Quantum mechanics and nuclear physics base.

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Quantum mechanics and nuclear physics base.
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CHAPTER 5
Wave Properties of Matter and Quantum Mechanics I

• 5.1 X-Ray Scattering
• 5.2 De Broglie Waves
• 5.3 Electron Scattering
• 5.4 Wave Motion
• 5.5 Waves or Particles?
• 5.6 Uncertainty Principle
• 5.7 Probability, Wave Functions, and the
  Copenhagen Interpretation
• 5.8 Particle in a Box

Louis de Broglie (1892-1987)
I thus arrived at the overall concept which
guided my studies for both matter and
radiations, light in particular, it is necessary
to introduce the corpuscle concept and the wave
concept at the same time. - Louis de Broglie, 1929
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5.1 X-Ray Scattering

• Max von Laue suggested that if x-rays were a form
  of electromagnetic radiation, interference
  effects should be observed.
• Crystals act as three-dimensional gratings,
  scattering the waves and producing observable
  interference effects.

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Braggs Law

• William Lawrence Bragg interpreted the x-ray
  scattering as the reflection of the incident
  x-ray beam from a unique set of planes of atoms
  within the crystal.
• There are two conditions for constructive
  interference of the scattered x rays

• The angle of incidence must equal the angle of
  reflection of the outgoing wave.
• The difference in path lengths must be an
  integral number of wavelengths.
• Braggs Law n? 2d sin ? (n integer)

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The Bragg Spectrometer

• A Bragg spectrometer scatters x rays from
  crystals. The intensity of the diffracted beam is
  determined as a function of scattering angle by
  rotating the crystal and the detector.
• When a beam of x rays passes through a powdered
  crystal, the dots become a series of rings.

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5.3 Electron Scattering
In 1925, Davisson and Germer experimentally
observed that electrons were diffracted (much
like x-rays) in nickel crystals.
George P. Thomson (18921975), son of J. J.
Thomson, reported seeing electron diffraction in
transmission experiments on celluloid, gold,
aluminum, and platinum. A randomly oriented
polycrystalline sample of SnO2 produces rings.
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5.2 De Broglie Waves
If a light-wave could also act like a particle,
why shouldnt matter-particles also act like
waves?

• In his thesis in 1923, Prince Louis V. de Broglie
  suggested that mass particles should have wave
  properties similar to electromagnetic radiation.
• The energy can be written as
• hn pc pln
• Thus the wavelength of a matter wave is called
  the de Broglie wavelength

Louis V. de Broglie(1892-1987)
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Bohrs Quantization Condition revisited

• One of Bohrs assumptions in his hydrogen atom
  model was that the angular momentum of the
  electron in a stationary state is nh.
• This turns out to be equivalent to saying that
  the electrons orbit consists of an integral
  number of electron de Brogliewavelengths

electron de Broglie wavelength
Circumference
Multiplying by p/2p, we find the angular momentum
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5.4 Wave Motion

• De Broglie matter waves should be described in
  the same manner as light waves. The matter wave
  should be a solution to a wave equation like the
  one for electromagnetic waves
• with a solution
• Define the wave number k and the angular
  frequency w as usual

It will actually be different, but, in some
cases, the solutions are the same.
Y(x,t) A expi(kx wt q)
and
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Electron Double-Slit Experiment

• C. Jönsson of Tübingen, Germany, succeeded in
  1961 in showing double-slit interference effects
  for electrons by constructing very narrow slits
  and using relatively large distances between the
  slits and the observation screen.
• This experiment demonstrated that precisely the
  same behavior occurs for both light (waves) and
  electrons (particles).

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Wave-particle-duality solution

• Its somewhat disturbing that everything is both
  a particle and a wave.
• The wave-particle duality is a little less
  disturbing if we think in terms of
• Bohrs Principle of Complementarity Its not
  possible to describe physical observables
  simultaneously in terms of both particles and
  waves.
• When were making a measurement, use the particle
  description, but when were not, use the wave
  description.
• When were looking, fundamental quantities are
  particles when were not, theyre waves.

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5.5 Waves or Particles?

• Dimming the light in Youngs two-slit experiment
  results in single photons at the screen. Since
  photons are particles, each can only go through
  one slit, so, at such low intensities, their
  distribution should become the single-slit
  pattern.

Each photon actually goes through both slits!
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Can you tell which slit the photon went through
in Youngs double-slit expt?
When you block one slit, the one-slit pattern
returns.
At low intensities, Youngs two-slit experiment
shows that light propagates as a wave and is
detected as a particle.
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Which slit does the electron go through?

