Transcript Quantum Theory_ch27 - bba-npreiser

Quantum Physics
Chapter 27
Birth of quantum physics
• The birth of quantum physics is attributed to Max
Planck's 1900 paper on blackbody radiation.
• Planck discovered the electromagnetic radiation
emitted by a perfect blackbody has an intensity per
unit wavelength that varies from wavelength to
wavelength, as each curve indicates. At the higher
temperature, the intensity per unit wavelength is
greater, and the maximum occurs at a shorter
wavelength.
• http://phet.colorado.edu/simulations/sims.php?sim=Blackbo
dy_Spectrum
• “Quantum mechanics is the description of
matter in all its detail on an atomic
scale….Because atomic behavior is so unlike
ordinary experience it is very difficult to get
used to and it appears peculiar and
mysterious to everyone, both to the novice
and to the experienced physicist. Even the
experts do not understand it the way they
would like to and it is perfectly reasonable
that they should not….” – Richard Feynman
• Classical – regular size objects traveling at normal
speeds
– Classical world is deterministic: Knowing the
position and velocity of all objects at a particular
time the future can be predicted using known laws
of force and Newton's laws of motion.
• Relativistic – regular, small or large objects traveling at
high speeds (close to c)
– Space and time are altered
• Quantum – Tiny objects (atoms or smaller)
– Atoms are very small ; about 0.5 nanometers
– Quantum world is probabilistic: Impossible to
know position and velocity with certainty at a
given time.
– Only probability of future state can be predicted
using known laws of force and equations of
quantum mechanics.
– Observer and observed are tied together
QUANTUM MECHANICS IS A
PROBABILISTIC THEORY OF NATURE
•
•
•
BEFORE OBSERVATION IT IS IMPOSSIBLE TO
SAY WHETHER AN OBJECT IS A WAVE OR A
PARTICLE OR WHETHER IT EXISTS AT ALL !!
IN THE COPENHAGEN INTERPRETATION OF
BOHR AND HEISENBERG IT IS IMPOSSIBLE IN
PRINCIPLE FOR OUR WORLD TO BE
DETERMINISTIC !
EINSTEIN, A FOUNDER OF QM WAS
UNCOMFORTABLE WITH THIS
INTERPRETATION - God does not play dice !
Stern-Gerlach Experiment
• http://phet.colorado.edu/simulations/sims.php?sim=SternGer
lach_Experiment
• In their experiment O. Stern and W. Gerlach measured the
intrinsic spin angular momentum of silver atoms and found it
to take only two discrete values, + h/2 and - h/2 commonly
called "spin up" and "spin down. In the experiment, the
magnet produces an inhomogeneous field. Due to their
intrinsic magnetic moments the silver atoms are deflected by
the inhomogeneous field. The field causes an actual
separation in space of atoms with spin up and spin down.
Particle or Wave
•
•
•
When we think of particles, we think of material objects like a ball, a car ,
a person…They can be located at a space point at a given time. They can
be at rest, moving or accelerating.
When we think of waves, we think of ocean waves, sound waves, radio
waves... Wavelength ,frequency, velocity and oscillation size defines
waves.
When particles collide they cannot pass through each other ! They can
bounce or they can shatter
•
Waves can pass through each other. As they pass
through each other they can enhance or cancel.
Waves and Particles
Spread in space and time
Waves
Can be superposed – show
interference effects
Pass through each other
Localized in space and time
Particles
Cannot pass through each other they bounce or shatter.
Light – Wave or Particle
• Early in the 20th century the photoelectric effect
changed the way physicists thought about light.
• In the photoelectric effect, light with a sufficiently high
frequency ejects electrons from a metal surface. These
photoelectrons, as they are called, are drawn to the
positive collector, thus producing a current.
• In 1905 Einstein presented an explanation of the
photoelectric effect that took advantage of Planck’s
work concerning blackbody radiation. It was primarily
for his theory of the photoelectric effect that he was
awarded the Nobel Prize in physics in 1921. In his
photoelectric theory, Einstein proposed that light of
frequency f could be regarded as a collection of
discrete packets of energy (photons), each packet
containing an amount of energy E given by: (where h is
Planck’s constant)
Light – Wave or Particle
• Thomas Young proved that light could behave as a
wave by doing the historic "Double Slit“ experiment
which it turns out was not done with a double slit at
all.
• "The experiments I am about to relate ... may be
repeated with great ease, whenever the sun shines,
and without any other apparatus than is at hand to
every one." – Thomas Young
• "...It will not be denied by the most prejudiced,"
Young chided his skeptical listeners, "that the fringes
[which are observed] are produced by the
interference of two portions of light." - Thomas
Young
• He was speaking in 1803 to a group of scientists that
believed in Newton’s view of light as that light is
made of tiny bullet-like particles, because it is
always observed (or so Newton thought) to travel in
straight beams…..
Electron – Particle or Wave?
• If light, which was thought to be a wave can act as
particle, can a particle act as a wave??
• In 1924, a French physicist named Louis de Broglie
suggested that, like light, electrons could act as both
particles and waves.
• In the 1920’s Clinton Davisson and George Thomson
both described experiments that confirmed the wave
nature of electrons and won the Nobel Prize in 1937.
• When electrons strike the screen a pattern of bright and
dark fringes is revealed (b) indicated interference, not a
pattern that would be expected if electrons were acting as
particles.
• Waves can exhibit particle-like characteristics, and
particles can exhibit wave-like characteristics.
Wave properties of Particles
• If Young’s double-slit experiment is performed
with electrons, an interference pattern in
observed demonstrating that electrons can
behave as waves.
• If a particle is a wave it must have a wavelength,
in order to derive the wavelength we start with
finding the wavelength of a photon.
• E = hf = hc/l and p = E/c = hc/cl = h/l so
• l = h/p
• De Broglie suggested that this formula is true
for any particle! Thus, the frequency and
wavelength of matter waves can be
h
determined. I.e. de Broglie wavelength of al 
mv
particle is
Sample Problem
• Calculate the de Broglie wavelength for a proton
(mp=1.67x10-27 kg ) moving with a speed of 1.00 x
107 m/s.
34
lp s
6.63 10 J  s 


