Here are solutions to midterm 2. I see that I did not write it in the solutions, but it is helpful for sp2 type integrals to recall that \(\langle \psi_{2s} | x | \psi_{2x} \rangle = 3a\) and also \(\langle \psi_{2s} | y | \psi_{2y} \rangle = 3a\).
Sunday, June 9, 2019
Friday, June 7, 2019
Study guide for final.
Points of emphasis on the final will include understanding how to model motion using superposition of eigenstates, understanding how to use degenerate states to create off-center states, understanding the cost of confinement and localization in quantum physics, understanding the nature of kinetic energy, and understanding tunneling in bound state systems. There will be no problems on scattering on the final.
Compared to the last midterm, there will be much less graphing of wave-functions.
You can bring the pages of notes you had for each of the two midterms (2 pages total) and add to that anything that we have covered since the 2nd midterm. Be prepared to calculate expectation values.
Added Monday morning: The final will encompass everything we have studied this quarter, however, there will be points of emphasis and de-emphasis that can help you focus your study. Mixed states and modeling time dependence using mixed states is a point of emphasis. In quantum it can be difficult to model motion. Making a particle go in a straight line was mathematically difficult because it involved a superposition of an infinite number of energy eigenstates. You won't be asked about that superposition. The only thing you could be asked about from our study of a free electron is how a moving electron has two kinds of kinetic energy, one associated with motion and another associated with confinement. I think this was covered in problem 4 of HW5.
In two and 3 dimensions, it is possible to model circular motion. Intriguingly the most elementary modeling of circular motion is easier modeling linear motion in quantum physics. Using just two appropriately chosen energy eigenstates, one can model an electron going back-and-forth or moving in a circle or even an ellipse. There are HW problems for the 1DHO, 2DHO, infinite square well and hydrogen that relate to this. Really understanding deeply how mixed states work and how they can relate to motion is an important skill in introductory quantum physics. We covered a number of examples of electrons that move back and forth. One of those involves tunneling, that is motion through a region where the energy of the electron, which does not have a specific value due to the electron being in a superposition state, is nevertheless definitely less that that of the potential energy in a region through which it moves. Understanding tunneling, how an electron can move from one well to another is important. If we look back at the difference between a finite square well and an infinite square well, we see that the main difference is that the electron wave function extends outside the finite square well. The reason for this is that the electron can lower its KE by becoming less localized (less confined). It is this tendency of an electron to do whatever it can reasonably do to become less localized that leads ultimately to tunneling. For example, the non-zero value of the ground state wave-function between two wells in a double well system is intimately related to the difference in energy between the GS and 1st excited state, which is directly related to tunneling times.
One of our main themes this quarter was the energy cost of localizing an electron. This is something that has no analogue in non-quantum physics. This energy cost comes directly from the \(-\hbar^2 \frac{\partial^2}{\partial x^2} \frac{1}{2m}\) term of the Schro Wave Equation. Solutions to the Wave Equaton are called wave-functions. They have to start from zero and end up and zero, so that they are normalizable, and the bending associated with getting them to do that yields a second derivative the integral of which is always positive. So there is an inherent, unavoidable cost to localizing an electron, which is the essence of what is called x,p uncertainty. Really it is the expectation values of x^2 and p^2 that are relevant to what we call x,p uncertainty. The cost of localization manifested itself in every bound state system we studied, and even for the free electron wave-packet. Anyway, a deep understanding of how localizing and/or confining an electron has a cost and how that depends on the electron's mass and \(\hbar\) is a good thing.
There will be a question about angular momentum similar to problem 8 from HW 7. A key thing for that problem is to know that \(L_z = \frac{\hbar}{i} \frac{\partial}{\partial \phi}\) and to understand the relationship between cartesian and spherical coordinates.
You will also be asked to calculate expectation values involving 1st excited states of hydrogen. Reviewing the post and video on sp2 states and Dirac notation again will be helpful for that because it helps teach the skill of organizing and evaluating expectation values and matrix elements (using Dirac notation).
