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MODERN PHYSICS
NICASTRO
and
IN THIS PRESENTATION, YOU WILL BE INTRODUCED TO THE FOLLOWING TOPICS:
IN THIS PRESENTATION, YOU WILL BE INTRODUCED TO THE FOLLOWING TOPICS:
1. QUANTUM WAVE PACKETS
IN THIS PRESENTATION, YOU WILL BE INTRODUCED TO THE FOLLOWING TOPICS:
1. QUANTUM WAVE PACKETS
2. UNCERTAINTY IN MEASUREMENT
IN THIS PRESENTATION, YOU WILL BE INTRODUCED TO THE FOLLOWING TOPICS:
1. QUANTUM WAVE PACKETS
2. UNCERTAINTY IN MEASUREMENT
3. MEASURING POSITION
IN THIS PRESENTATION, YOU WILL BE INTRODUCED TO THE FOLLOWING TOPICS:
1. QUANTUM WAVE PACKETS
2. UNCERTAINTY IN MEASUREMENT
3. MEASURING POSITION
4. MEASURING VELOCITY
IN THIS PRESENTATION, YOU WILL BE INTRODUCED TO THE FOLLOWING TOPICS:
1. QUANTUM WAVE PACKETS
2. UNCERTAINTY IN MEASUREMENT
3. MEASURING POSITION
4. MEASURING VELOCITY
5. HEISENBERG UNCERTAINTY
PRINCIPLE
IN THIS PRESENTATION, YOU WILL BE INTRODUCED TO THE FOLLOWING TOPICS:
1. QUANTUM WAVE PACKETS
2. UNCERTAINTY IN MEASUREMENT
3. MEASURING POSITION
4. MEASURING VELOCITY
5. HEISENBERG UNCERTAINTY
PRINCIPLE
6. WHY THE H.E.P. DOESN’T REALLY
APPLY TO “MACROSCOPIC OBJECTS”
IN THIS PRESENTATION, YOU WILL BE INTRODUCED TO THE FOLLOWING TOPICS:
1. QUANTUM WAVE PACKETS
2. UNCERTAINTY IN MEASUREMENT
3. MEASURING POSITION
4. MEASURING VELOCITY
5. HEISENBERG UNCERTAINTY
PRINCIPLE
6. WHY THE H.E.P. DOESN’T REALLY
APPLY TO “MACROSCOPIC OBJECTS”
7. THE COPENHAHEN INTERPRETATION
OF QUANTUM THEORY
IN THIS PRESENTATION, YOU WILL BE INTRODUCED TO THE FOLLOWING TOPICS:
1. QUANTUM WAVE PACKETS
2. UNCERTAINTY IN MEASUREMENT
3. MEASURING POSITION
4. MEASURING VELOCITY
5. HEISENBERG UNCERTAINTY
PRINCIPLE
6. WHY THE H.E.P. DOESN’T REALLY
APPLY TO “MACROSCOPIC OBJECTS”
7. THE COPENHAHEN INTERPRETATION
OF QUANTUM THEORY
8. QUANTUM FLUCTUATIONS
IN THE MACROSCOPIC WORLD,
WHAT DOES IT MEAN TO
MAKE
AN OBSERVATION, OR MAKE A
MEASUREMENT???
IN THE MACROSCOPIC WORLD,
WHAT DOES IT MEAN TO
MAKE
AN OBSERVATION, OR MAKE A
MEASUREMENT???
LET’S USE A BASEBALL
ANALOGY:
We “observe”
something by receiving
light waves that scatter,
reflect or are emitted by
objects.
Our eyes absorb this energy
and convert it into
information that our brain
processes.
When the pitcher throws
the ball, light rays reflect off
the ball and we, with the
help of our eyes and brain,
are able to locate the
position of the ball with a
great deal of certainty.
We can also determine (with some practice) the speed of
the ball, and the path it will take when it reaches the plate.
All this information is
received because
light rays reflect off
of the ball.
Of course, the actual speed of the ball might be
measured using more sophisticated devices (radar
gun, photocells, electronic timers, etc.).
Our brain processes
this information in
about one fourth of
a second, and sends
signals to our
muscles to swing
the bat at the
projected location
of the ball!
The baseball is “macroscopic”. The energy of the
light waves interacting with the ball does not
significantly effect its location, trajectory or speed.
Therefore, at this level and scale, where the position
and momentum of relatively large objects can be
accurately measured, the world is said to be
“DETERMINISTIC,” that is, capable of using basic laws
of physics (i.e., Newton’s Laws of Motion) to predict
future events.
THIS IS CALLED . . .
QUANTIZATION MEANS THAT WE CANNOT OBSERVE THE
UNIVERSE WITHOUT AFFECTING IT
QUANTIZATION MEANS THAT WE CANNOT OBSERVE THE
UNIVERSE WITHOUT AFFECTING IT
WHAT DOES THIS MEAN????
QUANTIZATION MEANS THAT WE CANNOT OBSERVE THE
UNIVERSE WITHOUT AFFECTING IT
WHAT DOES THIS MEAN????
At the QUANTUM (“microscopic”) level,
the universe is
NONDETERMINISTIC!
IF WE ARE TRYING TO LOCATE AN ELECTRON, OR
DETERMINE ITS MOTION (MOMENTUM), WE CAN
ONLY DO THIS BY CASTING LIGHT ON IT.
BEFORE
THE SHEER INTENSITY OF THE LIGHT WOULD
EASILY AFFECT OUR MEASUREMENT.
electron
AFTER
?
IF WE ARE TRYING TO LOCATE AN ELECTRON, OR
DETERMINE ITS MOTION (MOMENTUM), WE CAN
ONLY DO THIS BY CASTING LIGHT ON IT.
BEFORE
THE SHEER INTENSITY OF THE LIGHT WOULD
EASILY AFFECT OUR MEASUREMENT.
electron
BUT REMEMBER, EINSTEIN SAID THAT LIGHT
CONSISTS OF “PHOTONS” (LIGHT QUANTA), EACH
FREQUENCY HAVING ITS OWN ENERGY (hf).
AFTER
?
IF WE ARE TRYING TO LOCATE AN ELECTRON, OR
DETERMINE ITS MOTION (MOMENTUM), WE CAN
ONLY DO THIS BY CASTING LIGHT ON IT.
THE SHEER INTENSITY OF THE LIGHT WOULD
EASILY AFFECT OUR MEASUREMENT.
BUT REMEMBER, EINSTEIN SAID THAT LIGHT
CONSISTS OF “PHOTONS” (LIGHT QUANTA), EACH
FREQUENCY HAVING ITS OWN ENERGY (hf).
SO, WHAT IF WE MADE THE LIGHT SO DIM THAT
WE COULD SEND JUST ONE PHOTON TO DETECT
THE ELECTRON?
BEFORE
one photon
electron
IF WE ARE TRYING TO LOCATE AN ELECTRON, OR
DETERMINE ITS MOTION (MOMENTUM), WE CAN
ONLY DO THIS BY CASTING LIGHT ON IT.
THE SHEER INTENSITY OF THE LIGHT WOULD
EASILY AFFECT OUR MEASUREMENT.
BUT REMEMBER, EINSTEIN SAID THAT LIGHT
CONSISTS OF “PHOTONS” (LIGHT QUANTA), EACH
FREQUENCY HAVING ITS OWN ENERGY (hf).
SO, WHAT IF WE MADE THE LIGHT SO DIM THAT
WE COULD SEND JUST ONE PHOTON TO DETECT
THE ELECTRON?
BEFORE
one photon
electron
RECALL THAT PHOTONS OF
DIFFERENT FREQUENCIES ARE
PARTICLE-LIKE IN THEIR
INTERACTION WITH OTHER
OBJECTS, AND THAT EACH
PHOTON CONSISTS OF A TINY
“WAVE PACKET” WHICH IS
EITHER “COMPACT” (HIGH
FREQUENCY, HIGH ENERGY), OR
“SPREAD-OUT” (LOW
FREQUENCY, LOW ENERGY).
WAVE PACKETS
NARROW
*Photon well localized
*Short wavelength
*High frequency
BROAD
*Photon not localized
*Long wavelength
*Low frequency
HEISENBERG’S “QUANTUM MICROSCOPE”
Heisenberg devised this thought experiment to show how interactions
between an observer and the system under observation result in
unavoidable and unpredictable disturbances in the system
– a phenomenon that underlies Heisenberg’s Uncertainty Principle.
HEISENBERG’S “QUANTUM MICROSCOPE”
Heisenberg devised this thought experiment to show how interactions
between an observer and the system under observation result in
unavoidable and unpredictable disturbances in the system
– a phenomenon that underlies Heisenberg’s Uncertainty Principle.
The experiment attempts to measure the position and velocity (or momentum) of an
electron by shining light on it and detecting the scattered light.
But the light has both a wave and a particle nature, and to know precisely where the
light is, we need a wave packet of short-wavelength light.
BEFORE
detector
Light
source
electron
HEISENBERG’S “QUANTUM MICROSCOPE”
Heisenberg devised this thought experiment to show how interactions
between an observer and the system under observation result in
unavoidable and unpredictable disturbances in the system
– a phenomenon that underlies Heisenberg’s Uncertainty Principle.
The experiment attempts to measure the position and velocity (or momentum) of an
electron by shining light on it and detecting the scattered light.
But the light has both a wave and a particle nature, and to know precisely where the
light is, we need a wave packet of short-wavelength light.
Using such a packet, we
could determine the
electron’s position with high
precision.
AFTER
detector
Light
source
electron
?
HEISENBERG’S “QUANTUM MICROSCOPE”
Heisenberg devised this thought experiment to show how interactions
between an observer and the system under observation result in
unavoidable and unpredictable disturbances in the system
– a phenomenon that underlies Heisenberg’s Uncertainty Principle.
The experiment attempts to measure the position and velocity (or momentum) of an
electron by shining light on it and detecting the scattered light.
But the light has both a wave and a particle nature, and to know precisely where the
light is, we need a wave packet of short-wavelength light.
Using such a packet, we
AFTER
Light
source
electron
?
could determine the
electron’s position with high
precision.
detector But its correspondingly high
frequency (E=hf) means that
the wave packet carries high
energy, which imparts
momentum to the electron
in an unpredictable way,
disturbing its velocity.
HEISENBERG’S “QUANTUM MICROSCOPE”
Heisenberg devised this thought experiment to show how interactions
between an observer and the system under observation result in
unavoidable and unpredictable disturbances in the system
– a phenomenon that underlies Heisenberg’s Uncertainty Principle.
But the light has both a wave and a particle nature, and to know precisely where the
light is, we need a wave packet of short-wavelength light.
RESULT:
Good precision of its location, uncertainty about its velocity.
AFTER
detector
Light
source
electron
?
HEISENBERG’S “QUANTUM MICROSCOPE”
SO, WHAT IF WE USE A LOW-ENERGY PHOTON?
By using a low energy photons have a broad wave packet form (more “spread out”), so
it’s not precisely localized.
BEFORE
detector
Light
source
electron
HEISENBERG’S “QUANTUM MICROSCOPE”
SO, WHAT IF WE USE A LOW-ENERGY PHOTON?
By using a low energy photons have a broad wave packet form (more “spread out”), so
it’s not precisely localized.
RESULT:
the experiment measures velocity (momentum) precisely, but its position
remains uncertain.
AFTER
detector
Light
source
electron
?
HEISENBERG’S “QUANTUM MICROSCOPE”
BOTTOM LINE:
THE STANDARD INTERPRETATION OF QUANTUM PHYSICS IMPLIES
THAT IT IS MEANINGLESS TO TALK ABOUT A SUBATOMIC PARTICLE
AS HAVING SIMULTANEOUSLY BOTH A WELL-DEFINED VELOCITY AND
A WELL-DEFINED POSITION.
HEISENBERG’S “QUANTUM MICROSCOPE”
BOTTOM LINE:
THE STANDARD INTERPRETATION OF QUANTUM PHYSICS IMPLIES
THAT IT IS MEANINGLESS TO TALK ABOUT A SUBATOMIC PARTICLE
AS HAVING SIMULTANEOUSLY BOTH A WELL-DEFINED VELOCITY AND
A WELL-DEFINED POSITION.
THIS IS THE
“HEISENBERG UNCERTAINTY PRINCIPLE”.
MATHEMATICALLY THIS CAN BE STATED:
MATHEMATICALLY THIS CAN BE STATED:
The uncertainty in velocity
momentum (remember P = mv)
The uncertainty in position
Planck’s Constant
WHAT THIS MEANS
The uncertainty in velocity
momentum (remember P = mv)
Planck’s Constant
The uncertainty in position
AS
AND AS
∞
0
0
∞
This uncertainty leads to many strange things. For example, in a Quantum
Mechanical world, I cannot then predict where a particle will be with
100 % certainty. I can only speak in terms of probabilities.
For example, I can say that an atom will be at some location with a 99 %
probability, but there will be a 1 % probability it will be somewhere else
(in fact, there will be a small but finite probability that it will be found
across the Universe). This is strange.
This uncertainty leads to many strange things. For example, in a Quantum
Mechanical world, I cannot then predict where a particle will be with
100 % certainty. I can only speak in terms of probabilities.
For example, I can say that an atom will be at some location with a 99 %
probability, but there will be a 1 % probability it will be somewhere else
(in fact, there will be a small but finite probability that it will be found
across the Universe). This is strange.
BUT WHY DON’T WE OBSERVE THIS IN THE MACROSCOPIC
WORLD?
IF WE RESTATE THIS EQUATION, USING THE PRODUCT mv FOR MOMENTUM, THEN
m Δv Δx = h
Δv Δx = h
m
WHICH MEANS THAT THE LARGER THE
MASS, THE SMALLER THE
UNCERTAINTY.
THAT’S WHY MACROSCOPIC MASSES HAVE ALMOST NO UNCERTAINTY!!!
A SECOND FORM OF THE UNCERTAINTY PRINCIPLE IS
which perhaps has an even greater impact on our understanding of the
behavior of matter, involves the measurement of the energy of a particle, and
the time available to make the measurement. The shorter the time available,
the less accurate the energy measurement is.
If ΔE is the uncertainty in the results of our energy measurement, and
Δt the time we had to make the measurement,
then ΔE and Δt are related by
the variation above.
A SECOND FORM OF THE UNCERTAINTY PRINCIPLE IS
which perhaps has an even greater impact on our understanding of the
behavior of matter, involves the measurement of the energy of a particle, and
the time available to make the measurement. The shorter the time available,
the less accurate the energy measurement is.
If ΔE is the uncertainty in the results of our energy measurement, and
Δt the time we had to make the measurement,
then ΔE and Δt are related by
the variation above.
LET’S LOOK AT AN EXAMPLE:
A device that has become increasingly
important in research, particularly in the
study of fast reactions in molecules and
atoms, is the pulsed laser.
The lasers we have used in various
experiments are all continuous beam
lasers. The beam is at least as long as
the distance from the laser to the
wall.
A device that has become increasingly
important in research, particularly in the
study of fast reactions in molecules and
atoms, is the pulsed laser.
The lasers we have used in various
experiments are all continuous beam
lasers. The beam is at least as long as
the distance from the laser to the
wall.
The beam is at least as long as the
distance from the laser to the wall.
If we had a laser that we could
turn on and off in one nanosecond,
the pulse would be
1 foot or 30 cm long and contain
30cm
6 x 10– 5 cm/wavelength = 5 x 105wavelengths
Even a picosecond laser pulse which is 1000 times shorter, contains 500 wavelengths.
Some of the recent pulsed lasers can produce a pulse 500 times shorter than that,
only 2 femtoseconds (2 x 10– 15 seconds) long.
These lasers emit a pulse that is only one wavelength long.
If we had a laser that we could
turn on and off in one nanosecond,
the pulse would be
1 foot or 30 cm long and contain
30cm
6 x 10– 5 cm/wavelength = 5 x 105wavelengths
Even a picosecond laser pulse which is 1000 times shorter, contains 500 wavelengths.
Some of the recent pulsed lasers can produce a pulse 500 times shorter than that,
only 2 femtoseconds (2 x 10– 15 seconds) long.
These lasers emit a pulse that is only one wavelength long.
If we want to measure the energy of the photons in such a pulse, we only have 2
femtoseconds to make the measurement because that is how long the pulse takes to
go by us. In the notation of the uncertainty principle
Δt = 2 x 10–15 sec = 2 femtoseconds this is the time available to measure the
energy of the photons in our laser pulse
Let us suppose that the laser produces red photons whose wavelength is 6.2 x 10– 5 cm,
about the wavelength of the laser we have been using. According to our usual formula for
calculating the energy of the photons in such a laser beam we have
Ephoton = 12.4 x 10– 5 eV cm
6.2 x 10– 5 cm
= 2eV
Now let us use the uncertainty principle in the form ΔE ≥ h
Δt
to calculate the uncertainty in any measurement we would make the energy of the
photons in the 2 femtosecond laser beam. We have
ΔE ≥ h
Δt
≥
6.63 x 10– 34 J sec
2 X 10– 15 sec
≥ 3.31 x 10– 19 J
Converting ΔE from J to electron volts, we get
ΔE ≥ 3.31 x 10– 19 J
2 eV
The uncertainty in any energy measurement we make of these
photons is as great as the energy itself!
1.6 Å~ 10– 19 J/eV
≈
If we try to measure the energy of these photons, we expect the answers to
range from E – ΔE = 0 eV up to E + ΔE = 4 eV .
The uncertainty in energy is a consequence of the energy/time variation of
the Uncertainty Principle. The interaction time is known to a high degree of
precision.
If we try to measure the energy of these photons, we expect the answers to
range from E – ΔE = 0 eV up to E + ΔE = 4 eV .
The uncertainty in energy is a consequence of the energy/time variation of
the Uncertainty Principle. The interaction time is known to a high degree of
precision.
The same variation is vital in the field of QUANTUM ELECTRODYNAMICS,
where an apparent violation of energy conservation can be rationalized by an
interaction time within the uncertainty limits imposed.
This leads to the “creation” of “VIRTUAL PARTICLES” that last for a very short
time, but can explain the energies produced in a quantum process.
Such events are known as “QUANTUM FLUCTUATIONS” and they form the
basis for our understanding of particle interactions.
A Λ(1520) particle can be “created” if
the total energy (in the center of mass
system) of the incoming particles
equals the rest mass energy of the
Λ(1520).
A 2 eV photon suddenly creates a positronelectron pair. A short time later the pair
annihilates, leaving a 2 eV photon. In the long
term, energy is conserved
Quantum fluctuation. The uncertainty
principle allows such an object to suddenly
appear, and then disappear.
BUT ANOTHER ASPECT OF QUANTUM THEORY IS THAT PARTICLES, LIKE ELECTRONS,
HAVE NO REAL PHYSICAL SHAPE (LIKE A BASEBALL), AND HAVE A WAVE-LIKE
PROPERTY (NOT A PHYSICAL WAVE LIKE A WATER OR SOUND WAVE).
THIS WAVE-LIKE PROPERTY, A CONSEQUENCE OF WAVE-PARTICLE
COMPLIMENTARITY, SUGGESTS THAT THE ELECTRON IS MOST LIKELY WHERE THE
WAVES ARE “STRONGER”(LARGER AMPLITUDES). SO, INSTEAD OF BEING A “HARD
PARTICLE”, IT IS A PROBABILISTIC WAVE WHICH INDICATES THE “LIKELIHOOD” OF
THE ELECTRON’S PRESENCE IN SPACE.
BUT ANOTHER ASPECT OF QUANTUM THEORY IS THAT PARTICLES, LIKE ELECTRONS,
HAVE NO REAL PHYSICAL SHAPE (LIKE A BASEBALL), AND HAVE A WAVE-LIKE
PROPERTY (NOT A PHYSICAL WAVE LIKE A WATER OR SOUND WAVE).
THIS WAVE-LIKE PROPERTY, A CONSEQUENCE OF WAVE-PARTICLE
COMPLIMENTARITY, SUGGESTS THAT THE ELECTRON IS MOST LIKELY WHERE THE
WAVES ARE “STRONGER”(LARGER AMPLITUDES). SO, INSTEAD OF BEING A “HARD
PARTICLE”, IT IS A PROBABILISTIC WAVE WHICH INDICATES THE “LIKELIHOOD” OF
THE ELECTRON’S PRESENCE IN SPACE.
French physicist Louis deBroglie, in 1929,
suggested that if waves can have particle
properties, then particles should have wave
properties. The smaller the particle, the
more pronounced are its wave properties.
SO, AN ELECTRONS SHOULD HAVE DISTINCT WAVELIKE
PROPERTIES, LIKE FREQUENCY AND WAVELENGTH!
SO, AN ELECTRONS SHOULD HAVE DISTINCT WAVELIKE
PROPERTIES, LIKE FREQUENCY AND WAVELENGTH!
Planck’s Constant
“deBroglie
Wavelength
velocity
mass
THIS WOULD ANSWER THE QUESTION ABOUT WHY ELECTRONS DO NOT
LOSE ENERGY IN THEIR POSITION AROUND THE NUCLEUS OF AN ATOM,
AND THEREFORE WHY ATOMS EXIST!!!
IF AN ELECTRON, ENCIRCLING THE NUCLEUS OF ITS ATOM, HAS AN EXACT MULTIPLE (n) OF
WAVES, IT WOULD BE IN A STANDING WAVE PATTERN AROUND THAT NUCLEUS.
IF AN ELECTRON, ENCIRCLING THE NUCLEUS OF ITS ATOM, HAS AN EXACT MULTIPLE (n) OF
WAVES, IT WOULD BE IN A STANDING WAVE PATTERN AROUND THAT NUCLEUS.
(YOU REMEMBER
STANDING WAVE
PATTERNS FROM OUR
1ST SEMESTER WORK IN
WAVE THEORY.)
Length of SWP with six nodes
If this standing wave were twisted into a “circular” pattern, so that the circumference
Were exactly this length, then you would see something like this:
This would represent one possible “shell” location for electrons around a nucleus.
This would represent one possible “shell” location for electrons around a nucleus.
IT WOULD ALSO SERVE TO PREDICT THE SIZES OF “ALLOWABLE” SHELLS WHERE
ELECTRONS MAY BE FOUND (AT A DISTANCE “r” FROM THE NUCLEUS).
SINCE IT WOULD REPRESENT A “STANDING WAVE PATTERN,” ENERGY WOULD NOT
BE LOST, BUT “RECYCLED” WITHIN THE WAVE ITSELF.
IT WOULD ALSO SERVE TO PREDICT THE SIZES OF “ALLOWABLE” SHELLS WHERE
ELECTRONS MAY BE FOUND (AT A DISTANCE “r” FROM THE NUCLEUS).
HERE’S A DEMONSTRATION OF deBROGLIE WAVE PRODUCED BY OSCILLATING
A HOOP AT RESONANT FREQUENCIES.
Bohr’s atom showed electrons as particles similar to planets going
around the sun. de Broglie’s atom treats electrons more as waves with
wave patterns that fit symmetrically within the atom. In both cases, the
energy levels of the electrons must go up by an incremental quantity
(quantum).
Both of these played a role in the modern theory of the atom and the
structure of the electrons. As elements were built by adding one
proton and one electron at a time, the electrons would find a position
and shape that maximized their distance from each other (from
repulsion) yet kept them as close as possible to the positive nucleus
(attraction). The way they positioned themselves followed a fairly basic
pattern known as its quantum number.