Quiz and quiz solutions.

Here is a link to the quiz  and below that are my solutions to problems 1 and 2.  (These are intended to be a model for you in terms of how much work they show, presentation...  Also, I an interested in your comments, thoughts or questions about the quiz. Please do comment here if you have any thoughts or feelings about the quiz (or the class in general).  Everything is just below the break.

Homework 3 solutions link.

https://drive.google.com/file/d/0B_GIlXrjJVn4aGZWMlppVGdLcmc/edit?usp=sharing

As Eric pointed out, there is a mistake in the solution to problem 7 (and problem 8). For 7, the theta integrand is actually  \( sin^3 (\theta) \), which integrates to 4/3. This changes the answer from 1.5a to 3a. It also changes the answer for problem 8 to \( 2 \sqrt2 a \), which is about 2.8a, i.e., a little less than 3a.

Quiz on Tuesday.

The best way to prepare is to pick a HW problem and see if you can do it on just a blank piece of paper, not looking at any solution.  Reading solutions, or any reading, is not as good, except as a prelude to actually working a problem on a blank piece of paper (with your notes as your only aid).

Homework 4. Due Monday Feb 4 (3 PM).

(energy, oscillators, hybrids and L) 
Energy:
1. a) Calculate the expectation value of 1/r for an electron in the state \( \psi_{21z} \).
b) Use that to obtain the expectation value of the potential energy for an electron in this state.
c) evaluate that in eV.
d) using your knowledge of the total energy, infer the kinetic energy for an electron in this state.

2. a) Do the same thing for an electron in the state \( \psi_{200} \).
b) Summarize your results in one or two cogent sentences. Include a comparison to what you already know about the ground state.

Midterm is on Thursday, Feb 6.

In response to the first comment below, the goal of this midterm is to test, bring into focus and solidify, and perhaps enhance your understanding of quantum physics in the context of our 1D systems and the ground state and 1st excited states of the H atom electron. (The latter goals are just as important as the first.)

A page with states and integrals, as well as "constants" that will help you calculate things in eV, nm type units will be essential. Just one page. No worked problems.

For the 1D HO and infinite square well, bring 3 states each. For the H atom,  the gs and 1st excited states will be needed. There may also be finite square well questions. A good qualitative understanding of the finite square well is of value.

One can expect that there will be calculational problems, e.g., expectation values, numerical evaluation (getting something into eV, for example) and qualitative discussion.

I'll add more here as I think of it. Reworking HW problems is an excellent way to start preparing. This web site contains a lot that will help you. Reading a book is not generally an effective way to prepare.

The best way to prepare is to pick a HW problem, or a problem of your own design and of reasonable difficulty, and see if you can do it on just a blank piece of paper, not looking at any solution.  Reading solutions, or any reading, is not as good, except as a prelude to actually working a problem on a blank piece of paper (with your notes as your only aid).

Problem 9 & discussion of \( 2sp^2 \) states.

Hybridization of 1st excited states can lead to what are called \( sp^2 \) states. \( sp^2 \) states are hybrids of the 2px, 2py and 2s states (as in problem 9). These states are orthogonal and normalized, and they are part of an orthonormal basis of that spans the space of all 1st excited states of this potential.

\( sp^2 \) -related structures.

These are some important structures in which \( sp^2 \) bonding plays a significant role:

Also, here is a snippet from an intriguing site about bonding.  What we are doing is the "fairly high-powered stuff on the wave nature of electrons".
http://www.chemguide.co.uk/atoms/bonding/doublebonds.html#top

Problem 10 note.

For problem 10 (HW3) you will need to use del^2 in spherical coordinates to calculate the K.E.. Since Psi100 does not depend on theta or phi, I think you can use the r part which I believe is:
$$ (1/r^2) \frac {d} {dr} (r^2 \frac {d \psi_{100}} {dr}) $$

Contour Plots.

Here are two examples of contour plots of 2p states created using Wolfram Alpha.

Several things to notice:
1) for the x-y plane (where z=0) I write r as sqrt(x^2+y^2),
2) I left out the "a"s. That is, I set a=1. (so everything is in units of .053 nm)
3) The range is specified. (That is what the x, -6, 6, y, -6, 6 part does.)
(These are not off-center hybrids. They are pure p states; the second one is rotated 45% via a superposition of x + y.)

H-atom states and linear algebra concepts.

This post shows wave-functions of an electron in a hydrogen atom potential. In particular, it shows ground state and 1st excited state wave-functions.

