Particles Act Like Waves

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Particles Act Like Waves!
De Broglies Matter Waves

• ? h / p

Schrodingers Equation
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Announcements

• Schedule
• Today
• Atomic physics
• Uncertainty principle
• Hobson Ch. 14
• Quiz 6.
• Next time
• Start nuclear physics
• Hobson Ch. 15
• Homework
• HW 9 due today
• HW 10 due December 1
• Essay/Report
• Due Dec 6

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The Problem of the atom

• Last time we saw that experiments supported the
  picture that an atom is composed of light
  electrons around a heavy nucleus
• Problem if the electrons orbit the nucleus,
  classical physics predicts they should emit
  electromagnetic waves and loose energy. If this
  happens, the electrons will spiral into the
  nucleus!
• The atom would not be stable!
• What is the solution to this problem?

4
Bohrs Revolutionary Idea

• Can the new quantum theory explain the stability
  of the atom?
• If the energies can take on only certain discrete
  values, i.e., it is quantized, there would be a
  lowest energy orbit, and the electron is not
  allowed to fall to a lower energy!
• What is the role of Plancks Constant h?

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Plancks Constant h and the atom

• Bohr (and others) noted that the
  combination a0 (h/2p)2/ me2 has the units
  of length about the size of atoms
• Bohr postulated that it was not the atom that
  determined h, but h that determined the
  properties of atoms!
• Since the electron is bound to the nucleus by
  electrical forces, classical physics says that
  the energy should be E - (1/2) e2/a0
• If the radii are restricted to certain values,
  the the energy can only have certain values

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The Bohr Atom (NOT Correct in detail!)

• The allowed orbits are labeled by the integers n
  1, 2, 3, 4.
• The radii of these orbits can be determined from
  the quantization condition radius n2 a0
  n2 (h/2p)2/ me2
• The energy can only have the values En E1 /n2,
  E1 - (1/2)(e2 / a0)/n2
• The spectra are the result of transitions between
  these orbits, with a single photon (f E/h)
  carrying off the difference in energy E between
  the two orbits.

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Ideas agree with Experiment

• Bohrs picture
• The only stable orbits of the electrons occur at
  definite radii.
• When in these orbits, contrary to classical EM,
  the electrons do not radiate.
• The radiation we see corresponds to electrons
  moving from one stable orbit to another.
• Experiments (already known before 1912)
• Experiment Balmer had previously noticed a
  regularity in the frequencies emitted from
  hydrogen
• f f 0 ( (1/n2) - (1/m2)) where n and m are
  integers.
• Bohrs Theory Fits exactly using the value of h
  determined from other experimentsPhoton carries
  energy (hf) difference of stable orbits.

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Hydrogen Spectrum Balmer series
6.171
7.314
frequency (1014 Hz)
6.912
7.557
4.571

• Balmer Formula????f? f0 ( (1/n2) - (1/m2))
• 32.91 ( 1/4 - 1/9 ) 4.571
• 32.91 ( 1/4 - 1/16 ) 6.171
• 32.91 ( 1/4 - 1/25 ) 6.911
• 32.91 ( 1/4 - 1/36 ) 7.313
• 32.91 ( 1/4 - 1/49 ) 7.556

IT WORKS!
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Demonstration Spectra of different atoms

• Observe spectra of different gases
• Individual grating for each student
• Using interference - wave nature of light - to
  separate the different frequencies (colors)

6.171
7.314
Hydrogen
frequency (1014 Hz)
6.912
7.557
4.571
Neon - strong line in Red
Sodium - strong line in yellow (street lights)
Mercury - strong lines in red, blue (street
lights)
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Towards Understanding

• Bohr atom
• Quantized energy levels, allowed orbits
• deBroglie waves
• Particle acts like wave, wavelength depends upon
  momentum
• Obviously related, but unclear exactly how
• Erwin Schroedinger pulled it all together in 1926

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The Schrodinger Equation

• In 1926 Erwin Schrodinger proposed an equation
  which describes completely the time evolution of
  the matter wave ??

