PHYS 424 Notes

1. Overview
   Our task is to solve -[hbar²/(2m)]d²Y/dx² + V(x,t) Y = i hbar dY/dt for various useful potentials V.

   We will start with potentials that lead to the simplest and most general mathematical procedures. These potentials are a bit artificial, but can in fact be used to learn some interesting physics. We will procede to somewhat less artificial potentials which lead to substantially more complicated mathematics. However, completely realistic potentials can only be handled numerically. We will not undertake numerical procedures in this course.

   Our first and least damaging simplification is to take V(x,t) to be time independent, V(x,t)=V(x). In this case we can seek solutions in the form Y(x,t) = y(x)f(t). We will find a set of f(t), and any arbitrary time dependence of the wave function can be written as a linear combination of these functions. [This technique can be attempted any time we can write the differential equation as a sum of terms each depending on just one variable. We will use it again and again.] We will then look at a series of simple potentials depending on one spatial variable only:

   1. V(x)=0, 0<=x<=a and V(x)=infinity, otherwise.
   2. V(x)= (1/2)kx² (the harmonic oscillator)
   3. V(x)=constant
   4. A large, very narrow potential
   5. V(x)= - V₀, |x|<a, and V(x)=0, otherwise
   6. Eventually, we will consider potentials dependent on all of x, y, z.

   These potentials can be used to describe, very approximately

   1. Atomic Nuclei, although the harmonic oscillator does a better job
   2. The lowest levels of any broad potential
   3. Free particles
   4. Any short-range potential
   5. Metals

2. Stationary States
   

   -[hbar²/(2m)]d²Y/dx² + V(x,t) Y = i hbar dY/dt

   Take Y(x,t) = y(x)f(t) and substitute:

   -[hbar²/(2m)]d²y/dx² f + V(x,t)yf = i hbar ydf/dt

   and after dividing by yf

   -[hbar²/(2m)][1/y] d²y/dx² + V(x) = i hbar (1/f) df/dt

   The left-hand-side of the last equation is a function of x only; the right-hand-side is a function of t only, and the equation must be valid for all x and t. For instance we could have E+ax = E+bt for all x and t. The only way this can be true is for both sides to be a constant [in the example we must have a=0 and b=0]. Choosing the constant to be E, we have

   -[hbar²/(2m)][1/y] d²y/dx² + V(x) = i hbar (1/f) df/dt = E

   which is two equations

   -[hbar²/(2m)] d²y(x)/dx² + V(x)y(x) = Ey(x)
   i hbar df(t)/dt = Ef(t)

   The second of these is a first-order ordinary differential equation and therefore has a general solution with one arbitrary constant. Substituting

   f = C e ^-iwt

   gives the algebraic equation
   hbar w C e ^-iwt = E C e ^-iwt
   whose solution is w = E / hbar , giving

   Y(x,t) = C y(x) e^-iEt/hbar

   The constant C may be lumped into the still-arbitrary y(x). The constant E is still arbitrary, so there are an infinite number of solutions available. We will show later that any interesting time dependence of Y may be fit by sums over these functions.

   The equation for y(x),

   -[hbar²/(2m)] d²y(x)/dx² + V(x)y(x) = Ey(x)
   [p²/(2m) + V] y = Ey

   is known as the time-independent Schroedinger Equation, and its solution for a number of simple potentials is our next task. Meanwhile notice that the solutions Y rather obviously have definite energy E, and less obviously that for any expectation value of a function Q(x,p),

   <Q(x,p)> = ò Y^* Q(x,p) Y dx
   = ò y^* Q(x,p) y dx .

   Hence, when V is independent of t, all physics can be determined from the time-independent equation alone. An equation of this type, with operator y = number X y, is known as an "eigenvalue" problem; E is the eigenvalue and y is called the eigenfunction. In the most familiar examples the operator is a matrix, but it is useful to define "eigenvalue" to include more general linear problems. The solution process has some surprises in it, as we will see by example below.

Last revised 2005/02/11

Part 1, 2, 3, 4. 5, 6, 7, Schedule Outline