This video discusses linear algebra concepts related to question 9 from HW1 and relevant to understanding electron excited states in a hydrogen atom. These concepts include: orthogonality, linear independence, spanning, basis and orthonormal basis.

In Memoriam: Ivan Mercado

Although I barely know any of you yet, in a sense we are all part of a community -a UCSC community and the community of people who teach physics and study physics at UCSC. It is with great sadness that I learned that Ivan Mercado, a member of our community and a close friend to many of you, passed away tragically and unexpectedly last weekend. Please feel free to share comments, thoughts and feelings related to Ivan here.

Homework 3. due Friday, 2 PM.

1. For an electron in the ground state of a hydrogen atom:
a) calculate the expectation value of \(r^2\) . [This requires an integral over all 3D space. Don't forget to include the \( 4 \pi r^2 \) factor and to understand where it comes from.]
b) Calculate the expectation value of the potential energy.
c) Evaluate the expectation value of the potential energy in eV. What is its sign?
d) Think about the relationship between PE, KE and E for this state. Since you probably already know that the energy is -13.6 eV (1 Rydberg), can you guess what the KE must me?

Homework 2 solutions link

https://drive.google.com/file/d/0B_GIlXrjJVn4X2ZtWV9ScU05TWs/edit?usp=sharing 

Please feel free to discuss here and to ask questions about any problem/solution that you are not clear about.

Homework 2, due friday Jan 17, 2 PM.

This is our last assignment on 1D QM and, as such, it is important to providing you with a foundation to understand the 3D quantum that we will do next. I would suggest starting soon and spending as much time as you can on this assignment.

 Whenever units are needed, please use eV and nm as your default units.  Problem 5 is pretty difficult, but I think that if you can do it and understand it then you will really learn a lot about the nature of potential and kinetic energy in quantum systems. This should be useful later in this class and, I think, in future quantum physics classes you take.

1. For an electron in the ground state of a 1D harmonic oscillator, calculate the potential energy expectation value.

Finite Square Well Videos.

In class we have discussed the nature of the electron bound states of a finite square well potential. Our favorite well is the 0.613 nm wide well with walls 16 eV high. In the first video I talk about the bound states of this well. The second and third videos show how these states are obtained from the Schrodinger (Wave) equation for this system.
         We will refer to finite wells quite a bit. Part of their appeal for us is that a finite well has bound state wave-functions that extend beyond the confining potential (with an evanescent form, i.e., exponentially decreasing to zero) into a region of flat potential (often chose to be zero). This is similar to the electron bound states of a hydrogen atom potential. It means that we can put 2 finite square wells close together to model a molecule and and explore the origins of molecular binding, which are deeply quantum. We can also examine the nature of quantum tunneling phenomena (from one well to another).
         The first video is mainly phenomenological (a long word which means we don't derive stuff, we just look at how it appears), so it is a bit less difficult than the 2nd and 3rd. The 2nd and 3rd videos examine how the quantum wave-functions and allowed bound-state energies are obtained.

HW 1 solution notes.

Enclosed are some solutions notes for homework 1. I have made these a little short and compact in order to fit it onto 4 pages, so these are not model solutions. Your solutions might, ideally, be a bit longer and more informative. For problem 9 I am planning to make a short video to discuss those linear algebra related issues.
     I am thinking about adding a short video here to discuss the Linear Algebra related issues of Problem 9, which are relevant to understanding H atom states.

Homework 1 video.

This video show me working problems 1 and 8 from homework 1. Also, by inference, I think it shows you how to do problem 5, which was also very popular in the poll.

About this class.

Physics 101B meets on Tuesday and Thursday at 4:00 PM.
Instructor: Zack Schlesinger, ISB 243, zacksc@gmail.com
Office hours: TBA
TA: Tia Plautz, tiaplautz@gmail.com
Website: http://physics101b.blogspot.com/

Section: Our class section will meet on Wednesday from 12:20 to 1:50 in room ISB 165 or Thiman 1. At the section you can get help with HW problems and concepts.

Homework is due every Friday by 2 PM in the physics mailroom (ISB232). There will be a box in the mailroom for you to put it in (or you can put in in my mailbox). I would suggest budgeting about 15 hours per week for homework for this class. Ideally that might be over several days, giving you time to mull over concepts, and to let things percolate and "sink in".

Exams: Our final is on Friday, March 21st from 12-3:00 PM. Our midterm will be in February, probably sometime between the 4th and 13th.

Grading: Grading uses a weighting of: 20% HW, 35% midterm and 45% Final.

