Possible Rectangular Potential Barrier Transmission Problem

[HunterDX77M]

1. The Problem Statement
An electron of effective mass m* = 0.2m_e and energy E = 0.1 eV hits a barrier of height 0.4 eV and width t = 5 nm. What is the probability of transmission through the barrier? Use the simplest estimate which is an exponential function.

Homework Equations

I think the Schrödinger equation?

The Attempt at a Solution

First let me just say that the professor stated that the only prerequisites for this course were the first two introductory classes in physics, which at my university covers mechanics, fluids, electricity and magnetism. I have never taken any course in quantum mechanics or any advanced physics. After Googling around for a while, I learned that this was called a "Rectangular Potential Barrier" problem and requires using the Schrödinger equation(s) to solve. I'll be honest, the only thing I know about Schrödinger is the cat. Now having said that, I am completely at a loss for where to even begin on this problem. If someone could point me in the right direction (possibly towards some decent lecture notes on the topic) that would be great.

Thanks in advance!

 

[rude man]

www.qudev.ethz.ch/phys4/PHYS4_lecture05v4_2page.pdf

(This should post on the Advanced Phsics forum)

 

[HunterDX77M]

www.qudev.ethz.ch/phys4/PHYS4_lecture05v4_2page.pdf

(This should post on the Advanced Phsics forum)

Thank you for your link, rude man. I will look it over and see if it help me understand it.

Also, I was going to post this in the Advanced Physics section, but the sticky post there said if a problem requires plugging into a formula, then it's probably introductory physics. Based on the research I had done up to that point, it seemed like this just required a formula.

 

[HunterDX77M]

Matching the units

I found the following equation in the link:

T = e^{-2k_2 L}; k_2 = \sqrt{\frac{2m(U-E)}{\hbar}}

The Problem I am have, however, is that the units don't seem to cancel each other out. In the end T is simply a dimensionless quantity (probability), but that doesn't work out in the math. Take a look below for my reasoning:


<br /> m = 0.2 \times 9.109 \times 10^{-31} ~kg\\<br /> <br /> U = 0.4~ eV\\<br /> <br /> E = 0.1~ eV\\<br /> <br /> \hbar = 6.58 \times 10^{-16} ~eVs\\<br /> <br /> L = 5 ~nm<br /> <br /> \\ \\<br /> k_2 = \sqrt{\frac{2 \times 0.2 \times 9.109 \times 10^{-31} ~kg ~\times ~(0.4-0.1)~eV}<br /> {6.58 \times 10^{-16} ~eVs}}<br />

The eV in the numerator and denominator cancel each other leaving me with units of
\sqrt{kg/s}

When multiplied by a unit of distance (with the width L), this does not give me a dimensionless quantity. Am I doing something wrong here?

e^{nm ~\times ~ \sqrt{kg/s}}

 

[rude man]

Well, you are absolutely right, and the formula on that link is wrong.

I dug up a better one :
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/barr.html

This time I think you will be pleased with the dimensional consistency of the given equation. As you can see, the denominator under the sq. rt. needs to be (h-bar)^2, not h-bar.

I would still say that, given formula or not, you're dealing with a subject not typically a part of introductory physics, but what the heck.