Quantum Tunneling Probability for a rectangular barrier

[Stevenpd]

Hey everyone,

For my a research project on quantum tunneling I have to tie in math somehow, so I was wondering if anyone could help me learn to calculate the probability of a wave (preferably like an electron or electrical voltage) of some sort passing through a rectangular barrier, which is supposedly the most simple calculation in quantum tunneling. I have a somewhat decent calculus background because AP calc is almost over. I can't just get an equation and plug in numbers either, i'd like to be able to explain it somewhat. Thanks!

Steven

 

[LostConjugate]

They did not give you examples in class? Hint: Solve the shrodinger eq. The probability amplitude is the coefficient of the solution where X > L and L is where the potential returns to Vo

The mean square of the probability amplitude is the actual probability.

 

[LostConjugate]

Actually its not the simple because the velocity changes. It is the First Transmission coefficient minus the second reflection coefficient. That will give you the probability amplitude.

 

[Matterwave]

Calculus AB probably won't be enough math to do this derivation, unless you learned differential equations in it...which I don't recall learning to any detail in Calc AB.

BUT, you can nevertheless follow the derivation if you just take the solutions to the differential equations as given. I will point you to the wikipedia articles:
http://en.wikipedia.org/wiki/Quantum_tunneling#Schr.C3.B6dinger_equation_-_tunnelling_basics

and
http://en.wikipedia.org/wiki/Rectangular_potential_barrier

 

[Stevenpd]

Ok well actually we have learned differential equations somewhat. and I'm learning BC during my study hall period anyways so that helps.

But yea we haven't actually talked about any quantum mechanics in my physics course. this is all on my own. I could use some help just understanding a little background on all the symbols such as plank's constant or the hamiltonian constant etc.

 

[Stevenpd]

ok actually the schrodinger equation i guess really isn't as bad as I thought. Things are starting to come together for me. So i have a few questions then...

1. How can I calculate V(x) (potential energy)?

2. What exactly is the wavefunction? (I see its exp([strange symbol]x)... what's the strange symbol too? (the oval with an I through it))

3. Why do you take the 2nd derivative of the wave function?

4. What part of this equation is the actual probability a wave will pass through a given barrier?

and I'll probably have more... :) but these are ok to start off with.

 

[Matterwave]

1. V(x) is the potential, and it is given. For a rectangular barrier, it just has 2 or 3 values. For example, V(x) for rectangular barrier might be:

V(x)=0 for x<0 and x>L and V(x)=V for 0<x<L where V is just some constant. In the quantum case, V(x) acts like force does in the classical case. You star from V(x) and initial conditions and from there, you derive the dynamics of your system.

2. The wavefunction is a function which encodes all of the information that can be known about a particle (in general, it should include the spin and angular momentum, etc; however, in 1-D problems like the one you are trying to solve, these don't really matter). We usually denote a wavefunction by the greek symbol Psi $$\Psi$$ or $$\psi$$ when we are working in configuration space.

3. The second derivative of the wave function arises because the Hamiltonian operator (in 1-D) is:
$$H=\frac{p^2}{2m}+V(x)$$
Where the momentum operator is:
$$p=\frac{\hbar}{i}\frac{\partial}{\partial x}$$

When you square the momentum, you get a second derivative out of it.

4. The probability P that a particle is within some interval x to x+dx is given by:
$$P=\int_{x}^{x+dx} \bar{\psi}\psi dx$$
where the bar denotes taking a complex conjugate. The actual probability of tunneling is given by the transmission coefficient, which is derived within the articles.

I hope this is enough basics. You can try to see if the articles make better sense now.

 

[mannoos]

I would like to ask something about my observation. For a particle (with energy E) incident to a rectangular potential with length L and height Vo, there is the plot of transmission probability as a function of E/Vo. For various values of α [where α² = 8mL²Vo/hbar²] one can see that the lines have some common points.
For example for α = 16 there are two peaks until to have the same peak with α = 8
Furthermore, for α = 16 there are four peaks until to have common point with α = 8 for second time and α = 4 for first time.
Which means that as much times α is multiplied with 2, such more peaks has the line until to meet the line with the half value of a

Finally, apart from the common peak points they also have other common points at periodic values of E/Vo
So, my question is what are these common points represent and why there is this periodical repetition of them.