Wave-packet states. Tuesday class.

A wave-packet refers to a method for the creation of a single electron state using many single-electron energy eigenstate wave-functions together.

HW 5. Due Monday May 6. With solutions.

HW5 soln link
https://drive.google.com/file/d/1asYW5eZ73eUyvEtGpD9KUsHxAWnfuaWc/view?usp=sharing

I would recommend starting on this soon and working on it some each day so you have time to ruminate and ask questions as needed. I think that is what a lot of people are doing and having really good success with that approach. I don't believe that you can do this all at one time and really learn a lot. Just my 2 cents. 

Additionally, I think it will really help your success if you ask questions when you are stuck. You can ask questions here or by email. Actually, sometimes posting your question here and asking me by email is good. Engagement and discussion are encouraged for this assignment and all assignments.

1.  a) Show that \(\psi_k = e^{ikx}\) is an energy eigenstate of the Schrodinger equation (TISE) for a free electron: \( - \frac {\hbar^2}{2m} \frac {\partial^2} {\partial x^2} \psi (x) = E \psi(x) \) What is its energy eigenvalue, \(E_k\)?
 b) Also, show that \(\psi_k = e^{ikx} e^{-iE_kt/\hbar}\) is solves the Schrodinger wave equation for a free electron: \( - \frac {\hbar^2}{2m} \frac {\partial^2} {\partial x^2} \psi (x) = i\hbar \frac {\partial \psi(x)}{\partial t} \)

Discussion of length scales.

A "length scale", or characteristic length, is an important thing that has a significant role in physics. A number of people have asked about it. What does it mean?  Let's first summarize

Topic for April 25 class.

For tomorrow's class we will start a new topic, which is free particle propagation and scattering.
This involves non-local states.

Discussion of problems 5 & 6, HW4.

April 24: Thanks to some awesome work by students in this class, I have some information on wave-functions and their parameters for the 2-square-well potential.  This was obtained using the TISE, boundary conditions associated with the TISE, and the normalization condition:
***You don't have to derive these parameters yourself. You can use the ones provided here! The most interesting part of the problem is how you use this information.

For the ground state:
\(\psi_1(x) = 0.593 cos [1.13(x-1.47 nm)]\),   inside the well between x=0.5 nm and 2.5 nm.
\(\psi_1(x) = 0.139 cosh(b_1 x)\),    between the two wells (between x=-0.5 nm and +0.5 nm).
\(\psi_1(x) = 0.238 e^{-b_1 (x-2.5 nm)}\),    to the right  (beyond x= 2.5 nm).
where \(b_1= 2.57 nm^{-1}\).
There are a lot of interesting things to notice about this. Since the individual wells are -0.3 eV deep and 2 nm wide, comparison to the single well ground state could be interesting and provide some key insights into how the quantum world works. For example, I notice that the k value for this double-well ground state is a little smaller than that for a single well. Why is that?

For the 1st excited state:
\(\psi_2(x) = 0.62 cos [1.17(x-1.48 nm)]\),   inside the well between x=0.5 nm and 2.5 nm.
\(\psi_2(x) = 0.134 sinh(b_2 x)\),    between the two wells (between x=-0.5 nm and +0.5 nm).
\(\psi_2(x) = 0.253 e^{-b_2 (x-2.5 nm)}\),    to the right  (beyond x= 2.5 nm).
where \(b_2= 2.55 nm^{-1}\).

What do these wave-functions look like? How can we use them to help us with problems 5 & 6?  Let's discuss this here.  
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videos added on May 1&2: These videos illustrate how to think about and approach this problem.

Guide to the Midterm & solutions.

The first problem is an infinite square well problem that includes an expectation value and some graphing.

Equation sheet clarification.

On your equation sheet,

Questions and email

For most questions the best way to ask it is on the blog so that other people can see your question and my answer. If you need to email I am:
zacksc@gmail.com
Please email me there and not at some other email. You will get a much quicker reply that way and I will appreciate your using that email.

Midterm preparation post.

April 22, 3:30 PM
Things to bring to the exam:

HW 4. with solutions added at end

This assignment includes problems illustrating the nature of tunneling and exploring why the wave-function squared is known as a "probability density". 

Delta-function potential.

This video summarizes Thursday class, showing the ground state of an attractive delta function potential.

Finite square well: an alternate method to obtain k and b.

I showed this to some students today and they really liked it, so I thought I would share it here.

Expectation values and energy.

There was a really interesting question, from a person in the back row on the right, on Tuesday

Using Wolfram-Alpha

This video show a few examples of how you can use Wolfram Alpha.

Square Well boundary conditions.

This video shows how to use boundary conditions along with the time-independent Schrodinger equation

Time independent Schrodinger equation.

This video shows the derivation of the time independent Schrodinger equation (TISE)

HW 3. FSWs, probablity density, evanescent wave-functions, energies...Solutions added 4-20

In this HW set, we will begin to explore and learn about the nature and meaning of the wave function and why \(\psi^{*} \psi\) is called the probability density.

Thurs April 11 Class notes.

This shows the nature of the dependence of the size of the electron and its kinetic energy expectation value for an electron in the ground state of a 1D HO. The horizontal axis is k, the strength of the confining potential.

Harmonic Oscillator-April 9: pre and post class notes.

Today we will look at the nature of a quantum harmonic oscillator (HO) in 1 dimension (1D).

Notes on notation, reading and HW.