HW 8&9 solutions

HW 7 solutions

Review session for Final, Sunday 3:00 PM, ISB 231 (see also IP 2 soln notes 4 posts down

Extra credit

Anyone wanting to get extra-credit toward an A, may work on Wilson's problem.

For extra credit toward a B, you could do the same sort of analysis for the matrix:
0, i hbar, o
-i hbar, 0, 0
0, 0, 0

which is the matrix of Lz in the basis: 100, 010, 001 ; which spans the 3-fold degenerate 1st excited state manifold of the 3D H.O.

These should be handed in in Sunday and should be be well-organized, clear and attractive. The key thing is to look deeply at what the eigenvectors are telling you about the nature of the spatial states. The credit begins with that part, and the clear, cogent discussion that accompanies it.

Wilson's matrix

Dear Wilson,
. Here is the matrix I promised you. I hope you like it as much as I do. I think it is very beautiful.
. This is the matrix of Lz in the sub-basis: 101, 011, 110, 200, 020, 002; which spans the 6-fold degenerate 2nd-excited state manifold of the 3-dimensional harmonic oscillator. We worked on our choice for basis order, and calculating the matrix, in office hours today. Our guiding aesthetic --that influenced and guided our choice of basis-state order, was connection. We wanted to put states which "connected" next to each other as much as possible. I imagine you are pretty excited right now. I would be too in your shoes.
. If you feel inspired to do so, please solve for the eigenstates and then write a cogent summary of the nature of the states, their organization and meaning. One thing about this matrix is, I believe, that there are two states with (Lz) e.v. of zero? Do you get that too? What does that mean? How can you handle those to get maximum insight? Are some of the eigenstates related to flower states in any way? These are some of the questions I would like to see addressed in your report. I trust that you will also come up with other questions and resolutions.
. I would like to also invite everyone in class to do this, if they like. It provides an excellent opportunity to develop and exhibit your understanding of matrices, degeneracy and angular momentum in QM. Ideally I would suggest you do this by Sunday --you can hand it in at the review section-- that way you will be able to focus on reviewing 1D QM, the hydrogen atom states and degeneracies, and related topics after the Sunday review section.

Best regards,

-Zack

Summary: descent of QM

This picture, while somewhat whimsical, provides a perspective on the relationships between the different "realms" of quantum physics we have explored this quarter. On the left are the systems of higher dimensionality ( 2 and 3D); on the right, the 1D quantum systems. The key difference between those two classes (1D vs higher D) is degeneracy. In 1D there is usually only one eigenstate associated with a particular energy; in higher D there may be several eigenstates which have the same energy.
This is called degeneracy. The origin of degeneracy is symmetry. The degeneracies we see in the 3D harmonic oscillator and in the hydrogen atom eigenstates arises from the apparent and less apparent symmetries of these potentials. We say that the eigenstates which have the same energy belong to a particular degeneracy manifold.
. .For the 2D harmonic oscillator, for example, the two orthogonal states 10 and 01 span the 1st excited state degeneracy manifold. We also found, through our exploration via the Lz matrix, that the two states 01 +i 10 and 01-i 10 also span this 1st excited state degeneracy manifold and, moreover, that they are Lz eigenstates with e.v.'s of hbar and -hbar.
For the 3D harmonic oscillator there are three 1st excited states: 100, 010 and 001. You can show that these are all l=1 eigenstates, meaning that they are eigenstates of the operator L^2 with e.v. 2 hbar^2, however, they are not all eigenstates of Lz. Using the Lz matrix method (i.e. calculating the matrix of Lz, finding its eigenvectors and using them to create a new basis of this 3-fold degenerate manifold) one finds that the three orthogonal states: 100+i 010, 001, and 100-i 010 also span the 1st-excited state degeneracy manifold and that they are Lz eigenstates (with e.v.'s of: hbar, 0 and -hbar, respectively.)
Let's pause on this for now and look back at what we covered in 1d QM.

