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.