PHYS-2402
Sep. 22, 2016
Lecture 8
Announcement
HW3 (due 9/27)
Chapter 3
13, 17, 23, 25, 28, 31, 37, 38, 41, 44
Answer Keys for HW1 & 2
Course webpage
http://highenergy.phys.ttu.edu/~slee/2402/
Textbook
Chapter. 3Wave & Particles I
Outline:
• Blackbody Radiation (Plank; 1900; 1918*)• The Photoelectric Effect (Einstein; 1905; 1921*)• The Production of X-Rays (Rontgen;1901; 1901*)• The Compton Effect (Compton; 1927; 1927*)• Pair Production (Anderson; 1932; 1936*)• Is It a Wave or a Particle? à Duality?
EM-“Waves” behaving like “Particles”
Is It a Wave or a Particles?
Duality
Ch.3 E&M-Waves behaving like Particles
Ch.4 Particles behaving like Waves
Wave D>!
Particle D<<!
Is It a Wave or a Particles?
Duality
Ch.3 EM-Waves behaving like Particles
Ch.4 Particles behaving like Waves
Wave D>!
Particle D<<!
Double-slit Diffraction
Experiment
light
INDIVIDUAL
PHOTON
HITS
Although diffraction of light is
a wave phenomenon,
there is no smooth distribution of light
in the diffraction pattern,
but
the pattern is rather formed of many
individual hits of particles – the photons
A single photon
DOES NOT
get “disintegrated” in the
Diffraction process
to make a smooth
diffraction pattern
Coming back to that soon…
Electromagnetic
waves (light)
Particles
(photons)
Massive
ParticlesWaves
?
Chapter. 4
Wave & Particles II
Outline:
• A Double-Slit Experiment (watch “video”)• Properties of Matter Waves• The Free-Particle Schrödinger Equation• Uncertainty Principle
• The Bohr Model of the Atom
• Mathematical Basis of the Uncertainty Principle –
The Fourier Transform
“Matter” behaving as “Waves”
Electron passing through a wide slit arrive at a screen
one by one and produce a bright region essentially
the same width as the slit
Very strange
With a single narrow slit,
Electrons arrive one by one all over the screen.
They don’t avoid any points …
Very strange
Like a
Wave
…
But when 2 slits
are open,
there are certain
points at
which no
electron ever
arrives.
Each electron
wave is
destructively
interfering
with itself.
Electrons producing a double-slit interference pattern - one particle at a time
Actual double-slit pattern produced by electrons
Bragg’s Law• Bragg's law states that when X-rays hit an atom (in a x-tal), they
make the electronic cloud move as does any EM-wave. The movement of these charges re-radiates waves with the same frequency; this phenomenon is known as the Rayleigh scattering.
• These re-emitted wave fields interfere with each other either constructively or destructively, producing a diffraction pattern on a detector or film. The resulting wave interference pattern is the basis of diffraction analysis. [..excellent probe for small length scale]
Diffraction of a beam from multiple atomic planes
Bragg’s Law• The interference is constructive when the phase shift is a multiple of
2π; this condition can be expressed by Bragg's law,
• n is an integer determined by the order given, λ is the wavelength of the X-rays, d is the spacing between the planes in the atomic lattice, and θ is the angle between the incident ray and the scattering planes.
• Bragg's Law is the result of experiments into the diffraction of X-raysoff crystal surfaces at certain angles. Bragg's law confirmed the existence of real particles at the atomic scale, as well as providing a powerful new tool for studying crystals in the form of X-ray diffraction.
Bragg’s Law
Electron wave diffracted by AlMn quasicrystal De Broglie HypothesisThe E&M waves can be described using the language of quantum particles (photons). Q:: Can particles behave as waves?
De Broglie (1923) suggested that a plane mono-energetic wave is associated with a freely moving particle:
( ) ( )0i t kxx e ω −Ψ =Ψ
This wave travels with the phase velocity
This is a solution of the wave equation in 1-dimension:
2 22
2 2vt xψ ψ∂ ∂=
∂ ∂
vkω
=
for photon, E = pc and E = hfhf = pc => h = pc/f = pλ so, λ = h/p
We’ll apply the same logic which helped us to establish the relationship between p and λ for photons:
2 hk pπ
λ = =
- depends on the momentum rather then energy (e.g., for an object at rest, λ = infinity)
Compare with Compton wavelength of the particle
De Broglie wavelength
p - the object’s m/m
Chmc
λ =
4.2 Properties of Matter Waves• What properties characterize a wave?• λ, f, v, also A that varies with position/time (i.e. (x,t))• Wave function: we use different symbol for wave function
of different kinds of wave
(1) For transverse wave on string; y(x,t)-- string’s transverse “displacement”
(2) For E&M plane wave; E(x,t) &B(x,t)-- describe how the oscillating E- & B-field vary with (x,t)
(3) For matter wave;ψ(x,t) = probability amplitude|ψ(x,t)|2 è tell us probability of finding the particle(see next slide for some details)
Properties of
Matter Waves
Amplitude =
function of (x,t)),( tx!
Probability
Amplitude
Amplitude =
function of (x,t)
),( tx!
Probability
Amplitude
The probability density
to find
a particle
at coordinate x, at time t
2|),(| tx!
2|),(| txdxdtP != !!
The probability
to find
a particle
in an interval """"x, """"t
Integrate over """"x, """"t
x
t 2|),(|)( txdtxP != !
4.2 Properties of Matter Waves• Wave length
– De Broglie’s hypothesis: ç=è– λ = Wavelength of matter wave– Experimentally confirmed; e.g. x-tal diffraction– de Broglie got Nobel prize in 1929
2 hk pπ
λ = =C
hmc
λ =
• Frequency (of matter waves)– f = E/h– now, it’s open more convenient to express λ = h/p and f = E/h in
terms of the following quantities;k = 2π/λ (wave number), w = 2π/T (angular frequency)
– Another convenient definition: h = h/2π = 1.055 x 10-34 Js– express the fundamental wave-particle relationship as
!p = "!k E = !ω
Properties of
Matter Waves
Wavelengthp
h=!
De Broglie (1929)
Frequencyh
Ef =
Properties of
Matter Waves
Speed !fv =
BUT
v is NOT the velocity
of the massive particle,
it’s the velocity of the
Matter wave
It does give the wave speed
v = (E/h)(h/p) = E/p
c.f. for photon, E = pc
Electromagnetic
waves (light)
Particles
(photons)
Massive
ParticlesWaves
?