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.  
---------------

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.

Visualizing wave-functions.

Here is a post to help you picture what is going on with a quantum wave function as a function of time. These are gifs that help you see the oscillations of particular quantum states. The first two are energy eigenstates, so their oscillations are "pure". The 3rd gif shows a mixed (non-stationary) state where the time dependence involves two different frequencies and thus exhibits complex interference patterns that change as a function of time.

Fig 1. The wave-functions shown above are for the ground state and 1st excited state, respectively, of an electron in a harmonic oscillator potential. The wavefunction is complex, hence there are two lines in each  plot. The horizontal axis is x in nm. The vertical axis is wave-function amplitude in \(nm^{-1/2}\). The ground state and 1st excited state oscillation for an electron in an infinite square well would be similar, but with a different spatial shape.

-----------------------------------------------------------------------------

 Fig 2. Suppose an electron in an HO potential is in a quantum state such that its wave-function is an equal mix of the ground and 1st excited states. Shown here is the complex wave-function for that non-stationary state. The vertical axis is wave-function amplitude in \(nm^{-1/2}\).

Office hours and section times.

Michael Saccone is an excellent TA and I have worked with him in the past. He will be available to help you with quantum this quarter at the following times:
Tuesday: 10:25 to 11:25 AM, office hours in ISB 292
Tuesday: 3:20 to 5:20 PM, section in Thimann 391
Wednesday: 1:20 to 2:25 PM, office hours in ISB 292
Wednesday: 2:40 to 5:05 PM, section in ISB 235

Zack Schlesinger's office hours are 11:30-12:30 AM in ISB 243

Homework #2. 1D HO, due April 12. with solutions

For the one-dimensional (1D) harmonic oscillator

Notes from Class-April4 2019

Notes from class, Thursday, April 4. Please feel free to discuss, ask questions...

Notes for our 2nd class.

Thursday, April 4. (These notes are from before class (Thursday morning). Note the use of the "jump break".
Today we will look at stationary states* of an infinite square well ...

Homework #1 with solutions.

For quantum lengths and energies, please express your results in nm and eV.

Physics 139a Syllabus.

Grading will be based on HW (20%), midterm1 (25%), midterm2 (25%) and final exam (30%).
1st midterm date:April 23
2nd midterm date: May 28
final exam: Wednesday, June 12, 12-3:00 PM
Book: Griffiths, Intro to QM, 3rd edition. We will start with chapter 2.

Class is from 1:30 to 3:05 PM on Tuesday-Thursday. My regular office hours will be 11:30 to 12:30 PM on Wednesday and after class on some days as needed. My email is zacksc@gmail.com

Expect weekly homework.

Sections: TA Michael Saccone will have sections and office hours as follows:
Tuesday: 10:25 to 11:25 AM, office hours in ISB 292
Tuesday: 3:30 to 5:20 PM, section in Thimann 391
Wednesday: 1:20 to 2:25 PM, office hours in ISB 292
Wednesday: 2:40 to 5:05 PM, section in ISB 235

Welcome to 139a

In this class we will study the behavior of electrons. We do that using the Schrodinger wave equation. (I believe that equation was originally known as “the wave equation”. Later, it became known as the “Schrodinger equation”. Some notable things are:  it is a wave equation, it has an hbar multiplying the spatial derivative, and, the time derivative is a 1st derivative, not a 2nd derivative, which is important.)

In physics we often study the time evolution of things. The time dependence of an electron wave function can be obtained by writing the wave-function as a sum of eigenstates (energy eigenstates), which each have a time-dependent phase factor. (Phase is important in wave theories.) We will spend a lot of time in this class finding eigenstates, talking about eigenstates, and calculating things using eigenstates. We will learn about the eigenstates of infinite square wells, harmonic oscillators, finite square wells, delta function potentials and hydrogen atoms. We will use these eigenstates to model time-dependence and to calculate expectation values. We will also look at the eigenstates of a free electron and use them to make a wave packet which is spatially localized and moves at a characteristic velocity.

One of the really interesting aspects of quantum physics is the kinetic energy. When we study the traveling wave packet, we will see that there are two kinds of kinetic energy. One is associated with motion (like classical K.E.).  The other is a quantum kinetic energy, associated with confinement, which, like quantum spin, has no classical analogue.  This quantum kinetic energy plays a huge role in the phenomenology of electrons, atoms, molecules and lots of important things.

We will learn what determines the size of a hydrogen atom, why an electron in a finite square-well has a lower ground-state energy than an electron in a comparable infinite square well, as well as how the bound states of a one-dimensional harmonic oscillator and an infinite square-well are essentially similar. We will study the role of symmetry and degeneracy in influencing the nature of hydrogen atom eigenstates, chemical bonding, and the periodic table. We will introduce the puzzling and elusive concept of quantum spin, a property with no classical analogue. We will follow the inexorable path from de Broglie’s suggestion to Schrodinger’s wave equation and explore some of the interesting phenomenology that flows from that.

Here is a week-by-week outline, subject to revision:
Notes:  weeks 1-5. QM in one dimension (1D); weeks 1-8 focus on the theory of a single electron in different circumstances (i.e., potentials)

week 1. infinite square well, expectation values, kinetic energy and confinement
week 2. harmonic oscillator, emergence of a quantum length scale.
We will use a Hermite polynomial approach and not raising and lowering operators. We will do expectation value calculations for position, size, kinetic and potential energy and involving time dependence.  (Later in the quarter we will return to the harmonic oscillator potential and learn about raising and lowering operators.)
Also, the time-independent Schrodinger equation. We will derive that from the Schrodinger Wave equation using separation of variables.
week 3. finite square well, delta function. Relaxing the boundary allows the wave-function to spread out somewhat and reduces K.E.
A finite square well has some bound states, but not an infinite number. We will explore how to use the time-independent Schrodinger equation and boundary conditions to obtain low-energy eigenstates for the square well. We will look at the degree to which states extend outside the well in terms of probability density. We will examine the reduction of kinetic energy associated with de-localization and the price that is paid for that in P.E.
week 4. formalism, linear algebra, orthogonality, linear independence, how eigenstates form an orthonormal basis of a Hilbert space.
Please review the concepts and applications of orthogonality, linear independence and inner products. Understand what a basis is. Review Hermitian matrices and how their eigenstates can be used as a basis (orthogonal) (and that their eigenvalues are real). Review trace and how the trace of a hermitian matrix is invariant…
week 5. free electron, traveling wave packet, kinetic energy, scattering, tunneling
week 6. 2D harmonic oscillator, constructing and normalizing 2D states, angular momentum, symmetry, hydrogen atom state, the size of the hydrogen atom
week 7. hydrogen atom 2, excited states, sp2 wave-functions, interaction of electromagnetic radiation (light) and electrons, stimulated transitions between states
week 8. hydrogen atom 3, size, bonding, symmetry and the periodic table
week 9.  spin, states involving more than 1 electron
week 10. review