( - (h2 / 2m) ? 2 V) ????i h (d? /dt)
where m characteristic mass of particle
V potential energy function to describe
the forces
Newton
Schrodinger Given the force, find motion
Given potential, find wave F ma m (d2x/dt2)
(- (h2 / 2m) ?2 V) ????i
h (d? /dt) solution x f(t)
solution ? f(x,t)
Note Schrodingers equation is more difficult
to solve, but it is just as well-defined as
Newtons. If you know the forces acting, you can
calculate the potential energy V and solve the
Schrodinger equation to find ?.
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Key Results of Schrodinger Eq.

• The energy is quantized
• Only certain energies are allowed
• Agrees with Bohrs Idea in general
• Predicts the spectral lines of Hydrogen exactly
• Applies to many different problems - still one of
  the key equations of physics!
• The wavefunction is spread out
• Very different from Bohrs idea
• The electron wavefunction is not at a given
  radius but is spread over a a range of radii.

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What is ? ?

• Our current view was fully developed by Bohr from
  an initial idea from Max Born.
• Borns idea ? is a probability amplitude wave!
  ?2 tells us the probability of finding the
  particle at a given place at a given time.
• Leads to indeterminancy in the fundamental laws
  of nature goodbye Newtonian worldview!
• Uncertainty principles
• Not just a lack of ability to measure a property
  - but a fundamental impossibility to know some
  things
• Einstein doesnt like it
• The theory accomplishes a lot, but it does not
  bring us closer to the secrets of the Old One.
  In any case, I am convinced that He does not play
  dice.

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Probability interpretation for ?2

• The location of an electron is not determined by
  ?. The probability of finding it is high where
  ?2 is large, and small where ?2 is small.
• Example A hydrogen atom is one electron around a
  nucleus. Positions where one might find the
  electron doing repeated experiments

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The Uncertainty Principle

• Werner Heisenberg proposed that the basic ideas
  on quantum mechanics could be understood in terms
  of an Uncertainty Principle

where ?p and ?x refer to the uncertainties in the
measurement of momentum and position.
The constant h-bar has the approximate value
h 10 -34 Joule seconds
Similar ideas lead to uncertainty in time and
energy
?E ?t ³ (1/2) h/2p (1/2) h
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Uncertainty Principle and Matter Waves

• The uncertainty principle can be understood from
  the idea of de Broglie that particles also have
  wave character
• What are properties of waves
• Waves are patterns that vary in space and time
• A wave is not in only one place at a give time -
  it is spread out
• Example of wave with well-defined wavelength l
  and momentum p h/ l, but is spread over all
  space, i.e., its position is not well-defined

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The Nature of a Wave - continued

• Example of wave with well-defined position in
  space but its wavelength l and momentum p h/ l
  is not well-defined , i.e., the wave does not
  correspond to a definite momentum or wavelength.

0
Most probable position
Position x
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Quantum Tunneling

• In classical mechanics an object can never get
  over a barrier (e.g. a hill) if if does not have
  enough energy
• In quantum mechanics there is some probability
  for the object to tunnel through the hill!
• The particle below has energy less than the
  energy needed to get over the barrier

tunneling
Energy
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Example of Quantum Tunneling

• The decay of a nucleus is the escape of particles
  bound inside a barrier
• The rate for escape can be very small.
• Particles in the nucleus attempt to escape
  1020 times per second, but may succeed in
  escaping only once in many years!

tunneling
Radioactive Decay
Energy
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Example of Probability Intrinsic to Quantum
Mechanics

• Even if the quantum state (wavefunction) of the
  nucleus is completely well-defined with no
  uncertainty, one cannot predict when a nucleus
  will decay.
• Quantum mechanics tells us only the probability
  per unit time that any nucleus will decay.
• Demonstration with Geiger Counter

tunneling
Radioactive Decay
Energy
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Worldview

• Quantum mechanics has given us
• Probability waves we cant know exactly where a
  particle is at nor can we know exactly what its
  momentum is.
• Tunneling effects a particle is permitted to
  tunnel through a barrier. We can know the
  likelihood (probability) it will tunnel, but we
  cant know when it will tunnel!
• Recall the Newtonian worldview
• If we knew the state of the universe at some
  time, Newtonian physics fully explained how the
  universe would evolve. This led to a
  deterministic universe.
• The Newtonian worldview is annihilated by the
  quantum theory.
• Every single interaction is now random! We can
  calculate the probability for an event to occur,
  but we cant guarantee it will occur!
• Philosophical consequences of quantum theory run
  very deep, in part because of our inability to
  comprehend it.