1.67 10 kg 1.00 10
31
7
ms

 3.97 1014 m
• A non-relativistic electron and a non-relativistic
proton are moving and have the same de Broglie
wavelength. Which of the following are also the
same for the two particles: (a) speed, (b) kinetic
energy, (c) momentum, (d) frequency?
Sample Problem
• Determine the de Broglie wavelength for a
baseball (mass = 0.15 kg) moving at a speed of
13 m/s.
Wave Functions
•
•
•
•
For electron (and other particle) if we look for particle behaviour we see
particle and if we look for waves behaviour we see wave behaviour. The
behaviour you observe– depends on your method of observation.
As a result objects in quantum mechanics are described by a mathematical
functions called wave functions which are measures of probability.
In 1926 Schrödinger proposed a wave equation that describes the manner
in which matter waves change in space and time.
Schrödinger’s
wave equation is 2a key element in quantum mechanics:
 2   2  2  1  1


1

 

  V  E

sin


 r 2 rr r 2  2

2
2m 
•
•
•
 sin  
sin  
 

Schrödinger’s wave equation is generally solved for the wave function, Ψ
The wave function depends on the particle’s position and the time.
According to Max Born, the value of |Ψ|2 at some location at a given time
is proportional to the probability of finding the particle at that location at
that time
Solutions to Schroedinger’s Wave Equation
for Hydrogen
Solutions to Schroedinger’s Wave Equation
for Hydrogen – radial part
http://www.orbitals.com/orb/orbtable.htm
Solutions to Schroedinger’s Wave Equation
for Hydrogen – angular part
Schroedinger's Cat
• On June 7 of 1935, Erwin Schroedinger wrote to Albert Einstein to
congratulate him on what is now known as the EPR paper, a famous
problem in the interpretation of Quantum Mechanics. Soon thereafter, he
published what was to become one of the most celebrated paradoxes in
quantum theory:
• A cat is placed in a box, together with a radioactive atom. If the atom
decays, and the geiger-counter detects an alpha particle, the hammer hits
a flask of prussic acid (HCN), killing the cat. The paradox lies in the clever
coupling of quantum and classical domains. Before the observer opens the
box, the cat's fate is tied to the wave function of the atom, which is itself
in a superposition of decayed and undecayed states. Thus, said
Schroedinger, the cat must itself be in a superposition of dead and alive
states before the observer opens the box, ``observes'' the cat, and
``collapses'' it's wave function.
The Uncertainty Principle
• When measurements are made, the experimenter is always
faced with experimental uncertainties in the measurements
– Classical mechanics offers no fundamental barrier to
ultimate refinements in measurements
– Classical mechanics would allow for measurements with
arbitrarily small uncertainties
• Quantum mechanics predicts that a barrier to measurements
with ultimately small uncertainties does exist
– If a measurement of position of a particle is made with
precision Δx and a simultaneous measurement of linear
momentum is made with precision Δp, then the product of
the two uncertainties can never be smaller than h/4
The Uncertainty Principle
h
xp x 
4
• “The more precisely the position is
determined, the less precisely the momentum
is known in this instant, and vice versa.” -Heisenberg, uncertainty paper, 1927
• Another form of the principle deals with
energy and time:
h
Et 
4
• Mathematically,
Wave Packet
• A way to
visualize a
wave packet is
to construct
one by adding
waves of
different
wavelengths.
Wave Packet
• A larger range of wavelengths gives a more localized
wave or a wider the spread of wavelengths
contributes to a smaller x.
h
Et 
4
Wave Packet
• If the think of the wave as localized in time, we could say a
wave packet that is more confined in time requires a wider
distribution of frequencies.
• If we divide both sides by h, we get ft > 1/4
• https://phet.colorado.edu/en/simulation/fourier
• If the think of the wave as localized in position, we could say a
wave packet that is more confined in space requires a wider
distribution of wavelengths.
h
h
xp x 
4
l
mv
A localized wave or wave packet:
• A moving particle in quantum theory:
Spread in position
x
Spread in momentum
Superposition of waves of different
wavelengths makes a packet.
p
h
l
mv
Narrower the packet , more the spread in momentum
Basis of Uncertainty Principle
Thought Experiment – the Uncertainty Principle
• A thought experiment for viewing an electron with a powerful microscope
• In order to see the electron, at least one photon must bounce off it
• During this interaction, momentum is transferred from the photon to the
electron
• Therefore, the light that allows you to accurately locate the electron
changes the momentum of the electron
• If we wish to locate the electron more accurately, we must use light with a
smaller wavelength. Consequently, the photons in the light beam will
have greater momentum, since some of the photon’s momentum will be
transferred, the momentum change of the electron will be greater.
THE MOMENTUM OF A PHOTON AND THE
COMPTON EFFECT
• The phenomenon in which an X-ray photon is
scattered from an electron, with the scattered
photon having a smaller frequency than the incident
photon, is called the Compton effect.
Compton Effect
• Compton showed that the difference between
the wavelength λ’ of the scattered photon and
the wavelength λ of the incident photon is
related to the scattering angle θ by:
Sample Problem - macroscopic uncertainty
• A 50.0-g ball moves at 30.0 m/s. If its speed is
measured to an accuracy of 0.10%, what is the
minimum uncertainty in its position?
p  m  v   m  v  v 



