Lecture 17

Review

L17.P1

Schrödinger equation

The general solution of Schrödinger equation in three dimensions (if V does not dependon time)

is

where functions are solutions of time-independent Schrödinger equation

If potential V is spherically symmetric, i.e. only depends on distance to the origin r,

spherical harmonics

then the separable solutions are

where and are solutions of radialequation

Hydrogen-like atom energy levels:

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Identical particles

Bosons and fermions

In classical mechanics, you can always identify which particle is which. In quantummechanics, you simply can't say which electron is which as you can not put any labels onthem to tell them apart.

There are two possible ways to deal with indistinguishable particles, i.e. to constructtwo-particle wave function from single particle wave functions and that is noncommittal to which particle is in which state:

Symmetric

Antisymmetric

Therefore, quantum mechanics allows for two kinds of identical particles: bosons (for the "+" sign) and fermions (for the "-" sign). N-particle states are constructed in the same way, antisymmteric state for fermions (which can be easily written as Slater determinant) and symmetric state for bosons; the normalization factor is . In our non-relativistic quantummechanics we accept the following statement as an axiom:

All particles with integer spin are bosons,all particles with half-integer spin are fermions.

Note that it is total wave function that has to be antisymmetric. Therefore, for example if spatial wave function for the electrons is symmetric, then the corresponding spin state has to be antisymmteric. Note: make sure that you can add angular momenta and know what are singlet and triplet states.

L17.P2

From the above, two identical fermions can not occupy the same state. It is called Pauli exclusion principle.

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L17.P3

Perturbation theory

General formalism of the problem:

Suppose that we solved the time-independent Schrödinger equation for some potentialand obtained a complete set of orthonormal eigenfunctions and correspondingeigenvalues .

This is the problem that we completelyunderstand and know solutions for.

We mark all these solutions and the Hamiltonian with " " label.

Now we slightly perturb the potential:

The problem of the perturbation theory is to find eigenvalues and eigenfunctions of theperturbed potential, i.e. to solve approximately the following equation:

using the known solutions of the problem

Nondegenerate perturbation theory

We expand our solution as follows in terms of perturbation H'

The first-order correction to the energy is given by:

First-order correction to the wave function is given by

Note that as long as m ≠n, the denominator can not be zero as long as energy levels arenondegenerate. If the energy levels are degenerate, we need degenerate perturbation theory( consider later).

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L17.P4

Degenerate perturbation theory

Suppose now that the states are degenerate, i.e. have the same energy .

How to calculate first-order energy correction E1?

In the case of n-fold degeneracy, E1 are eigenvalues of n x n matrix

The second-order correction to the energy is

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L17.P5

Notes on the exam preparation & exam taking:

1. Make sure that you know, understand and can use all formulas and concepts from this lecture.

2. Make sure that you can solve on your own and without looking into any notes any problem done in class in Lectures or from homeworks (if integrals are complicated, use Maple, Matematica, etc.)

3. During exam, look through all the problems first. Start with the one you know best and the one that is shortest to write a solution for.

4. Make sure that you read the problem very carefully and understand what is being asked. If you are unsure, ask me.

5. To save time, make sure you are not repeating the same calculations. For example, if you need to do several similar integrals, make sure that you are not redoing the ones you have already done.

If you are out of time and you have not finished, write an outline of what you would do to finish the problem if you had time.

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Problem 1

Three electrons are confined into a one-dimensional box of length a. The confinement potential is

Ignore the Coulomb interaction between electrons. What is the ground state? Find its energy and write the corresponding wave function(s). Is it degenerate?

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Problem 1

Three electrons are confined into a one-dimensional box of length a. The confinement potential is

Ignore the Coulomb interaction between electrons. What is the ground state? Find its energy and write the corresponding wave function(s). Is it degenerate?

Solution

Electrons have spin 1/2. Therefore, the eigenstates and eigenvalues are

Each energy level can contain, therefore, two electrons. There are two possible configurations for the ground state:

that correspond to the wave functions

The ground state energy is

It is degenerate (d=2).

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Problem 2

Consider a particle in the two-dimensional infinite potential well:

The particle is subject to the perturbation

where C is a constant. Calculate first-order corrections to the energies of the ground state and first excited state.

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Problem 2

Consider a particle in the two-dimensional infinite potential well:

The particle is subject to the perturbation

where C is a constant. Calculate first-order corrections to the energies of the ground state and first excited state.

Solution

The lowest-order energy wave functions and energies are given by

The ground state nx=ny=1 is non-degenerate

Therefore, the first-order correction to the energy is

The first excited state is doubly degenerate:

Therefore, we need to use degenerate perturbation theory to find first-order energy correction to thefirst excited state, i.e. we need to find eigenvalues of matrix

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P2

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Problem #3

Calculate the Fermi energy for noninteracting electrons in a two-dimensional infinite squarewell. Let be the number of free electrons per unit area.

3

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Possibly useful formulas

Solutions for one-dimensional infinite square well of width a:

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