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Title: Quantum mechanics and nuclear physics base.


1
Quantum mechanics and nuclear physics base.
2
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
3
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.

4
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)

5
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.

6
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.
7
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)
8
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
9
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
10
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).

11
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.

12
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!
13
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.
14
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.
15
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 .

16
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

17
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.
18
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
19
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.
20
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

21
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

22
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.

23
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
24
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)
25
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)
26
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.
27
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.

28
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.

29
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.

30
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
31
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
32
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.
33
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.

34
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)
35
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

36
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
37
Deriving the Schrodinger Equation using operators
The energy is
Substituting operators E KV
Substituting
38
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
39
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)
40
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.

41
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
42
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.

43
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.

44
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
45
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
46
The Parabolic Potential Well
47
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.

48
Analysis of the Parabolic Potential Well
As the quantum number increases, however, the
solution approaches the classical result.
49
The Parabolic Potential Well
  • The energy levels are given by

The zero point energy is called the Heisenberg
limit
50
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

51
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

52
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.

53
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
54
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.
55
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.

56
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.

57
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.

58
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
59
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.
60
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.
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