 50.0 102 kg 1.0 103  30 m s  1.5 102 kg  m s
h
6.63 1024 J  s
32
x 


3.5

10
m
3
4  p  4 1.5 10 kg  m s


Early hints of quantum nature of atoms….

Discrete Emission and Absorption spectra


When excited in an electrical discharge, atoms emitted radiation only at
discrete wavelengths
Different emission spectra for different atoms
l (nm)
• Geiger-Marsden (Rutherford) Experiment (1911) conclusion: Most of
atomic mass is concentrated in a small region of the atom
• “It was almost as incredible as if you fired a 15-inch shell at a piece of
tissue paper, and it came back to hit you!”
--E. Rutherford (on the ‘discovery’ of the nucleus)
Rutherford Experiment
Bohr Model – Planetary Model
• In 1913 Bohr introduced a model of the
atom similar to a solar system with
electrostatic force holding the electrons
in orbit instead of gravity.
• With his model he was able to calculate
the emission lines of atomic hydrogen
• BUT there was a BIG PROBLEM:
 As the electron moves in its circular orbit,
it is ACCELERATING.
 As you learned earlier, accelerating charges
radiate electromagnetic energy.
 Consequently, an electron would
continuously lose energy and spiral into
the nucleus in about 10-9 sec.
de Broglie ….
• de Broglie came up with an explanation for why the
angular momentum might be quantized in the
manner Bohr assumed it was. de Broglie realized that
if you use the wavelength associated with the
electron, and assume that an integral number of
wavelengths must fit in the circumference of an
orbit, you get the same quantized angular momenta
that Bohr did
• http://physics.bu.edu/~duffy/semester2/c37_deBrog
lie.html
Heisenberg
• “The more precisely the position is determined, the less
precisely the momentum is known in this instant, and vice
versa.” --Heisenberg, uncertainty paper, 1927
• Heisenberg in trying to resolve the issues with the Bohr model
developed his uncertainty theory and his the "matrix" form of
quantum mechanics
• Why doesn’t the quantum electron collapse into the nucleus,
where its potential energy is lowest?
– The more confined it gets, the bigger p spread it has, from Heisenberg
Uncertainty. More p2/2m means more KE.
– So there’s a tradeoff between lowering PE and raising KE.
Schrödinger
• I knew of [Heisenberg's] theory, of course, but I felt
discouraged, not to say repelled, by the methods of
transcendental algebra, which appeared difficult to
me, and by the lack of visualizability. -Schrödinger in
1926
• Most physicists were slow to accept "matrix
mechanics" because of its abstract nature and its
unfamiliar mathematics. They gladly welcomed
Schrödinger's alternative wave mechanics when it
appeared in early 1926
Dirac
• After Schrödinger showed the equivalence of
the matrix and wave versions of quantum
mechanics, and Born presented a statistical
interpretation of the wave function, Jordan in
Göttingen and Paul Dirac in Cambridge,
England, created unified equations known as
"transformation theory." These formed the
basis of what is now regarded as quantum
mechanics.
Sources
• bohr.winthrop.edu/faculty/mahes/link_to_webpages
/courses/phys202/lecture29.1.ppt
• www.physics.uci.edu/~silverma/quantum.ppt
• http://www.lassp.cornell.edu/ardlouis/dissipative/Sc
hrcat.html
• http://www.juliantrubin.com/bigten/youngdoubleslit
.html
• http://www.aip.org/history/heisenberg/p08.htm
• online.physics.uiuc.edu/courses/phys214/spring06/Le
ctures/Lect11.ppt