• Shine light on the double slit and observe with a
  microscope. After the electron passes through one
  of the slits, light bounces off it observing the
  reflected light, we determine which slit the
  electron went through.
• The photon momentum is
• The electron momentum is

Need lph lt d to distinguish the slits.
Diffraction is significant only when the aperture
is the wavelength of the wave.
The momentum of the photons used to determine
which slit the electron went through is enough to
strongly modify the momentum of the electron
itselfchanging the direction of the electron!
The attempt to identify which slit the electron
passes through will in itself change the
diffraction pattern! Electrons also propagate as
waves and are detected as particles.
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5.6 Uncertainty Principle Energy Uncertainty

• The energy uncertainty of a wave packet is
• Combined with the angular frequency relation we
  derived earlier
• Energy-Time Uncertainty Principle .

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Momentum Uncertainty Principle

• The same mathematics relates x and k Dk Dx
  ½
• So its also impossible to measure simultaneously
  the precise values of k and x for a wave.
• Now the momentum can be written in terms of k
• So the uncertainty in momentum is
• But multiplying Dk Dx ½ by h
• And we have Heisenbergs Uncertainty Principle

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How to think about Uncertainty
The act of making one measurement perturbs the
other. Precisely measuring the time disturbs the
energy. Precisely measuring the position
disturbs the momentum. The Heisenbergmobile. The
problem was that when you looked at the
speedometer you got lost.
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Kinetic Energy Minimum

• Since were always uncertain as to the exact
  position, , of a particle, for
  example, an electron somewhere inside an atom,
  the particle cant have zero kinetic energy

The average of a positive quantity must always
exceed its uncertainty
so
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5.7 Probability, Wave Functions, and the
Copenhagen Interpretation

• Okay, if particles are also waves, whats waving?
  Probability
• The wave function determines the likelihood (or
  probability) of finding a particle at a
  particular position in space at a given time

The probability of the particle being between x1
and x2 is given by
The total probability of finding the particle is
1. Forcing this condition on the wave function is
called normalization.
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5.8 Particle in a Box

• A particle (wave) of mass m is in a
  one-dimensional box of width l.
• The box puts boundary conditions on the wave. The
  wave function must be zero at the walls of the
  box and on the outside.
• In order for the probability to vanish at the
  walls, we must have an integral number of half
  wavelengths in the box
• The energy
• The possible wavelengths are quantized and hence
  so are the energies

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Probability of the particle vs. position

• Note that E0 0 is not a possible energy level.
• The concept of energy levels, as first discussed
  in the Bohr model, has surfaced in a natural way
  by using waves.
• The probability of observing the particle between
  x and x dx in each state is

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The Copenhagen Interpretation

• Bohrs interpretation of the wave function
  consisted of three principles
• Heisenbergs uncertainty principle
• Bohrs complementarity principle
• Borns statistical interpretation, based on
  probabilities determined by the wave function
• Together these three concepts form a logical
  interpretation of the physical meaning of quantum
  theory. In the Copenhagen interpretation,
  physics describes only the results of
  measurements.

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Quantum Mechanics

• 1 The Schrödinger Wave Equation
• 2 Expectation Values
• 3 Infinite Square-Well Potential
• 4 Finite Square-Well Potential
• 5 Three-Dimensional Infinite-Potential Well
• 6 Simple Harmonic Oscillator
• 7 Barriers and Tunneling

Erwin Schrödinger (1887-1961)
A careful analysis of the process of observation
in atomic physics has shown that the subatomic
particles have no meaning as isolated entities,
but can only be understood as interconnections
between the preparation of an experiment and the
subsequent measurement. - Erwin Schrödinger
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Opinions on quantum mechanics
I think it is safe to say that no one understands
quantum mechanics. Do not keep saying to
yourself, if you can possibly avoid it, But how
can it be like that? because you will get down
the drain into a blind alley from which nobody
has yet escaped. Nobody knows how it can be like
that. - Richard Feynman
Those who are not shocked when they first come
across quantum mechanics cannot possibly have
understood it. - Niels Bohr
Richard Feynman (1918-1988)
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6.1 The Schrödinger Wave Equation

• The Schrödinger wave equation in its
  time-dependent form for a particle of energy E
  moving in a potential V in one dimension is
• where i is the square root of -1.
• The Schrodinger Equation is THE fundamental
  equation of Quantum Mechanics.

where V V(x,t)
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General Solution of the Schrödinger Wave Equation
when V 0

• Try this solution

This works as long as
which says that the total energy is the kinetic
energy.
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General Solution of the Schrödinger Wave Equation
when V 0

• In free space (with V 0), the general form of
  the wave function is
• which also describes a wave moving in the x
  direction. In general the amplitude may also be
  complex.
• The wave function is also not restricted to being
  real. Notice that this function is complex.
• Only the physically measurable quantities must be
  real. These include the probability, momentum and
  energy.

