Matrices, eigenvectors, eigenvalues, and the double dot quantum well.

Related to HW8, the video below discusses eigenvectors for the matrix problems and also problem 2, the double dot quantum well. Please feel free to post any questions here. Also here is a page related to the problem 2 showing the basis states and one energy eigenstate. If you don't fully understand any part of problem 2, please ask about it. The linked paper in problem 2 is where our notation comes from.

Raising and lowering operators.

These videos show how raising and lowering operators are defined and used for the one-dimensional harmonic oscillator (1DHO). It uses the same notation as Griffiths, chapter 2.
The two key equations you need in order to use the operators are:
\(a_- | n \rangle = \sqrt{n}\: |n-1 \rangle \),
\(a_+ | n \rangle = \sqrt{n+1} \: |n+1 \rangle \).

On the limitations of quantum mechanics

     Note added: Quantum mechanics can do a lot. It is the basis for understanding lasers, semiconductors, light-emitting diodes (LEDs), most of chemistry and essentially all of electronics (your phone, computer, MRI magnets, etc.).  However, some things elude an understanding starting from the Schrodinger equation. Notably, biological systems and many other complex macroscopic things. Why is that?
    Since most things are made of electrons, protons and neutrons, and the Schrödinger equation* provides the theory of electrons, protons, and neutrons, one might imagine that with hard work and enough computing power one could explain all "everything". That is,  biology, brain chemistry, psychology, etc using quantum mechanics. Indeed many people in the 1950s, 1960s and 1970s thought that it would be possible to explain everything, including life, using quantum mechanics and high-powered computers. Then in 1972 Philip Anderson published a paper called “More is different”**, in which he emphasized fundamental limits to reductionism and argued that due to something called emergence, and emergent phenomena, it is essentially impossible to solve the Schrödinger equation in many complex systems. Thus he argued that although the Schrodinger equation is essentially the theory of everything on earth, it actually tells us very little about most things that we view as important. This latter point was emphasized and elucidated by Robert Laughlin in about 2007 in a paper entitled (ironically) "The theory of everything". (Laughlin is a Stanford physicist who won a nobel prize for his theory of fractional quantization, a phenomena in which two-dimensional electrons lose their fermionic nature and become "1/3" fractional quasiparticles instead. His theory involved a made-up wave-function with exponents of 1/3 where there should have been exponents of 1 or -1 for ordinary fermions.)

Also of interest, Anderson received the Nobel Prize in 1977 for his contributions to the theory of electron localization.  (As you know, electron localization is very important.)


** From wikipedia:  “Anderson has also made conceptual contributions to the philosophy of science through his explication of emergent phenomena. In 1972 he wrote an article called "More is Different" in which he emphasized the limitations of reductionism and the existence of hierarchical levels of science, each of which requires its own fundamental principles for advancement.[16]
A 2006 statistical analysis of scientific research papers by José Soler, comparing number of references in a paper to the number of citations, declared Anderson to be the "most creative" amongst ten most cited physicists in the world.[17]

* When we say Schrodinger equation here, that could really mean Schrodinger or Dirac equation. The Dirac wave equation is the relativistic version of the Schrodinger equation and is is useful and necessary for understanding heavy elements and most of magnetism, which is a relativistic phenomenon.

HW 8. With solutions.

Solutions below.
Homework 8 includes spin problems and problems using spin matrices. Basically, what are called two level systems.

Notes and guide to Midterm.

This midterm will focus on the material we have covered since the last midterm.

sp2 states and Dirac Notation

This post contains a video which describes and uses Dirac notation. It shows a calculation of the expectation values of x and y which explains how sp2 hybridization permits the construction of three states at 120 angles to each other. It also includes a contour plot, in figure 1, that shows the nature of the probability density for one of the three sp2 states. Each sp2 state is equivalent to the other two, but with its orientation rotated by 120 degrees. That is why each of the three in-plane sp2 states has the same amount of \(\psi_{2s}\) mixed in. Different amounts of \(\psi_{2x}\) and \(\psi_{2y}\) are mixed in to create the angles (orientations) you will see in this video.

 Figure 1. Contours of constant probability density are show for the x-oriented sp2 state.

Introduction to spin.

The first video here provides an introduction to spin. The Pauli matrices

Review & spin.

This week I would like to do some review of quantum in one-dimension, finish our work on hydrogen atom electron states, and then beginning our study of quantum spin. Our emphasis will be on spin 1/2, which is covered in Griffiths in section 4.4.1 and 4.4.3. We will be using 2x2 Pauli spin matrices and "spinor" notation. Comments and thoughts on what you would like to learn more about or review are welcome!

Transition from spatial states to spin.