1D quantum physics: highlights:
Let’s start this summary from a phenomenological perspective and focusing on solutions of the time-independent Schrodinger equation in 1 dimension. Our key focus there was on calculating quantized energies for particular potentials and examining the nature the energy eigenstates.

o Bound states are created by an attractive potential. They tend to have a characteristic length scale that depends on the strength or spatial extent of the potential.
o Confinement leads to kinetic energy. This is most evident and clearly illustrated in the nature of ground states. Kinetic energy is generally of the form hbar^2/m a^2, where a is a characteristic length scale. The 2nd derivative term in the Schrodinger equation means that there is a cost to bending the quantum state function. Psi(x) has to rise high enough that the integral of (area under) Psi^2 is 1 and then come back to zero. When this happens over a short range of x, there is a high cost in kinetic energy.
o Excited states have nodes. For a given potential, you will find that if you order eigenstates according to their energy, from lowest to highest, they will also be ordered according to the number of nodes they have.
o Some 1D potentials have an infinite number of bound states; some have a finite number of bound states
o Potentials that have a finite number of bound states also have unbound (extended) states. These are not normalizable and they are characterized by a continuous “quantum number”, usually called k or q, rather than a discreet quantum number such as n. [ For example, for a constant potential there are zero bound states and the extended states can be written as exp[ikx], where k is any real number.]
o Potentials we have studied include:
1. Attractive delta function: 1 bound state
2. Finite square well: a finite number of bound states
3. Infinite square well: infinite number of bound states
4. Harmonic oscillator: infinite number of bound states
o For a given potential, all the eigenstates together span the space of all state functions

The above list essentially talks about characteristics of the spatial eigenstates (the eigenstates of the time-independent Schrodinger equation), however one of the most important capabilities of the quantum theory is the ability to calculate and predict the time evolution of the state function.

Usually one is given the state function at a particular time, e.g., t=0. The state function at any later time can then be calculated with no uncertainty. This is done by writing the state function at t=0 as a linear superposition of energy eigenstates. The time dependence for each energy eigenstate is exp[-iEnt/hbar], where En is the energy of the nth eigenstate.. Thus one obtains the time evolution of the state function.

The belief that time evolution of the state function can be obtained in this way can be viewed as the central dogma of quantum mechanics. The analogy to classical mechanics is as follows:
* In classical mechanics one is given the initial conditions, usually position and velocity, and one then calculates the position and velocity as a function of time. The time evolution of position and velocity is governed by Newton's relation: F=ma.
* In quantum mechanics one is given the initial conditions in the form of a state function and one calculates the time evolution of this state function. The time evolution of the state function is governed by the Schrodinger equation.
There is no intrinsic uncertainty in either system. For simple, solvable systems, if the position and velocity are precisely specified at t=0, and the potential (force) is know, then the position and velocity in the future can be obtained with precision. The same is true for the state function. If the state function is precisely specified at t=0, then, for a simple, solvable system, the future state function can be obtained with precision.

IP 2 solution notes: finding Lz eigenstates and exploring degeneracy manifolds

Here are some notes related to a solution of the IP2 problem, which involved exploring and elucidating the degeneracy manifolds of the 2D harmonic oscillator by finding Lz eigenstates from states constructing as products of 1D HO energy eigenstates.
What are we doing here? What is this problem all about? We can step back and take a broader perspective on this whole endeavor. This is not simple, but it may be helpful. Our central conundrum is associated with the realization that whenever there is degeneracy there are an infinite number of ways to span the degeneracy manifold. If we only look at one of them, then we have no perspective on this essential aspect, which is critical to understanding QM in more than 1D.

The things we do, like getting x,y,z states from the Psi_n,l,m and, on the other hand, getting Lz eigenstates (m) from x,y states of the 2D HO, are all related to gaining perspective in this way. Without this our perspective would be very limited.