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.
Tuesday, January 28, 2014
Saturday, January 25, 2014
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.
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.
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).
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).
Thursday, January 23, 2014
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.
Wednesday, January 22, 2014
\( sp^2 \) -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
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:
$$ (1/r^2) \frac {d} {dr} (r^2 \frac {d \psi_{100}} {dr}) $$
$$ (1/r^2) \frac {d} {dr} (r^2 \frac {d \psi_{100}} {dr}) $$
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.)
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.)
Sunday, January 19, 2014
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.
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.
Saturday, January 18, 2014
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.
Wednesday, January 15, 2014
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?
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?
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