just wants a simple expression for the potential.

 

[TSny]

Does anyone else have any ideas of what form of Hamiltonian you would use given the wave function of a periodic system, such as a wire?

My interpretation of the problem is that you have a particle that is free to move along the length of the wire (along the x axis, 0<x<L). But the particle is not free to move in directions perpendicular to the axis of the wire (y and z direction). Being "free" along the x-axis should tell you something about the x-dependence of the potential energy function. Likewise, "harmonic" in the y and z directions tells you about the y and z dependence of the potential energy function. You should be able to write a mathematical expression for the potential energy function V(x, y, z) based on this information. Then you can write down the complete Hamiltonian.

Note that you are also supposed to impose a periodic boundary condition in the x direction on the wave function. This periodic boundary condition on the wavefunction is used to help write the expressions for the wavefunctions. But this boundary condition is not used to construct the form of the Hamiltonian.

 

[Simon Bridge]

ahh ok thanks! and classically? it's constant, just kQ/r would this fit to the potential term in the Hamiltonian?

No, that would be the potential for a point charge.
You need to review classical electrostatics.
What is the potential inside a conductor?

 

[rwooduk]

No, that would be the potential for a point charge.
You need to review classical electrostatics.
What is the potential inside a conductor?

My interpretation of the problem is that you have a particle that is free to move along the length of the wire (along the x axis, 0<x<L). But the particle is not free to move in directions perpendicular to the axis of the wire (y and z direction). Being "free" along the x-axis should tell you something about the x-dependence of the potential energy function. Likewise, "harmonic" in the y and z directions tells you about the y and z dependence of the potential energy function. You should be able to write a mathematical expression for the potential energy function V(x, y, z) based on this information. Then you can write down the complete Hamiltonian.

Note that you are also supposed to impose a periodic boundary condition in the x direction on the wave function. This periodic boundary condition on the wavefunction is used to help write the expressions for the wavefunctions. But this boundary condition is not used to construct the form of the Hamiltonian.

inside a conductor its constant, but there is no numerical value given in the question. i think i see where this is going, if it is "free" then there would be no potential so effectively the potential term would be zero. does that sound correct?

thanks for the replies!

 

[TSny]

inside a conductor its constant, but there is no numerical value given in the question. i think i see where this is going, if it is "free" then there would be no potential so effectively the potential term would be zero. does that sound correct?

Free to move in the x direction means that there is no x-component of force acting on the particle. So, the potential energy will not depend on x. However, the particle is trapped inside the wire by y and z components of force. "Harmonic potentials in the y and z directions" means that these force components are modeled as obeying Hooke's law. This should tell you how the potential energy depends on the y and z coordinates. Think harmonic oscillator.

 

[rwooduk]

Free to move in the x direction means that there is no x-component of force acting on the particle. So, the potential energy will not depend on x. However, the particle is trapped inside the wire by y and z components of force. "Harmonic potentials in the y and z directions" means that these force components are modeled as obeying Hooke's law. This should tell you how the potential energy depends on the y and z coordinates. Think harmonic oscillator.

So is the potential just a constant or zero in the x direction? if it's a constant no numerical vlaue is given in the question so how would i obtain a numerical value for the eigen value? yes, thanks already applied the Hamiltonian for a SHO to the y and z eigen states.

 

[TSny]

You can just take the potential energy of the particle to be independent of x and zero for all points on the x-axis in the region 0 < x < L.

Since no numbers are given, you will express your answers in terms of parameters such as the mass of the particle, the length of the wire, and the "force constant" of the harmonic potential.

You should find that Schrodinger equation can be solved by separation of variables: $\psi(x, y, z) = X(x)Y(y)Z(z)$.

 

[rwooduk]

Ok that's great, thanks everyone for all the help with this question!

 

[rwooduk]

I'm going to complete this thread after getting confirmation from the lecturer... "There is no potential term. The Hamiltonian is only equal to the kinetic term (there is no approximation in this statement). "