## Wavefunction Wfor Electron Probability Density PitW1 Traveling and Standing Waves

The behavior of atomic scale particles is guided by a wavefunction, W(r,t), which is usually a complex number. The probability of finding the particle is given by the square of the absolute value

of the wavefunction. This quantity P is a probability density, so that the chance of finding the particle in a particular small region dxdydz is Pd%dydz.

Complex Numbers x+iy. The complex number is a notation for a point in the xy plane, where the symbol "i" acts like a unit vector in the y direction, formally obtained by rotating a unit vector along the %-axis in the ccw direction by je/2 radians. Thus i2=~l. The complex conjugate of the complex number, % + iy, is obtained by changing the sign of y, and is thus x-iy. The absolute value of the complex number is the distance r from the origin to the point x,y, namely r - <*? + y2)1/2 = {(x + iy) (x-iy)]1'2. (4.9)

A convenient representation of a complex number is rexp(i0) = r(co$0 + isin#), where 0= tan"1 (y/x). (4,10)

The wavefunction should be chosen so that P is normalized. That is,

if the integral covers the whole region where the electron or other particle may possibly exist.

There are other properties that a suitable wavefunction must have, as we will later discuss. Similarly, a many-particle ¥/(r1,r2,...,rn ,t) and probability P(r1}r2,...,rn ,t) can be defined.

The wavefunction for a beam of particles of identical energy p2/2m in one dimension is a traveling wave

W(x,t) = L~1/2exp(ikx-i(ot) = L~1/2[cos(kx-a)t) + isin (kx-wt)], (4.12)

where fc = 2jt/A and oj=2kv. According to the DeBroglie relation (4.6)

where h = h/2n. Similarly, from (4.7), a) = 2iiv=E/h. (4.14)

The normalization gives one particle in each length L, along an infinite %-axis. A point of fixed phase (such as a peak in the real part of the wave) moves as % = (w/k)t, so (co/k) is called the phase velocity vph=(a)/k) (4.15)

of the wave. Note that

W(x,t) = r1/2e(ikx-i£,t) = ri/2exp(ife%_iwt) (4.16)

has a constant absolute value at any describing a particle equally likely to be at any position on the infinite %-axis. There is no localization in this wavefunction since the momentum is perfectly described, implying Ax =

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