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Normalization and Probability

• The probability P(x) dx of a particle being
  between x and x dx is given in the equation
• The probability of the particle being between x1
  and x2 is given by
• The wave function must also be normalized so that
  the probability of the particle being somewhere
  on the x axis is 1.

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Properties of Valid Wave Functions

• Conditions on the wave function
• 1. In order to avoid infinite probabilities, the
  wave function must be finite everywhere.
• 2. The wave function must be single valued.
• 3. The wave function must be twice
  differentiable. This means that it and its
  derivative must be continuous. (An exception to
  this rule occurs when V is infinite.)
• 4. In order to normalize a wave function, it must
  approach zero as x approaches infinity.
• Solutions that do not satisfy these properties do
  not generally correspond to physically realizable
  circumstances.

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Time-Independent Schrödinger Wave Equation

• The potential in many cases will not depend
  explicitly on time.
• The dependence on time and position can then be
  separated in the Schrödinger wave equation. Let
• which yields
• Now divide by the wave function y(x) f(t)

The left side depends only on t, and the right
side depends only on x. So each side must be
equal to a constant. The time dependent side is
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Time-Independent Schrödinger Wave Equation
We integrate both sides and find where C is an
integration constant that we may choose to be 0.
Therefore
But recall our solution for the free particle In
which f(t) e -iw t, so w B / h or B hw,
which means that B E ! So multiplying by
y(x), the spatial Schrödinger equation becomes
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Time-Independent Schrödinger Wave Equation
This equation is known as the time-independent
Schrödinger wave equation, and it is as
fundamental an equation in quantum mechanics as
the time-dependent Schrodinger equation. So
often physicists write simply where
is an operator.
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Stationary States

• The wave function can be written as
• The probability density becomes
• The probability distribution is constant in time.
  
• This is a standing wave phenomenon and is called
  a stationary state.

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6.2 Expectation Values

• In quantum mechanics, well compute expectation
  values. The expectation value, , is the
  weighted average of a given quantity. In general,
  the expected value of x is

If there are an infinite number of possibilities,
and x is continuous
Quantum-mechanically
And the expectation of some function of x, g(x)
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Momentum Operator

• To find the expectation value of p, we first need
  to represent p in terms of x and t. Consider the
  derivative of the wave function of a free
  particle with respect to x
• With k p / h we have
• This yields
• This suggests we define the momentum operator as
  .
• The expectation value of the momentum is

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Position and Energy Operators
The position x is its own operator. Done. Energy
operator The time derivative of the
free-particle wave function is Substituting
w E / h yields The energy operator is The
expectation value of the energy is
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Deriving the Schrodinger Equation using operators
The energy is
Substituting operators E KV
Substituting
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6.3 Infinite Square-Well Potential

• The simplest such system is that of a particle
  trapped in a box with infinitely hard walls
  thatthe particle cannot penetrate. This
  potential is called an infinite square well and
  is given by
• Clearly the wave function must be zero where the
  potential is infinite.
• Where the potential is zero (inside the box), the
  time-independent Schrödinger wave equation
  becomes
• The general solution is

x
0
L
where
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Quantization

• Boundary conditions of the potential dictate
  that the wave function must be zero at x 0
  and x L. This yields valid solutions for
  integer values of n such that kL np.
• The wave function is
• We normalize the wave function
• The normalized wave function becomes
• These functions are identical to those obtained
  for a vibrating string with fixed ends.

x
0
L
½ - ½ cos(2npx/L)
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Quantized Energy

• The quantized wave number now becomes
• Solving for the energy yields
• Note that the energy depends on integer values of
  n. Hence the energy is quantized and nonzero.
• The special case of n 1 is called the ground
  state.

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6.4 Finite Square-Well Potential

• The finite square-well potential is

The Schrödinger equation outside the finite well
in regions I and III is
yields
Letting
Considering that the wave function must be zero
at infinity, the solutions for this equation are
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Finite Square-Well Solution

• Inside the square well, where the potential V is
  zero, the wave equation becomes where
• The solution here is
• The boundary conditions require that
• so the wave function is smooth where the
  regions meet.
• Note that the wave function is nonzero outside
  of the box.

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Penetration Depth

• The penetration depth is the distance outside the
  potential well where the probability
  significantly decreases. It is given by
• The penetration distance that violates classical
  physics is proportional to Plancks constant.

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6.6 Simple Harmonic Oscillator

• Simple harmonic oscillators describe many
  physical situations springs, diatomic molecules
  and atomic lattices.