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Important Quantum Effects in Our WorldI Lasers
Usually light is emitted by an excited atom is in
a a random direction - light from many atomsgoes
in all directions direction and energy have
uncertainty for light emitted from any one atom
What is special about a Laser??
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Important Quantum Effects in Our WorldI Lasers
- continued
Lasers work because of the quantum properties of
photons -- one photon tends to cause another
tobe emitted one photon cannot be
distinguished from another
Excited Atoms
Many Photons
One Photon
If there are many excited atoms, the photons can
cascade -- very intense, collimated light is
emitted forming a beam of precisely the same
color light
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Important Quantum Effects in Our WorldI Lasers
- continued
Since photons cannot be distinguished, which atom
emitted a given photon is completely uncertain
But that means The direction and energy can be
very certain!
If there are many excited atoms, the photons can
cascade -- very intense, collimated light is
emitted forming a beam of precisely the same
color light
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Important Quantum Effects in Our World
Superconductivity Discovered in 1911 by K.
Onnes Completely baffling in classical physics
Explained in 1957 by Bardeen, Cooper And
Shrieffer at the Univ. of Illinois. (Bardeen is
the only person to win two Nobel Prizes in the
same field!)Due to all the electrons acting
together to form a single quantum state --
electrons flow around a wire like the electrons
in an atom!
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Demonstration
High - Temperature SuperconductorsDiscovered
in 1987 (Nobel Prize) (Still not understood!)
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Summary

• Niels Bohr (1912) realized the significance that
  the quantization could explain the stability of
  the atom
• Schrodinger (1926) Equation for wave function
  ?(x,t) for a particle --- Still Today the Basic
  Eq. of Quantum Mechanics. Explains all of
  Chemistry!
• ( ?(x,t) ) 2 is probablity of finding the
  particle at point x and time t. More about this
  later.
• Heisenberg showed that quantum mechanics leads to
  uncertainty relations for pairs of variables
• Quantum Theory says that we can only measure
  individual events that have a range of
  possibilities
• We can never predict the result of a future
  measurement with certainty
• More next time on how quantum theory forces us to
  reexamine our beliefs about the nature of the
  world

?E ?t ³ h/2
?p ?x ³ h/2
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(Extra) Example Harmonic Oscillator

• Classical situation Mass attached to a spring.
• The spring exerts a force on the mass which is
  proportional to the distance that the spring is
  stretched or compressed. This force then
  produces an acceleration of the mass which leads
  to an oscillating motion of the mass. The
  frequency of this oscillation is determined by
  the stiffness of the spring and the amount of
  mass.
• Quantum situation suppose F is proportional to
  distance, then potential energy is proportional
  to distance squared. Solutions to
  Schrodinger Eqn

What is shown here? Possible wave functions
?(x) at a fixed time t! How does this change in
time? They oscillate with the classical
frequency! What distinguishes the different
solutions? The Energy! (Classically this
corresponds to the amplitude of the oscillation)
Note not all energies are possible! They are
quantized!
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(Extra) Example Hydrogen atom

• Potential Energy is proportional to 1/R (since
  Force is proportional to 1/R2). What are the
  solutions to Schrodingers equation and how are
  they related to Bohrs orbits?

Radial Wavefunctions for the Hydrogen Atom (
vertical lines ? Bohr radii )

• The Bohr orbits correspond to the solutions
  shown which have definite energies.
• The energies which correspond to these wave
  functions are identical to Bohrs values!
• For energies above the ground state (n1), there
  are more than one wave function with the same
  energy.
• Some of these wave functions peak at the value
  for the Bohr radius for that energy, but others
  dont!