MIT 量子力学作业4

Homework 4

3.23 Electrical, Optical and Magnetic Properties of Materials, Fall 2014

Due on Monday, October 13 at noon

in the problem set box outside 13-4138

PROBLEM 1.

Consider a particle in a 1D potential box of length L.

1)Quickly derive the eigenstates of this problem and their respective energies.

2)If a particle is in the ground state of such a well and the potential then instantly double its size

(length: 2L). What are the eigenstates of the new potential? Project the initial wave function on these harmonics.

3)Using the relevant harmonics (after justifying why they are relevant), describe the motion of the

particle. What is the period of this movement?

4)After a certain amount of time, the particle is measured at the fourth energy state. Calculate the

energy change of the particle. Does it mean that conservation of energy is not respected in quantum mechanics?

PROBLEM 2.

Below are represented two eigenstates of the Hydrogen atom. For each of them, answer the following questions:

1)What is the number of angular nodes? What is the number of radial nodes?

2)What is the energy of the orbital? What is the principle quantum number n? What is the

degeneracy of this energy level?

3)What is the l quantum number? What is the name of the orbital (eg: 1s, 2p, etc.)?

Figure 1: orbital 1 and orbital 2

PROBLEM 3.

The Hückel model treats planar hydrocarbons by considering only the unsaturated C atoms (i.e. those C atoms for which not all of the outer shell electrons are already involved in σ-bonding with H or C atoms.) Hückel ‘s technique is based on the variational method using a basis of p z orbitals localized on these unsaturated C atoms, neglecting all overlap integrals and making the further simplifications:

1) For each of the structures above, identify the unsaturated C atoms, which contain the pz orbital

considered in the Hückel model.

2) For each of these structures, write out the Hückel determinantal equation (i.e. the matrix whose

determinant gives the energy levels of the system).

3) Solve for the energy levels, and sketch an energy level diagram including any degeneracies.

PROBLEM 4. Variational principles

The idea behind using the variational principal is to find a function as close as possible to the ground state of the system by starting with a family of functions expected to reasonably contain a good approximation the actual ground state.

1) Consider the (1D) harmonic oscillator: H ̂=p²̂2m +12mω²x²

̂. First, we will consider a rational function as a possible eigenstate.Ψa (x )=

1x²+a (a is a parameter). Is Ψa normalized? Calculate E a =<Ψa |H

̂|Ψa >. Minimize this energy with respect to the parameter a . What is the value of a that gives the energy minimum? What is the minimum energy? Calculate

ΔE E o =E a −E 0E o

, where E 0 is the true ground state energy.

2) Now, let’s consider another test function Ψa (x )=e −ax². Following the same process as before, what is the closest state to the ground state? What is the value of ΔE E o ?

PROBLEM 5.

Consider a cubic crystal where atoms of mass M 1 lie on one set of planes and atoms of mass M 2 lie on planes interleaved between those of the first set. Let a represent the repeat distance of the lattice in the direction normal to these planes. Consider waves that propagate in a symmetry direction for which