Consider the Taylor expansion of a potential
function
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Simple Harmonic Oscillator
Consider the second-order term of the Taylor
expansion of a potential function Substitutin
g this into Schrödingers equation Let
and which yields
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The Parabolic Potential Well
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The Parabolic Potential Well

• Classically, the probability of finding the mass
  is greatest at the ends of motion and smallest at
  the center.
• Contrary to the classical one, the largest
  probability for this lowest energy state is for
  the particle to be at the center.

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Analysis of the Parabolic Potential Well
As the quantum number increases, however, the
solution approaches the classical result.
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The Parabolic Potential Well

• The energy levels are given by

The zero point energy is called the Heisenberg
limit
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6.7 Barriers and Tunneling

• Consider a particle of energy E approaching a
  potential barrier of height V0, and the potential
  everywhere else is zero.
• First consider the case of the energy greater
  than the potential barrier.
• In regions I and III the wave numbers are
• In the barrier region we have

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Reflection and Transmission

• The wave function will consist of an incident
  wave, a reflected wave, and a transmitted wave.
• The potentials and the Schrödinger wave equation
  for the three regions are as follows
• The corresponding solutions are
• As the wave moves from left to right, we can
  simplify the wave functions to

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Probability of Reflection and Transmission

• The probability of the particles being reflected
  R or transmitted T is
• Because the particles must be either reflected or
  transmitted we have R T 1.
• By applying the boundary conditions x ? 8, x
  0, and x L, we arrive at the transmission
  probability
• Note that the transmission probability can be 1.

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Tunneling

• Now we consider the situation where classically
  the particle doesnt have enough energy to
  surmount the potential barrier, E lt V0.

The quantum mechanical result is one of the most
remarkable features of modern physics. There is a
finite probability that the particle can
penetrate the barrier and even emerge on the
other side! The wave function in region II
becomes The transmission probability that
describes the phenomenon of tunneling is
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Tunneling wave function
This violation of classical physics is allowed by
the uncertainty principle. The particle can
violate classical physics by DE for a short time,
Dt h / DE.
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Analogy with Wave Optics

• If light passing through a glass prism reflects
  from an internal surface with an angle greater
  than the critical angle, total internal
  reflection occurs. However, the electromagnetic
  field is not exactly zero just outside the prism.
  If we bring another prism very close to the first
  one, experiments show that the electromagnetic
  wave (light) appears in the second prism The
  situation is analogous to the tunneling described
  here. This effect was observed by Newton and can
  be demonstrated with two prisms and a laser. The
  intensity of the second light beam decreases
  exponentially as the distance between the two
  prisms increases.

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Potential Well

• Consider a particle passing through a potential
  well, rather than a barrier.
• Classically, the particle would speed up in the
  well region because
• K mv2 / 2 E V0
• Quantum mechanically, reflection and transmission
  may occur, but the wavelength decreases inside
  the well. When the width of the potential well is
  precisely equal to half-integral or integral
  units of the wavelength, the reflected waves may
  be out of phase or in phase with the original
  wave, and cancellations or resonances may occur.
  The reflection/cancellation effects can lead to
  almost pure transmission or pure reflection for
  certain wavelengths. For example, at the second
  boundary (x L) for a wave passing to the right,
  the wave may reflect and be out of phase with the
  incident wave. The effect would be a cancellation
  inside the well.

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Alpha-Particle Decay

• The phenomenon of tunneling explains
  alpha-particle decay of heavy, radioactive
  nuclei.
• Inside the nucleus, an alpha particle feels the
  strong, short-range attractive nuclear force as
  well as the repulsive Coulomb force.
• The nuclear force dominates inside the nuclear
  radius where the potential is a square well.
• The Coulomb force dominates outside the nuclear
  radius.
• The potential barrier at the nuclear radius is
  several times greater than the energy of an
  alpha particle.
• In quantum mechanics, however, the alpha
  particle can tunnel through the barrier. This is
  observed as radioactive decay.

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6.5 Three-Dimensional Infinite-Potential Well

• The wave function must be a function of all three
  spatial coordinates.
• Now consider momentum as an operator acting on
  the wave function. In this case, the operator
  must act twice on each dimension. Given

So the three-dimensional Schrödinger wave
equation is
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The 3D infinite potential well
Its easy to show that
where
and
When the box is a cube
Try 10, 4, 3 and 8, 6, 5
Note that more than one wave function can have
the same energy.
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Degeneracy

• The Schrödinger wave equation in three dimensions
  introduces three quantum numbers that quantize
  the energy. And the same energy can be obtained
  by different sets of quantum numbers.
• A quantum state is called degenerate when there
  is more than one wave function for a given
  energy.
• Degeneracy results from particular properties of
  the potential energy function that describes the
  system. A perturbation of the potential energy
  can remove the degeneracy.