Feel free to write anything related to what you have learned in Chapter 2 or wanted to learn but did not learn enough.

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 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>

\sum_{n=a}^{b}: sum n = a to b
\int_{x=a}^{b}: integrate x = a to b
\frac{a}{b}: fraction a/b
\hbar: hbar
\infty: infinity symbol
\pi: Greek letter pi
\omega: Greek letter omega
“_a” or “_{abc}”: subscript “a” or “abc”
“^a” or “^{abc}”: superscript “a” or “abc”

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