130_notes.dvi

Griffiths 1.2, 1.3

Cohen-Tannoudji et al. Chapter

4.1 Derivations and Computations

4.1.1 Review of Complex Numbers

This is a simple review, but, you must make sure you use complex numbers correctly. One of the
most common mistakes in test problems is to forget to take the complex conjugate when computing
a probability.

A complex numberc=a+ibconsists of a real partaand an imaginary partib. (We chooseaand
bto be real numbers.)iis the square root of -1.

The complex conjugate ofcisc∗=a−ib. (Just change the sign of all thei.)

The absolute square of a complex number is calculated by multiplying it by its complex conjugate.

|c|^2 =c∗c= (a−ib)(a+ib) =a^2 +iab−iab+b^2 =a^2 +b^2

This give the magnitude squared of the complex number. The absolute square is always real.

We will use complex exponentials all the time.

eiθ = cosθ+isinθ

e−iθ = cosθ−isinθ

You can verify that the absolute square of these exponentials is always 1. They are often called a
phase factor.

We can write sinθ=e

iθ−e−iθ

2 i and cosθ=

eiθ+e−iθ

2.

As with other exponentials, we can multiply them by adding the exponents.

eikxe−iωt=ei(kx−ωt)

4.1.2 Review of Traveling Waves

A normal traveling wave may be given by

cos(kx−ωt).

The phase of the wave goes through 2πin one wavelength inx. So the wavelengthλsatisfies

kλ= 2π.