Basically, if you understand localization, tunneling, hydrogen atom states, expectation value calculations and mixed states, you should be good.
Compared to the last midterm, there will be much less graphing of wave-functions.
You can bring the pages of notes you had for each of the two midterms (2 pages total) and add to that anything that we have covered since the 2nd midterm. Be prepared to calculate expectation values.
Added Monday morning: The final will encompass everything we have studied this quarter, however, there will be points of emphasis and de-emphasis that can help you focus your study. Mixed states and modeling time dependence using mixed states is a point of emphasis. In quantum it can be difficult to model motion. Making a particle go in a straight line was mathematically difficult because it involved a superposition of an infinite number of energy eigenstates. You won't be asked about that superposition. The only thing you could be asked about from our study of a free electron is how a moving electron has two kinds of kinetic energy, one associated with motion and another associated with confinement. I think this was covered in problem 4 of HW5.
In two and 3 dimensions, it is possible to model circular motion. Intriguingly the most elementary modeling of circular motion is easier modeling linear motion in quantum physics. Using just two appropriately chosen energy eigenstates, one can model an electron going back-and-forth or moving in a circle or even an ellipse. There are HW problems for the 1DHO, 2DHO, infinite square well and hydrogen that relate to this. Really understanding deeply how mixed states work and how they can relate to motion is an important skill in introductory quantum physics. We covered a number of examples of electrons that move back and forth. One of those involves tunneling, that is motion through a region where the energy of the electron, which does not have a specific value due to the electron being in a superposition state, is nevertheless definitely less that that of the potential energy in a region through which it moves. Understanding tunneling, how an electron can move from one well to another is important. If we look back at the difference between a finite square well and an infinite square well, we see that the main difference is that the electron wave function extends outside the finite square well. The reason for this is that the electron can lower its KE by becoming less localized (less confined). It is this tendency of an electron to do whatever it can reasonably do to become less localized that leads ultimately to tunneling. For example, the non-zero value of the ground state wave-function between two wells in a double well system is intimately related to the difference in energy between the GS and 1st excited state, which is directly related to tunneling times.
One of our main themes this quarter was the energy cost of localizing an electron. This is something that has no analogue in non-quantum physics. This energy cost comes directly from the \(-\hbar^2 \frac{\partial^2}{\partial x^2} \frac{1}{2m}\) term of the Schro Wave Equation. Solutions to the Wave Equaton are called wave-functions. They have to start from zero and end up and zero, so that they are normalizable, and the bending associated with getting them to do that yields a second derivative the integral of which is always positive. So there is an inherent, unavoidable cost to localizing an electron, which is the essence of what is called x,p uncertainty. Really it is the expectation values of x^2 and p^2 that are relevant to what we call x,p uncertainty. The cost of localization manifested itself in every bound state system we studied, and even for the free electron wave-packet. Anyway, a deep understanding of how localizing and/or confining an electron has a cost and how that depends on the electron's mass and \(\hbar\) is a good thing.
There will be a question about angular momentum similar to problem 8 from HW 7. A key thing for that problem is to know that \(L_z = \frac{\hbar}{i} \frac{\partial}{\partial \phi}\) and to understand the relationship between cartesian and spherical coordinates.
You will also be asked to calculate expectation values involving 1st excited states of hydrogen. Reviewing the post and video on sp2 states and Dirac notation again will be helpful for that because it helps teach the skill of organizing and evaluating expectation values and matrix elements (using Dirac notation).
Basically, if you understand localization, tunneling, hydrogen atom states, expectation value calculations and mixed states, you should be good.
Tuesday, June 4, 2019
Physics 139b...
Courses offered in the fall of 2019 which are related to this class are, primarily, Physics 139b and a course on quantum computing. I think they will both be really interesting classes.
139b will be taught by Wolfgang Altmannshofer and is the direct sequel to this class. As such, Prof Altmannshofer and I have discussed explicitly what will be covered in each class
139b will be taught by Wolfgang Altmannshofer and is the direct sequel to this class. As such, Prof Altmannshofer and I have discussed explicitly what will be covered in each class
Monday, June 3, 2019
HW 9. operators. With solutions
Please turn this in to Michael Saccone's mailbox in the physics department mail room by 5 PM on Friday.
