Download Quantum Mechanics

Quantum Physics
When we consider the motion of objects on the atomic
level, we find that our classical approach does not work
very well.
For example, quantum physics describes how electrons
surround the nucleus of the atom and other subatomic
actions.
Therefore, for understanding motion on the microscopic
scale we must use Quantum Mechanics.
"I think I can safely say that nobody understands quantum mechanics."
- Richard P. Feynman
"I don't like it, and I'm sorry I ever had anything to do with it"
- Erwin Schrödinger
"Anyone who is not shocked by the quantum theory has not understood it."
- Niels Bohr
We know some small particles like electrons, protons, and neutrons.
Quantum physics even describes the particles which make these particles!
(The model of an atom that you were taught in high-school is a
approximation). The electrons don't orbit like planets; they form blurred
clouds of probabilities around the nucleus. Protons and neutrons? They're
each made of three quarks, each with its own 'flavor' and one of three
'colors'. Lets not forget the gluons, the even smaller particles that hold this
mess together when they collect and form glueballs. The quantum model of
the atom is much more complex than the traditional model. The world of
subatomic particles is a very bizarre one, filled with quantum probabilities
and organized chaos. For example, the exact position and velocity of an
electron is very hard to find because attempts to "see" it involve bouncing
other particles off of it. By doing this, you've just changed the electron's
velocity, so your data is useless. What quantum physics does is give us the
statistical probability of the electron's location at any one moment.
Blackbody Radiation
One of the earliest indications that classical physics was
incomplete came from attempts to describe blackbody
radiation.
A black body is a theoretical object
that absorbs 100% of the radiation
that hits it. Therefore it reflects
no radiation and appears perfectly
black.
Blackbody radiation is the emission
of electromagnetic waves from the surface of an object.
The distribution of blackbody radiation depends only the
temperature of the object and is independent of the
material.
The Blackbody Distribution
The intensity spectrum emitted
from a blackbody has a
characteristic shape.
The maximum of the intensity is
found to occur at a wavelength given
by Wien’s Displacement Law:
fpeak = (5.88 × 1010 s-1·K-1)T
T = temperature of blackbody (K)
Notice the plot is versus frequency on these
plots, but since it is related to wavelength
via, f λ = c
λmaxT = 2.90 × 10 m K
−3
Example: The solar radiation curve approximates closely
a blackbody at 5900K. What region of the spectrum is
the sun most intense?
2.90 × 10−3 m K
−7
λmax =
5900 K
= 4.915 × 10 m
= 491.5nm
Consequences: What part of the spectrum is this?
2.90 × 10−3 m K
T=300K
= 9.667 × 10−6 m
λmax =
300 K
= 9.667 microns
What portion of the spectrum is this?
2.90 × 10−3 m K
T=3 million K
= 9.667 × 10−10 m
λmax =
6
3. × 10 K
=9.667Å = 0.97nm
What portion of the spectrum is this?
2.90 × 10−3 m K
T=3K
= 9.667 × 10−3 m
λmax =
3K
=9.667mm
What portion of the spectrum is this?
The Ultraviolet Catastrophe
Classical physics can
describe the shape of the
blackbody spectrum only at
long wavelengths. At short
wavelengths there is
complete disagreement.
This disagreement between
observations and the
classical theory is known as
the ultraviolet catastrophe.
Planck’s Solution
In 1900, the German physicist Max Planck was able to
explain the observed blackbody spectrum by assuming
that it originated from oscillators on the surface of the
object and that the energies associated with the
oscillators were discrete or quantized:
En = nhf
n = 0, 1, 2, 3…
n is an integer called the quantum number
h is Planck’s constant: 6.62 × 10-34 J·s
f is the frequency
Planck’s idea of quanta was the beginning of quantum physics. The
smallest packet of energy a electron can give or absorb. (Like you
can’t take an elevator to the 32nd ½ floor). With electrons any
energy is a whole number of quantas.
The correct mathematical work to Planck’s work was later done by
Erwin Schroedinger, an Austrian scientist.
Quantization of Light
Einstein proposed that light itself
comes in chunks of energy, called
photons. Light is a wave, but also a
particle. The energy of one photon
is
E = hf
where f is the frequency of the light
and h is Planck’s constant.
Useful energy unit: 1 eV = 1.6 × 1019 J
Quantum Mechanics
The essence of quantum mechanics is that
certain physical properties of a system (like
the energy) are not allowed to be just any
value, but instead must be only certain
discrete values.
Example
(a) Find the energy of 1 (red) 650 nm photon.
(b) Find the energy of 2 (red) 650 nm photons.
The PhotoElectric Effect
When light is incident on
a surface (usually a
metal), electrons can be
ejected. This is known
as the photoelectric
effect.
Around the turn of the century, observations of the
photoelectric effect were in disagreement with the
predictions of classical wave theory.
