The Electron as a Wave

1
The Electron as a Wave

• Chapter 2 of Solymar

2
Introduction

• Electrons as waves
• Difference between particles and waves
• Waves interfere with each other
• Particles can be counted

3
Thought Experiment

• Difference between particles and waves (Figs. 2.1
  2.2)
• If we fire bullets at a screen with two slits
• When one slit is closed and the other is open
  Number of bullets P1 P2
• When both slits are open P12 P1 P2 (simple
  superposition)
• If waves propagate through a screen with two
  slits
• When one slit is closed and the other is open
  Wave intensity
• When both slits are open P12 ? P1 P2
  (interference)

4
de Broglie Wavelength

• Question
• Do electrons behave like bullets or waves?
• de Broglie suggested that electrons have wavelike
  properties with a wavelength inversely
  proportional to the momentum
• h Planck constant 6.6?10-34 Js
• Wavelike properties demonstrated through electron
  diffraction

5
Electron Diffraction

• An electron beam incident on a crystal (Fig. 2.3)
• Davisson and Germer did it on a nickel crystal
• The periodic array of atoms reflected electrons
• Reflected electrons showed an interference
  pattern
• Constructive interference when the path
  difference satisfy
• Destructive interference when
• n 1, 2, 3,

6
l for Crystal Diffraction

• Not any wavelength would work for crystal
  diffraction
• Upper limit for l
• Lower limit for l
• The angle difference between two successive
  maxima
• The detectors resolution is limited, so a large
  Dq is desired
• Wavelength should be close to d, say 0.1 nm

7
Energy of the Electrons

• Wavelength should be around 0.1 nm
• Electron velocity
• Accelerating voltage

8
Macroscopic Objects

• Questions
• Why macroscopic objects dont have wavelike
  properties?
• Where is the boundary between microscopic objects
  with wavelike properties and macroscopic objects
  with no wavelike properties?
• Everything has wavelike properties
• Electrons, neutrons, protons, you, and me
• For a bullet, m 10-3 kg, v 103 m/s
• Practically, there is no way to measure this wave
  amplitude!
• Macroscopic objects do have wavelike properties.
  Its just we are not capable of measuring them!

9
Electron Microscope

• The wave nature of light allows an optical
  microscope to magnify a small object
• To have better magnification, a shorter
  wavelength is required
• To have even shorter wavelengths, we can use
  electrons
• Electron wavelengths are thousands of times
  shorter than violet light
• Electrons can be focused by electric fields
• Magnification in electron microscope allows
  observation of molecules (nm)
• Two major types scanning electron microscope
  (SEM) and transmission electron microscope (TEM)

10
Some Properties of Waves

• Wave function
• Phase velocity
• Velocity of wave with a single frequency
• When several waves are superimposed
• an and wn are related to kn
• Or in a continuum case
• a(k) and w are functions of k
• Consider t 0

11
A Special Case

• For t 0
• Consider the case of Fig. 2.4
• a(k) 1 for
• a(k) 0 outside this interval

12
Wave Packet

• The solution plotted in Fig. 2.5
• The wave function is essentially contained in a
  packet
• Define width where amplitude drops to 0.63 of its
  maximum
• The relation between spread in wave number and
  space
• When wave spreads more in wave number, its more
  limited in space, and vice versa

13
Time Varying Case

• Consider time varying case, still limit a(k) to
  Dk
• Two velocities
• Phase velocity velocity of the central
  frequency
• Group velocity (w - wo)t - (k - ko)z constant

14
Applications to Electrons

• The electron is described with a wave packet
• As a wave
• As a particle
• Differentiate
• Integrate
• This is de Broglie equation

15
Electrons Location

• The location of the electron is the location of
  the wave packet
• The location of the wave packet is Dz
• Uncertainty about the location of the electron
• This is the popular form of the uncertainty
  principle
• If Dz 1 nm
• for electrons
• for bullets

16
HW Assignment

• 2.1 2.3 2.4 2.5