Download Modern Physics (PHY 251) Lecture 18

Modern Physics (PHY 251)
Lecture 18
Joanna Kiryluk
Fall Semester Lectures
Department of Physics and Astronomy, Stony Brook University
• Matter waves
• Rutherford Scattering
• Models of an Atom
Duality of light and matter …
1
Matter Waves
De Broglie: the dual (wave-particle)
behavior of radiation applies to matter
(symmetry of nature)
• A photon has a light wave associated with it that
governs its motion
• A particle with non-zero rest mass has an
associated matter wave that governs its motion.
https://www.youtube.com/watch?v=Vsm2ubwuA_g
https://www.youtube.com/watch?v=JIGI-eXK0tg
(documentary/biography < 5.33min)
2
Matter Waves
De Broglie: the total energy of an entity E (momentum p) is
related to the frequency n (wavelength l) of the wave associated
with its motion:
Note:
E = hν , p = h λ
for massive (rest mass >0) particles:
ln=c
predicts the de Broglie wavelength l of a matter wave
λ = h p associated with the motion of a particle with momentum p.
h ≈ 6.6 ×10 −34 J ⋅ s
h ≈ 4.1×10 −15 eV ⋅ s
3
Example: What’s the de Broglie’s wavelength (in meters) of
a) a baseball of m=1kg moving at a speed of v=10m/s?
b) an electron whose kinetic energy is 100 eV (i.e. non-relativistic)?
m0= 0.511 MeV/c 2
E = hν , p = h λ
1eV ≈ 1.6 ×10 −19 J
h ≈ 6.6 ×10 −34 J ⋅ s
h ≈ 4.1×10 −15 eV ⋅ s
The wave nature of light/matter propagation is not revealed by experiments when the important
dimensions of the apparatus used are very large compared to the wavelength of light/matter. 4
Double-Slit Experiment:
cannot predict where electron would land
(Quantum particle)
or Light source
The same result!
http://freevideos.astronomycentral.net/video/Xmq_FJd1oUQ/Quantum-Physics-made-simple-WaveParticle-Duality-Animation.html
https://www.youtube.com/watch?v=M4_0obIwQ_U (Episode: Quantum Leap)
Interactive simulations (play@ home):
http://phet.colorado.edu/en/simulation/quantum-wave-interference
7
5
Group work:
1eV ≈ 1.6 ×10 −19 J
h ≈ 6.6 ×10 −34 J ⋅ s
h ≈ 4.1×10 −15 eV ⋅ s
6
Matter Waves – Do they exist? Yes!
1927 Experiments by Thompson (UK), Davisson and Germer
(USA): diffraction of electron beams confirmed de Broglie
relation
λ=h p
Constructive
Destructive
interference
7
Matter Waves – Do they exist? Yes!
1927 Experiments by Thompson (UK), Davisson and Germer
(USA): diffraction of electron beams confirmed de Broglie
relation
λ=h p
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/bragg.html
8
@home (practice)
In an experiment an electron beam of 54 eV kinetic energy falls on a single
crystal of nickel. The Bragg angle is q = 65o .
We know the lattice spacing d= 0.91x10-10 m from X-rays experiment.
The electron’s rest mass is 0.511 MeV/c2. Calculate electron’s wavelength
using:
a) Bragg’s law (assume n=1, i.e. the most intense diffraction maximum)
b) de Broglie’s wavelength
Do they agree?
Useful formulae:
E = hν , p = h λ
9
Source: Ch. Baily
10
Duality of light and matter: Summary
https://www.youtube.com/watch?v=FlIrgE5T_g0
21:49min – 26.07min
Niels Bohr, principle of complementarity:
§ The wave and particle models are complementary.
§ If a measurement proves the wave character of radiation or
matter, then it is impossible to prove the particle character in the
measurement and conversely.
§ Our understanding of radiation or matter is incomplete unless we
take into account measurements which reveal the wave aspects
and those which reveal particle aspects.
The radiation and matter are neither waves nor particles.
Probability interpretation of the wave-particle duality:
the link between wave and particle models.
Born, Schrodinger (will be discussed in the upcoming lectures)
12
Quantum Mechanics (Future Lecture)
Schroedinger’s theory of quantum mechanics
The theory specifies the laws of wave motion that
particles of any microscopic system obey.
§ Maxwell theory of electromagnetism: e-m waves
§ Schroedinger theory: wave functions
but what is really
waving and how???
Need a quantitative relation between the properties of particle
and properties of the wave function that describes the wave.
How does the wave governs the particle?
13
Quantum Mechanics (Future Lectures)
Born’s interpretation of the wave function
Probability density function connects the properties of the wave function
and the behavior of the associated particle:
∗
P ( x, t ) = Ψ ( x, t ) Ψ ( x, t ) = Ψ ( x, t )
2
Ψ ∗ ( x, t ) - complex conjugate of Ψ ( x, t )
Born’s postulate: if at the instant t, a measurement is made to locate
the particle associated with the wave function Ψ ( x, t ), then the
probability P ( x, t ) dx that the particle will be found at a coordinate
2
between x and x+dx is equal to Ψ ( x, t ) dx
∞
∞
∫ P ( x, t )dx = ∫ Ψ ( x, t )
−∞
2
dx = 1
−∞
Normalized, i.e. probability to find
the particle somewhere is 100%
14
Bohr Model of the Atom
22
Models of the Atom
Ernest Rutherford – “the father of nuclear physics”
§ 1911: Rutherford proposed the existence of
a massive nucleus as a small central part of
an atom
J.J. Thompson’s
Plum Pudding Model (1904)
Rutherford’s
Planetary Model (1911)
15
Rutherford experiment
§ Rutherford designed an experiment to use a particles as atomic
bullets (then new technology).
§ The “gold-foil experiment” was performed in 1909 by Hans Geiger
and Ernest Marsden (under Rutherford’s direction)
16
Rutherford experiment
§ Rutherford designed an experiment to use a particles as atomic
bullets (then new technology).
§ The “gold-foil experiment” was performed in 1909 by Hans Geiger
and Ernest Marsden (under Rutherford’s direction)
Radium
17
Rutherford experiment
“It was quite the most incredible event that has ever
happened to me in my life. It was almost as incredible as if
you fired a 15-inch shell at a piece of tissue paper and it
came back and hit you. On consideration, I realized that this
scattering backward must be the result of a single collision,
and when I made calculations I saw that it was impossible
to get anything of that order of magnitude unless you took a
system in which the greater part of the mass of the atom
was concentrated in a minute nucleus. It was then that I had
the idea of an atom with a minute massive center, carrying a
charge.”
— Ernest Rutherford
18
Interactive Simulations: Rutherford Scattering
http://phet.colorado.edu/en/simulation/rutherford-scattering
Download it and run it!
19
Rutherford scattering
Coulomb force
Nα
α
Ekin
≈ 8MeV
Au
n (target density)
dΩ = 2π sin θ dθ
dN
For a given number of incoming a particles Na what is dN,
where dN is the number of scattered a particles into an angle
dW at angle q.
dN dσ
=
Nα ⋅ n
(last lecture)
dΩ dΩ
dσ
D2
20
=
4
Rutherford scattering - Differential Cross Section
Here a nucleus = point like particle (i.e. no structure)
dN dσ
=
Nα ⋅ n,
dΩ dΩ
2
2 &2
#
&
#
dσ
1
zZe
1
=%
×
( ×%
(
dΩ $ 4πε 0 ' $ 2Mv 2 ' sin 4 (θ 2 )
Coulomb interaction
dA ( r sin θ ) ×
dΩ = 2 =
r
dΩ = 2π sin θ dθ
In the spherical
coordinate system
21
Result: Rutherford model of Atom
Summary: How protons, electrons and neutrons were discovered:
(10 min)
https://www.youtube.com/watch?v=kBgIMRV895w
Rutherford type of experiment has been applied with higher
energy beams to study structure of nuclei and nucleons
22
2nd midterm exam:
Tuesday, November 8
Material – Lectures 12,13,14,15,16,17
PLACE (NOT P118):
• Physics P130 (R01)
• Light Engineering 152 (R02 and R03)