Question

In elementary quantum mechanics, the square well is used to model the behavior of a bound particle in which one or more forces (external potentials, interaction with other particles, etc) prevent or restrict its ability to move about. We have seen in class that the solutions to the Schrodinger equation in and around the quantum well result in a series of eigenvector wavefunctions with distinct energy levels. In this assignment, we will use MATLAB to create the system and experiment with different dimensions and barriers.

In MATLAB, model the finite square well scenario by building the Hamiltonian matrix and using the "eig" function to solve for the eigenvectors and eigenvalues. Experiment with potential barrier values of 500eV and 1000eV, and plot the resulting waveforms superimposed over the square well. What were the confined-state energy levels? What physical significance do the eigenvalues and eigenvectors have? Do some research on the Kronig-Penney model, in which a repeating series of finite square wells is used to model the behavior of particles within a crystalline material, and discuss the significance. What is different about the behavior of electrons in a single atom versus that in a solid material?

Answer 1

Hamiltonian operator operates on the wavefunction to produce the energy, which is a number, (a quantity of Joules), times the wavefunction. Such an equation, where the operator, operating on a function, produces a constant times the function, is called an eigenvalue equation. The function is called an eigenfunction, and the resulting numerical value is called the eigenvalue. Eigen here is the German word meaning self or own.

It is a general principle of Quantum Mechanics that there is an operator for every physical observable. A physical observable is anything that can be measured. If the wavefunction that describes a system is an eigenfunction of an operator, then the value of the associated observable is extracted from the eigenfunction by operating on the eigenfunction with the appropriate operator. The value of the observable for the system is the eigenvalue, and the system is said to be in an eigenstate.

According to Kronig -penney model, an electron is considered to move in a square well potential.

In the square well, the potential energy of an electron is zero at the positive ion site of the lattice and maximum between any two positive ion sites.

So the electron moves in the form of a periodic lattice i.e., in the form of rectangular potential barrier.

Atoms: In case of electrons of a single atom, they occupy a discrete set of energy levels and they have a set of quantized energies but not continuous energies.

Solids:

In case of solids, the potential energy different electrons will be different. The energy of the more tightly bound electrons changes little and they remain localized and their possible energy values now fall into bands of allowed values separated by gaps.

Consider a single atom of a particular species. This atom has a discrete set of energy levels. When another atom of the same species is brought close to the first atom, each of the energy levels becomes degenerate. But the interaction between the atoms breaks the degeneracy and the level splits into two separate energy levels. If this procedure is repeated >10²⁰ times, each energy level will have split into >10²⁰ level, effectively forming a continuous band of energy states.