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This section covers how to solve the Schroedinger Eqn by separating x and t using the technique called separation of variables. This is possible because the potential energy V(x) is usually a function of only x, and not t.

Using the Schroedinger Eqn, boundary condition(s) and normalization condition, we will find still a number of (sometime infinite) solutions each of which is expressed as a product of a function of x and a function of t. The latter takes a form of <math>e^{-i\omega t}</math>, where <math>\omega = \frac{E}{\strike h}</math>. The spatial part depends on the potential energy involved in the problem, so at this point, we cannot write it explicitly. We will be spending the rest of the class figuring out this solution for samples of potentials.

The infinite square well is the special case of a potential well where the potential inside the well is 0 and the potential outside is infinite. There is zero probability for a particle to be outside the well. Solutions for ψ:

• must be a solution to the Schrodinger equation
• must be continuous

Also, ψ will not be smooth at the edges of the well.

Steady State Solutions: ψn=sqrt(2/L)*sin(nπx/L) (<math>\psi_n(x)=\sqrt{\frac{2}{L}}\sin{\frac{n\pi x}{L}}</math>) (where L is the width of the well, n is an integer)

The General Solution: Ψ(x,t)=∑c(n)*sqrt(2/L)*sin(nπx/L)*e^[(-i*n^2*π^2*hbar*t/(2*m*L^2)] <math>\psi(x,t)=\sum_{n=1}^{\infty}c_n\sqrt{2/L}\sin{\frac{n\pi x}{L}}\exp{-i\frac{n^2\pi^2\hbar}{2 m L^2}t}</math>

The <math>c_n</math>'s are just coefficients of the <math>\psi_n</math>'s as they are sumed, so that some of them are weighted more than others. In class, we discussed how <math>|c_n|^2</math> can be thought of as a “probability” that the particle is in state n. Really, the wavefunction describing the particle is composed of a little bit of many states, and the coefficients (the <math>c_n</math>'s) dictate how much state each contributes to the general solution of the particle.

The ENERGY of a state is contained in the time-dependent piece's exponent: <math>E_n=\frac{\hbar^2 k_n^2}{2 m}=\frac{n^2\pi^2\hbar^2}{2 m L^2}</math>

First of all, math expressions must be preceded by

<math> and followed by </math>

Here are some expressions that you may find useful.

\sum_{n=a}^{b}: sum n = a to b <math>\sum_{n=a}^{b}</math>
\int_{a}^{b}{f(x) dx}: integrate f(x) from x = a to b <math>\int_{a}^{b}{f(x) dx}</math>
\partial: partial derivative, <math>\partial</math>
\frac{a}{b}: fraction a/b <math>\frac{a}{b}</math>
\hbar: hbar <math>\hbar</math>
\infty: infinity symbol <math>\infty</math>
\pi: Greek letter pi <math>\pi</math>
\omega: Greek letter omega <math>\omega</math>
“A_a” or “A_{abc}”: subscript “a” or “abc” <math>A_{abc}</math>
“A^a” or “A^{abc}”: superscript “a” or “abc” <math>A^{abc}</math>

• blank line will indicate a new paragraph. If you want to force “new line” w/o having a new paragraph, add “\\”
• to make a list headed with a bullet mark, two blank spaces + “*” + another blank.
• to make a numbered list, two blank spaces + “-” + another blank.
• to make a headline of various sizes, click on “H1,” “H2,” etc. and fill in the space between “='s”. Down to H3 will be included in the Table of Contents at the right top. Since the chapter heading is H1, Section should probably be H2. Here are examples.