Website: Our class website will be used extensively and is a critical part of this class. A HW assignment (due Friday, Jan 10) and some review materials are posted below this post. Your participation via comments and questions on the website will be noted and appreciated!
url: http://physics101b.blogspot.com/

Content:
We will start with bound states of one-dimensional (1D) systems (square well, harmonic oscillator) with emphasis on wave-functions (of energy eigenstates) and expectation value calculations. This provides essential background for our next topic: the hydrogen atom where our emphasis will again be on wave-functions and expectation value calculations. Our 3rd topic will be spin and quantum statistics. These 3 topics provide the foundation for understanding our next topics: quantum origins of the periodic table, molecular binding and electrons in crystals.  Following this will be the study of semiconductor physics, lasers, quantum spectra and other topics in modern physics.

Summary: (not set in stone)
    1D quantum systems  (week 1)
    H atom                        (week 2 and 3)
    Spin                            (week 3)
    Periodic Table            (week 4)
    Molecules                  (week 5)
    Solids                         (week 6)
    Semiconductors
     & Lasers                   (week 7)
    Quantum spectra       (week 8)


Notes: The topics of this class are deeply interconnected. To understand the hydrogen atom one needs a basic understanding of 1D quantum bound states and expectation values. To understand the periodic table it is essential to understand the hydrogen atom states and degeneracies, as well as electron spin. Quantum kinetic energy is critical to understanding molecules and solids. Etc.  These connections make the beginning of this class particularly important, as it provides the foundation for what follows. Working homework problems will be a critical part of this class. I would recommend looking at each HW assignment soon after it is first posted (Sunday/Monday) and allowing yourself a few days and maybe 10-15 hours to complete each assignment. Homework is the backbone of the class and provides the means for you to develop understanding and to prepare for the exam problems, which will be similar to the HW problems.

This class provides a great opportunity for us to begin to examine and explore the world of quantum physics, particularly physics at the nanometer scale. Quantum physics is probably one of the most important and unusual breakthroughs in the history of science. Before the wave nature of the electron was revealed, that is, that electron behavior follows a wave equation known as the Schrodinger equation, there was no real model or understanding of the nature of an atom, the organizing principles of the periodic table, why atoms stick together to form molecules and solids, superconductivity, lasers, how photosynthesis begins, etc. Quantum physics remains an active area of research, particularly in novel materials where competing interactions between electrons confound theoretical efforts and sometimes lead to amazing and unexpected new phenomena. In this class we will begin the journey to understand this interesting world.

Homework 1: due Friday, Jan 10

Homework is due Friday by 2 PM in the Physics Dept mailroom across the hall from room ISB 231. There will be box to put your HW in from about noon to 2 PM. (If you come earlier or don't see the box, then you can put it in my mailbox.) 

Please do not hesitate post questions or comments here. I encourage you to question anything that seems incorrect or unclear to you. Your questions and comments are strongly encouraged and appreciated. I am also interested to get feedback on whether you find some of these problems too difficult or too easy. Feel free to respond to other students questions and comments. Peer-to-peer dialogue can be very valuable.

This assignment focuses on math-related things which will be relevant to something we will cover in the near future. Feel free to use Wolfram-alpha for integrals, help with graphing, etc.. Graphs should be hand sketched --not too big (or too small). (Two relationships that did not make it onto this assignment are: exp(i*theta)= cos(theta) + i sin(theta), and (e^a)*(e^b) = e^(a+b).)

Waves:
Imagine a string attached to fixed posts at either end. The string's simplest motions, are standing waves, called normal modes, in which the string moves at one particular frequency.  For problem 1 you are asked to sketch the string displacement profile at a time of maximum displacement (e.g., t=0) as a function of x.
1. a) For a wave on a string with fixed ends at x=-L/2 and L/2, sketch the lowest frequency mode.
b) Write a mathematical expression for this lowest frequency mode (at a time of maximum overall displacement).
c) Sketch the next lowest frequency mode of a wave on a string (with fixed ends at x=-L/2 and L/2).
d) Write a mathematical expression for y(x) for this mode (at maximum displacement).

Graphing: (Things to notice: overall shape, node location, asymptotic behavior. Please label and/or put scales on axes.)
2. (These are relevant to square well states.)
a) sketch a graph of cos(pi x/L) from x = -L/2 to L/2.
b) sketch a graph of sin(kx) from x = -pi/k to pi/k.
c) sketch a graph of cos(.7 pi x/L) from x = -L/2 to L/2. What is its value at x=L/2?
d) sketch a graph of B e^{-(x-L/2)/a} from x= L/2 to (L/2 + 4a). what is its value at x=L/2? At x=(L/2)+2a ?