A particle in a time-independent scalar potential

If , independent of time, then solutions to the Schroedinger equation of the type exist.  They are called stationary solutions or stationary states.  They correspond to states in which the particle has a well defined energy   is a solution of the time-independent Schroedinger equation.  The time-independent Schroedinger equation is an eigenvalue equation of the linear operator H.

"Square" potentials

Assume V(x)=V=constant in certain regions of space.  In such a region the Schroedinger equation yields .

  1. Let E>V:
    The most general solution is , with A and A’ complex constants.
  2. Let E<V:
    The most general solution is , with B and B’ complex constants.
  3. Let E=V: , , with C and C’ complex constants.
    (Note: A solution exists in the classically forbidden region.)

How does the wavefunction behave at a point where V is discontinuous, i.e. at a step?

  1. At a finite step the boundary conditions are that and are continuous.
  2. At an infinite step is discontinuous, but it has a finite discontinuity.  Therefore remains continuous as

The WKB approximation for bound states

Consider a one-dimensional problem.  We want to find the energies of the bound states of a particle in a given potential well.  The WKB approximation requires that  or .  Here denote an integral over one complete cycle of the classical motion and   .  The WKB method for bound states therefore leads to the Wilson-Sommerfeld quantization rule except that n is replaced by .  It leads to a quantization of the classical action  .

The Harmonic Oscillator

Consider a particle of mass m moving in a potential with .

Define , where . 

do not commute. .

Define  .   Then     .

Define the number operator . Then .

.

.

The eigenvectors of N are {|fn>} with eigenvalues {n}, where n is a non-negative integer.

The eigenvectors of  are {|fn>} with eigenvalues {n+}.

The eigenvectors of H  are {|fn>} with eigenvalues {}.

The results of operating with a or aT on |fn> are given by

.  The matrices representing a and aT in the {|fn>} basis are

.

The order of the basis vectors is

.

The eigenfunctions of the Harmonic oscillator

.

is the product of  and a polynomial of degree n and parity (-1)n called a Hermite polynomial.

 bx.

The Hermite polynomials Hn(z) are defined by

or  .

.

The generating function of the Hermite polynomials is .

The recurrence relations are ,  ,  and  .

The differential equation satisfied by the Hermite polynomials is .

Normalization:

Orthogonality: