Download 9/25 - SMU Physics

Particles (matter) behave as waves
and the Schrödinger Equation
1.
2.
Comments on quiz 9.11 and 9.23.
Topics in particles behave as waves:
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today
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3.
4.
The (most powerful) experiment to prove a wave: interference.
Properties of matter waves.
The free-particle Schrödinger Equation.
The Heisenberg Uncertainty Principle.
The not-unseen observer (self study).
The Bohr Model of the hydrogen atom.
The second of the many topics for our class projects.
Material and example about how to prepare and make a
presentation (ref. Prof. Kehoe)
Review: properties of matter waves
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The de Broglie wavelength of a particle:   h p
The frequency: f  E h
The h-bar constant:  h 2
The connection between particle and wave:
 p k
 E 
Wave number and angular frequency:
k  2 
  2 T  2 f
The free-particle Schrödinger Equation
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The matter waves:


2
2m
 2 Ψ  x,t 
x 2
i
Ψ  x,t 
t
The interpretation of the matter wave function:
probability density = Ψ  x,t 

2
The plane wave solution and verification:
Ψ  x,t   Aei kx t 
2
k

 2 Aei kx t 
p2
2
i  kx t 
i  kx t 
i  kx t 



ik
Ae

Ae

Ae


2m
x 2
2m
2m
2m
2
2
Aei kx t 
i
i
t
Erwin
Schrödinger,
1887-1961,
Austrian physicist,
shared 1933
Nobel Prize for
new formulations
of the atomic
theory.
 i  Aei kx t  
 Aei kx t 
2
p2
 2 Aei kx t 
Aei kx t 
 E   , 
i
2
2m
2m
x
t
Quantum but classical account
for energy E, not relativistic
Understand this plane wave
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How many of you reviewed the discussions
about waves in mechanics?
Complex exponential:
Ψ  x,t   Aei kx t   Acos  kx  t   iAsin  kx  t 
ReΨ  Acos  kx  t   Asin  kx  t   2
ImΨ  Asin  kx  t 

Probability to find the particle:
probability density = Ψ  x,t   Ψ*  x,t  Ψ  x,t 
2
= Aei kxt  Aei kxt   A2
Equal probability to find the particle anywhere
 the location of this particle is uncertain, although the
momentum of this particle is certain. Why? p  k
The Heisenberg Uncertainty Principle
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Particle-wave duality  uncertainties
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Plane wave of free electron:
Ψ  x,t   Aei kx t 
Momentum p  k is certain.
 Location (where to find the particle)
is not certain (equal probability).
so  p x  0    ?
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On the detection screen:
Location is known (measured).
 Momentum ( p x ) is not certain.
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Uncertainty? Review standard deviation:
p 
i  pi  p  ni
i ni
with p 
i pi ni
i ni
The Heisenberg Uncertainty Principle
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Because of a particle’s wave nature, it is theoretically
impossible to know precisely both its position along
on axis and its momentum component along that
axis; Δx and cannot be zero simultaneously. There is
a strict theoretical lower limit on their product:
 px  x 
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2
This is called the Heisenberg uncertainty principle
(Nobel Prize 1932).
Show example 4.4 and 4.5 (student work).
Solar system
Atom model analogous
to the solar system is
wrong
Electron waves
in an atom
Werner Heisenberg
(1901 – 1976),
German physicist.
Nobel Prize in 1932
for the creation of
quantum mechanics.
Example 4.6, an application of
the uncertainty principle
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Find the ground state of a
hydrogen atom (student work).
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The classical mechanics approach.
Eclassical  

rmost probable
8 0 r
The quantum mechanics approach.
Ematter wave 

e2
2
2mr 2

e2
4 0 r
The ground state, the minimum
mechanical energy for the electron.
2
dE
e2
 3 
0
2
dr
mr
4 0 r
4 0 2
 5.3 1011 m  0.053nm
r
2
me
 Emin  
me4
32 
2
2
0
2
 2.2 1018 J  13.6eV
Emin
The energy-Time Uncertainty Principle and
the discussions in section 4.5
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The energy-time uncertainty Principle:
 E t 
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In particle physics, we estimate some particle’s lifetime by measuring its
energy (mass) uncertainty.
example: the particle π0 has a mass of 134.98 MeV, decays into two
photons. Its mean lifetime τ = 8.4×10-17 sec, derived from its width of
0.0006 MeV in its mass measurement.
Et
t
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2
 6.6 1022 MeV  s
0.0006MeV 111017 s
By measuring the energy spread (uncertainty) of an emitted photon, we
estimate the time an atom stays at a certain excited state.
The Not-Unseen Observer: please read section 4.5 after the class.
Discuss with me in office hours if you have questions about this
section.
The Bohr Model of the hydrogen atom
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The classical approach:
2
1 2
e
e2
v2
KE

mv


m

2
8 0 r
4 0 r 2
r
1
E  U  KE  

E can take any value,
8 0 r for r is continuous.
e2
Bohr postulates: the electron’s angular momentum L
may only take the values:
Ln
where n  1, 2,3...
Because L  mvr We have mvr  n
e2
v2
m
Coulomb force hold the electron in place:
2
4

r
r
0
2
 4 0  n 2
 r
me 2
4 0  2

 0.0529 nm
For n = 1, the Bohr radius a0 
me 2
1
The energy E  
e2
8 0 r

me4
2  4 0 
2
2
1
1


13
.
6
 eV 
n2
n2
Niels Henrik David
Bohr (1885 – 1962),
Danish physicist.
Nobel Prize in 1922
for work on atomic
structure.
The Bohr Model of the hydrogen atom
Hydrogen spectrum
n
E  
E
n3
2 c
e2
8 0 r


n2
, c  197 MeV  fm
me4
2  4 0 
2
2
1
1


13
.
6
 eV 
n2
n2
1 1 
 R  2  2  R  1.0972 107 m 1

1 n 
1
n2
Review questions
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If a particle is confined inside a boundary of
finite size, can you be certain about the particle’s
velocity at any given time?
Why the Bohr’s hydrogen model is flawed?
If you have problem in understanding example
4.6, you need to see me in my office hour.
Preview for the next class
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Text to be read:
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In chapter 5:
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Section 5.1
Section 5.2
Section 5.3
Section 5.4
Questions:
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How would you generalize the Schrödinger equation we have
discussed in chapter 4 to include conservative forces?
Give an example of a classical bound state.
If a particle falls into a potential well (a quantum well), how
do you determine whether it is bounded?
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Class project topic, 2 and
how to prepare for a presentation
Quantum physics in renewable energy.
How to prepare for a presentation:
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Reference:
http://www.physics.smu.edu/kehoe/3305F07/3305Present.pdf
http://www.physics.smu.edu/kehoe/hep/WhyMass.pdf
Homework 6, due by 10/2
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4.
Problem 13 on page 134.
Problem 16 on page 134.
Problem 43 on page 137.
Problem 54 on page 138.