Observations of the Photoelectric Effect
• No electrons are emitted if the frequency of the
incident photons is below some cutoff value,
independent of intensity.
• The maximum kinetic energy of the emitted
electrons does not depend on the light
intensity.
• The maximum kinetic energy of the emitted
electrons does depend on the photon
frequency.
• Electrons are emitted almost instantaneously
from the surface.
The Photoelectric Effect Explained
(Einstein 1905, Nobel Prize 1921)
The photoelectric effect can be understood as follows:
• Electrons are emitted by absorbing a single photon.
• A certain amount of energy , called the work function,
W0, is required to remove the electron from the
material.
• The maximum observed kinetic energy is the
difference between the photon energy and the work
function.
Kmax = E – W0
E = photon energy
"just shows emission" E = W0 means Kmax is zero
Cutoff Frequency f0 = W0 /h
The work function is a property of the individual metal
Element
Aluminum
4.3
Carbon
Work
Functio
n
Element
Nickel
5.1
5.0
Silicon
4.8
Copper
4.7
Silver
4.3
Gold
5.1
Sodium
2.7
(eV)
Work
Functio
n
(eV)
Application: Photocells, Solar Cells
If the energy of a photon is high
enough it can break bonds in
molecules/materials
Walker Problem 25, pg. 1008
Zinc and cadmium have photoelectric work
functions given by WZn = 4.33 eV and WCd = 4.22
eV, respectively. (a) If both metals are illuminated
by UV radiation of the same wavelength, which one
gives off photoelectrons with the greater maximum
kinetic energy? Explain. (b) Calculate the
maximum kinetic energy of photoelectrons from
each surface if λ = 275 nm.
The Mass and Momentum of a Photon
Photons have momentum, but no mass.
We cannot use the formula p = mv to find
the momentum of the photon. Instead:
hf h
p=
=
c λ
The Wave Nature of Particles
We have seen that light is described sometimes
as a wave and sometimes as a particle.
In 1924, Louis deBroglie proposed that particles
also display this dual nature and can be
described by waves too!
The deBroglie wavelength of a particle is related
to its momentum:
λ= h/p
(Use p = γm0v if the velocity is large.)
That is why you can make an atom laser!
Example
If an electron has a speed of 1.00 × 106
m/s, what is its wavelength?
Example
The maximum momentum of electrons at the
Jefferson Lab accelerator in Newport News is 6
GeV/c.
(a) What is the wavelength of those electrons?
(b) Why is the wavelength well suited to the study of
nuclear physics?
We can investigate wave properties of
particles through interference/diffraction
- X-rays
- Electrons
- Neutrons
- Atoms like
hydrogen or
helium
Bragg equation
2d sin θ = mλ
m = 1,2,3...
Heisenberg Uncertainty Principle
To describe a particle a physicist would refer four properties, the
position of the electron, its momentum, its energy, and the time.
Heisenberg showed that no matter how accurate the
instruments used, quantum mechanics limits the precision
when two properties (let´s built two pairs: momentum-position
or energy-time) are measured at the same time.
"The more precisely
the POSITION is
determined,
the less precisely
the MOMENTUM is known"
WERNER HEISENBERG (1901 - 1976)
Determinism of Classical
Mechanics
• Suppose the positions and speeds of all particles in
the universe are measured to sufficient accuracy at
a particular instant in time
• It is possible to predict the motions of every particle
at any time in the future (or in the past for that matter)
“An intelligent being knowing, at a given instant of time, all forces
acting in nature, as well as the momentary positions of all things of
which the universe consists, would be able to comprehend the
motions of the largest bodies of the world and those of the smallest
atoms in one single formula, provided it were sufficiently powerful
to subject all the data to analysis; to it, nothing would be uncertain,
both future and past would be present before its eyes.”
Pierre Simon Laplace
Measuring the position
of an electron
• Shine light on electron and detect reflected
light using a microscope
• Minimum uncertainty in position
is given by the wavelength of the
light
• So to determine the position
accurately, it is necessary to use
light with a short wavelength
Measuring the momentum
of an electron
• By Planck’s law E = hc/λ, a photon with a short
wavelength has a large energy
• Thus, it would impart a large ‘kick’ to the electron
• But to determine its momentum accurately,
electron must only be given a small kick
• This means using light of long wavelength!
Fundamental Trade Off …
• Use light with short wavelength:
accurate measurement of position but not momentum
• Use light with long wavelength:
accurate measurement of momentum but not position
h
ΔxΔp ≥
=h
2π
The more accurately you know the position (i.e.,
the smaller Δx is) , the less accurately you
know the momentum (i.e., the larger Δp is);
and vice versa
Heisenberg’s Uncertainty Principle
involving energy and time
h
ΔEΔt ≥
=h
2π
• The more accurately we know the energy of a body,
the less accurately we know how long it possessed
that energy
• The energy can be known with perfect precision (ΔE
= 0), only if the measurement is made over an infinite
period of time (Δt = ∞)