Please click this link to see solutions.
To do this you would need to first watch the videos in the post on raising and lowering operators. (May 29). The notation is the same as the book. The ground state is denoted as: \(|0\rangle\).
Please click this link to see solutions.
To do this you would need to first watch the videos in the post on raising and lowering operators. (May 29). The notation is the same as the book. The ground state is denoted as: \(|0\rangle\).
Homework plan for this week.
HW8: HW8 is due Wednesday night. I am going to grade HW 8 myself. Please submit that electronically
by emailing a pdf or good quality scan of your work to
physics139ahw@gmail.com.
HW9 will be pretty short. HW 9 will be graded by one of the TAs so that has to be a paper copy, as in the past. Please turn that in on Friday by 5:00 PM to Michael Saccone’s mailbox in the physics department mailroom.
HW9 will be pretty short. HW 9 will be graded by one of the TAs so that has to be a paper copy, as in the past. Please turn that in on Friday by 5:00 PM to Michael Saccone’s mailbox in the physics department mailroom.
Sunday, June 2, 2019
HW 8 guide. More notes on problem 2. Using overlap integrals in problem 2.
(see also the post: on Matrices, eigenvectors ...
Problem 2 involves a single electron in a double dot quantum well. There is a paper entitled:
"Quantum Coherence in a One-Electron Semiconductor Charge Qubit" that provides some pictures and background for this problem. Even though this problem is focused on the position of an electron (not its spin), it uses a notation involving spin matrices, which we covered recently. This sort of notation is common in physics as a way to model what are called two-level systems. That means systems which have only a total of two energy-eigenstates, or, more often, systems where we ignore (project out) all states except the lowest two. (The two of lowest energy.) Treating the double dot well as a two-level system tends to be a very good approximation because there are two states close to each other in energy and all other states have much higher energies.
Problem 2 involves a single electron in a double dot quantum well. There is a paper entitled:
"Quantum Coherence in a One-Electron Semiconductor Charge Qubit" that provides some pictures and background for this problem. Even though this problem is focused on the position of an electron (not its spin), it uses a notation involving spin matrices, which we covered recently. This sort of notation is common in physics as a way to model what are called two-level systems. That means systems which have only a total of two energy-eigenstates, or, more often, systems where we ignore (project out) all states except the lowest two. (The two of lowest energy.) Treating the double dot well as a two-level system tends to be a very good approximation because there are two states close to each other in energy and all other states have much higher energies.
Thursday, May 30, 2019
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.
Wednesday, May 29, 2019
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:
\(a_- | n \rangle = \sqrt{n}\: |n-1 \rangle \),
\(a_+ | n \rangle = \sqrt{n+1} \: |n+1 \rangle \).
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.
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.
Tuesday, May 28, 2019
HW 8. With solutions.
Solutions below.
Homework 8 includes spin problems and problems using spin matrices. Basically, what are called two level systems.
Homework 8 includes spin problems and problems using spin matrices. Basically, what are called two level systems.
Thursday, May 23, 2019
Notes and guide to Midterm.
This midterm will focus on the material we have covered since the last midterm.
Wednesday, May 22, 2019
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.
Tuesday, May 21, 2019
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!
Monday, May 20, 2019
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
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
Friday, May 17, 2019
Lecture on the origin of size: length scales in quantum systems.
This video explores the origin of the size of the hydrogen atom.
Wednesday, May 15, 2019
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.
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.
Tuesday, May 14, 2019
Problems for summer, atoms, screening...
Here are some possibly interesting problems that we may not have time to fit in this course.
Monday, May 13, 2019
May 14 class & additional notes.
Tuesday's class is important. We will cover H atom states and things you need to know
Sunday, May 12, 2019
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
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Midterm 2 solutions
Here are solutions to midterm 2. I see that I did not write it in the solutions, but it is helpful for sp2 type integrals to recall that \(\...
