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1. Potential Problems in One Dimension --- Energy Eigenvalue Problems
1.1 Let's Talk --- Matching Conditions
In this talk we give the mathematical basis for demanding conditions that are imposed on the solution of time independent Schrodinger equation and its derivative.
In this talk we give the mathematical basis for demanding conditions that are imposed on the solution of time independent Schrodinger equation and its derivative.
1.2 Boundary condition at a rigid wall
We derive the boundary condition on a rigid wall as a limit of the boundary condition on a the wave function at a point where the potential has a finite jump discontinuity. It is shown that there is no restriction on the derivative of the energy eigenfunction. The only boundary condition is that the eigenfunction must vanish.
We derive the energy eigenvalues and eigenfunctions of a particle in a box of size \(L\).
1.4 Square Well Energy Eigenvalues and Eigenfunctions
The energy eigenvalue problem for a particle in a square well is solved. The energy eigenvalues are solutions of a transcendental equation which can be solved graphically.
The energy levels and eigenfunctions of a particle in a cubical box with rigid boundaries are derived.
1.6 Harmonic Oscillator ---- Eigenvalues and Eigenfucntions
The steps for obtaining energy eigenvalues and eigenfunctions are given for a harmonic oscillator. The details can be found in most text books, e.g. Schiff,"Quantum Mechanics"
2. Delta Function Potential
2.1 Delta Function Potential --- An overview
An overview of three methods to compute the energies and eigenfunctions of an attractive Delta function potential are given.
2.2 Delta Function Potential as Limit of Square Well .
A square well potential of range \(L\) and depth \(V_0\) tends to the Dirac delta function potential of strength \(g\) in the limit \(L\to 0\) and \(V_0\to \infty\) with the product \(V_0L\) held constant equal to \(g\). Using this fact we find the bound state solution of the delta function potential.
2.3 Dirac Delta Function Potential -Direct integration of the Schr\"{o}dinger equation
The energy eigenfunctions and eigenvalues for a particle in delta function potential are derived. It is found that, for an attractive delta function potential there is only one bound state.
2.4 Solution of Dirac delta function problem in momentum space
We solve for the bound state energy eigenvalue and eigenfunction of delta function potential using momentum space methods.
3. Barrier Penetration
3.1 Reflection and Transmission from a Potential
The reflection and transmission coefficients for a potential problem in one dimension are defined. For this purpose it is sufficient to know the behavior of the wave function at large distances. To set up the problem one needs to impose a suitable boundary condition on the wave function at large distances. It is shown that, for real potentials, the probability conservation implies that the reflection and transmission coefficients add to unity.
3.2 Reflection and Transmission Through A Square Barrier
The problem of transmission through a square barrier is solved and reflection and transmission coefficients are found.
3.3 The S -matrix in One Dimensional Potential Problems
S- matrix is defined for a particle incident on a potential in one dimension. The transformation properties of the S-matrix under time reversal and parity are given.
4. General Properties of Solution in 1 Dimension
4.1 LET's TALK --- NATURE OF ENERGY SPECTRUM
For a potential problem in one dimension there are three types of energy levels. These are (a) discrete, (b) continuous and non degenerate, and (c) continuous doubly degenerate energy eigenvalues. In this talk we explain the thumb rules to find out which of this cases apply for a given potential and a specified energy value.
4.2 Short Examples --- Nature of Energy Spectrum
We give several examples to explain how to obain the nature of energy spectrum without solving the problem completely.
4.3 General Properties of Motion in One Dimension
A discussion of nature of energy eigenvalues and eigenfunctions are discussed for general potentials in one dimension. General conditions when to expect the energy levels to be degenerate, continuous or form bands are given. Also the behaviour of eigenfunctions under parity and for also for large distances etc. are discussed.