This week, as a natural progression in this class, we are going to transition from studying spatial wave-functions to studying spin. This is part of a transition to more formal aspects of this class in which we use operator formalism. Will we use operators, e.g., raising and lowering operators, in both our study of spin and in our reexamination of the harmonic oscillator using operator formalism.
     This survey asks how abrupt you would like that transition to be? Are you tired of spatial states and would you like to transition to spin states ASAP? Or we you prefer a bit of review of 1, 2 and 3D QM before we transition to studying spin?

https://docs.google.com/forms/d/e/1FAIpQLSe-nDBcycI4qKQNaTEK2Fpsfb-UTew2VdcNBbp4-YaFMN2tSA/viewform?usp=sf_link

Lecture on the origin of size: length scales in quantum systems.

This video explores the origin of the size of the hydrogen atom.

Guide to HW 7.

1. Notes on energy of single electron states in an H atom potential:
The ground state is an energy eigenstate with an energy of about -13.6 eV.
1st excited states have an energy of about -13.6/4= -3.4 eV.

Problems for summer, atoms, screening...

Here are some possibly interesting problems that we may not have time to fit in this course.

May 14 class & additional notes.

Tuesday's class is important. We will cover H atom states and things you need to know

HW 7 with solutions.

Please turn in HW 7 to Michael Saccone's mailbox in the physics department mailroom by 6 PM on Monday,  May 20

Midterms: our 2nd midterm is on May 28. 1st midterm histogram is below.

This shows each score from the midterm of April 23.

Reading assignment

Your reading assignment for this class is to read in depth, so that you can recall it, everything on this blog. I really believe that what is in this blog with help you understand the material, and help you do the homework for this class more efficiently, using less of your time and gaining a deeper understanding of the material and of why you are doing particular HW problems.  I believe that the class as a whole is not taking advantage of the information and videos on this blog as much as would be ideal and strategic for you.

I think that part of an effective strategy for HW assignments for this class is to try to think about them for an hour or so each day starting when it is assigned and to pay attention to and follow any commentary on this blog. This could make the time that you spend actually doing the problems more productive and successful and give you time to ask questions if needed. This may be different from what works for other classes.

Section times and office hours.

Tuesday-Thursday. QM in 2 dimensions. Includes wave-functions and notes.

Guide to HW 6 & solutions

Please turn in HW 6 to Michael Saccone's mailbox in the physics department mailroom by 6 PM on Monday,  May 13

This guide discusses each of the problems in HW6, including a rough estimate of how long they could take to complete. This HW involves 3 parts:
A) problems 1 and 2 are directed creating a basis for understanding the meaning of the

HW 6

Please turn this HW in to Michael Saccone's mailbox in the physics department mailroom by 6 PM on Monday,  May 13. 
 
"What does the wave-function mean?". The knowledge you gain from problems 1 & 2 will help us address that essential question soon.
Problem 3 is relevant to your understanding of what an energy eigenstate is.
Problems 5-10 provide an introduction to quantum mechanics in 2 dimensions, motion in 2D, and the meaning and importance of degeneracy.

How is it going with HW this week?

I am thinking that the HW this week is not so excessively long as last week's assignment. This week's HW has a problem involving a traveling wave packet (divided into parts 1-5), a problem involving a wave-packet where the expectation value of x is zero all the time (modeling an electron that is not moving), and a scattering problem.

Modeling electrons.

Here is an electron in a wave-packet state that is a superposition of free-electron energy-eigenstate wave-functions and their time dependence. The smooth line shows the wave-function squared (in units of \(nm^{-1}\) and the oscillating curves, there are two of them, show the real and imaginary parts of the wave-function in units of \(nm^{-1/2}\). Although it says crystal wave-packet, it is actually a free-electron wave-packet (I think you can tell from the smoothness of the wave-function squared, which would be textured by the crystal lattice potential in a crystal.) This is calculated from the Schrodinger equation and a superposition of free electron eigenstates just like you are doing in HW5. I think if you plot your results it will look pretty much like this.

Special Note on homework.

Some people pointed out that last weeks HW was too long and I agree with that.  That's particularly unfortunate because problem 5/6 is so iconic and important to your understanding of quantum theory. So, what Michael and I  decided is to give you the opportunity to turn in problem 5/6 again (the tunneling problem from HW4) for extra credit as part of HW5. 
     Additionally, I added some videos and more discussion of problems 5&6 in the post "Discussion of problems 5 & 6, HW4".

Using mathjax here. (like latex)

As people are beginning to comment more, it might be helpful to use the mathjax, which is pretty much latex, enabled here. Feel free to practice that here. slash paren takes you in and out of math mode as illustrated here: