Download wave function - Purdue Physics

Quantum “Regime”
• At the atomic scale, classical physics ideas aren’t
sufficient. All particles have wavelike nature, all
waves are at the same time composed of particles.
• Usually this is not apparent, as we shall see – large
scale or heavy objects tend to have such very short
wavelengths that they appear “crisp”, as we normally
experience
• The word “quantum” arose originally when Max
Planck deduced that to explain the spectrum of
wavelengths of EM radiation emitted by a black body
at various temperatures, it was necessary to
postulate that EM energy is QUANTIZED in little
packages of size directly proportional to their
frequency.
Introduction
Waves vs. Particles
• Waves produce an
interference pattern
when passed through a
double slit
• Classical particles
(bullets) can pass
through only one of the
slits, and no
interference pattern will
be formed
Section 28.1
Waves vs. Particles
• Waves produce an interference
pattern when passed through a
double slit
• Quantum particles can produce
the interference pattern.
Experimental proof:
• Let photons come to the slits so
slowly that the detectors can keep
track of the individual discrete
arrival times. NONETHELESS,
the photons go preferentially to
the bright fringes, not just straight
line paths!!!
Section 28.1
Interference with Electrons
• Electrons can be used in a double slit experiment
• The blue lines show the probability of the electrons striking particular
locations
• Yet we know that electrons are also particles (because they are
charged, among other properties). “Wave-particle duality”
• Photons (light quanta) are neutral, and so their dual quantum nature
is less obvious – but still detectable as per previous slide. Section 28.1
Interference with Electrons, cont.
• The probability pattern of the electrons has the same
form as the variation of light intensity in the double-slit
interference experiment
• The experiment shows that electrons undergo
constructive and destructive interference at certain
locations on the screen
• The experiment also shows aspects of particle-like
behavior since the electrons arrive one at a time at the
screen, and also don’t just go in straight lines.
• In principle, you can slowly build up an interference
pattern over time even if only one electron (or photon)
per second arrives at the screen, for example. Weird!
Section 28.1
Work Function
• Already in the 1880s,
studies of what happens
when light is shone onto
a metal gave some
results that could not be
explained with the wave
theory of light
• The work function, Wc
is the minimum energy
required to remove a
single electron from a
piece of metal
Section 28.2
Work Functions of Metals
Section 28.2
Photoelectric Effect
• One way to extract
electrons from a metal
is by shining light onto it
• Light striking a metal is
absorbed by the
electrons
• If an electron absorbs
an amount of light
energy greater than Wc,
it is ejected off the metal
• This is called the
photoelectric effect
Section 28.2
Photoelectric Effect, cont.
• No electrons are
emitted unless the
light’s frequency is
greater than a critical
value ƒc
• When the frequency is
above ƒc, the kinetic
energy of the emitted
electrons varies linearly
with the frequency
Section 28.2
Photoelectric Effect, Explanation
• Trying to explain the photoelectric effect with the
classical wave theory of light presented two
difficulties
• Experiments showed that the critical frequency is
independent of the intensity of the light
• Remember, the E field in a light wave goes as the
square root of the intensity. You’d think a big enough
E field would manage to eject electrons. Wrong!
• Below the critical frequency, there are no ejected
electrons no matter how great the light intensity
• Also the KE of ejected electrons depends on the light
frequency, NOT intensity. Both phenomena lead to the
same conclusion.
Section 28.2
Photons
• Einstein proposed that light carries energy in
discrete quanta, now called photons
• Each photon carries a parcel of energy Ephoton = hƒ
• h is a constant of nature called Planck’s constant
• h = 6.626 x 10-34 J ∙ s
TINY, measures the scale
of quantum phenomena
• A beam of light should be thought of as a collection
of photons (as well as a wave, which it also is!)
• Each photon has an energy dependent on its
frequency
• If the intensity of monochromatic light is increased,
the number of photons is increased, but the energy
carried by each photon does not change
Section 28.2
Momentum of a Photon
• A light wave with energy E also carries a certain
momentum
pphoton
E hƒ h
= =
=
c
c
l
• “Particles” of light called photons carry a discrete
amount of both energy and momentum
• Photons, unlike classical particles, have no mass.
• Therefore they can travel with velocity exactly c in
vacuum (and ONLY at the velocity c)
Section 28.2
Blackbody Radiation
• Blackbody radiation is emitted over a range of
wavelengths
• To the eye, the perceived color of the cavity is
determined by the wavelength at which the radiation
intensity is largest
Section 28.2
Blackbody Radiation, Classical
• The blackbody intensity curve has the same shape
•
•
•
•
•
for a wide variety of objects
Electromagnetic waves can form standing waves as
they reflect back and forth inside the oven’s cavity
The frequencies of the standing waves follow the
pattern ƒn = n ƒ where n = 1, 2, 3, …
There is no limit to the value of n, so the frequency
can be infinitely large
But as the frequency increases, so does the energy
Classical theory predicts that the blackbody intensity
should become infinite as the frequency approaches
infinity
Section 28.2
Blackbody Radiation, Quantum
• The disagreement between the classical predictions
•
•
•
•
•
and experimental observations was called the
“ultraviolet catastrophe”
Planck proposed solving the problem by assuming
the energy in a blackbody cavity must come in
discrete parcels
Each parcel would have energy E = h ƒn
His theory (formula) fit the experimental results, but
gave no theoretical reason why the formula worked
Planck’s work is generally considered to be the
beginning of quantum theory.
Einstein gave the theoretical underpinning in 1905.
Section 28.2
Wave-like Properties of Particles
• The notion that the properties of both classical
waves and classical particles are present at the
same time is also called wave-particle duality
• The possibility that all particles are capable of wavelike properties was first proposed by Louis de Broglie
• De Broglie suggested that if a particle has a
momentum p, its wavelength is
h
l=
p
• His doctoral thesis is said to have been only two
pages long!! Probably an apocryphal story.
Section 28.3
Electron Interference
• To test de Broglie’s
hypothesis, an
experiment was
designed to observe
interference involving
classical particles
• The experiment showed
conclusively that
electrons have wavelike
properties
• The calculated
wavelength was in good
agreement with de
Broglie’s theory
Section 28.3
Wavelengths of Macroscopic Particles
• From de Broglie’s equation and using the classical
expression for kinetic energy
h
h
l= =
p
remember KE = ½ mv2
2m(KE )
• As the mass (and/or KE) of the particle (object)
increases, its wavelength decreases
• In principle, you could observe interference with
baseballs
• Has not yet been observed -- and an experiment
showing baseball interference is unlikely to be seen
anytime soon – but using atoms, yes, that’s been seen
Section 28.3
Electron Spin
• Electrons have another
quantum property that
involves their magnetic
behavior
• An electron has a
magnetic moment, a
property associated with
electron spin
• Classically, the
electron’s magnetic
moment can be thought
of as spinning ball of
charge
Section 28.4
Electron Spin, cont.
• The spinning ball of
•
•
•
•
charge acts like a
collection of current loops
This produces a magnetic
field
It acts like a small bar
magnet
Therefore, it is attracted to
or repelled from the poles
of other magnets
And experiences
magnetic torque
Section 28.4
Electron Spin, Direction
• When a beam of properly prepared electrons passes
near one end of a bar magnet, there are two directions of
deflection observed
• Two orientations for the electron magnetic moment are
possible
• Classical theory predicts the moment may point in any
direction
Section 28.4
Electron Spin, Direction, cont.
• Classically, the electrons should deflect over a range
•
•
•
•
of angles
Observing only two directions of deflection indicates
there are only two possible orientations for the
magnetic moment
The electron magnetic moment is quantized in
direction with only two possible values
The magnitude of the electron’s magnetic moment
is also quantized
Pick any axis in 3-D. The spin of the electron is +the standard amount along the chosen direction.
Super weird, but that’s the way it works
Section 28.4
Wave Function
• In the quantum world, the motion of a particle-wave
is described by its wave function
• The wave function can be calculated from
Schrödinger’s equation
• Developed by Erwin Schrödinger, one of the inventors
of quantum theory
• Schrödinger’s equation plays a role similar to
Newton’s laws of motion: it tells how the wave
function evolves with time, just as Newton’s laws tell
classically how a particle’s position, momentum, and
energy evolve with time.
• In many situations, the solutions of the Schrödinger
equation are mathematically similar to standing
waves
Section 28.5
Wave Function Example
• An electron is confined to
•
•
•
•
a particular region of
space
A classical particle would
travel back and forth
inside the box
The wave function for the
electron is described by
standing waves
Two out of many possible
standing waves are
shown (B)
[C], intensity, is the square
of the [B] amplitude
pattern
Section 28.5
Wave Function Example, cont.
• The wave function solutions correspond to electrons with
•
•
•
•
different kinetic energies
The wavelengths of the standing waves are different
• Given by de Broglie’s equation
After finding the wave function, one can calculate the position
and velocity of the electron
• But does not give a single value
The wave function allows for the calculation of the probability
of finding the electron at different locations in space
That’s how sparse electrons can slowly build up a wave
interference pattern on a screen – each electron has a
probability of arriving different places on the screen: on
average more arrive in the “bright” directions, etc.
Section 28.5
Heisenberg Uncertainty Principle
• Wavy quantum effects place fundamental limits on
•
•
•
•
the precision of measuring position or velocity
Standing waves in the box are the electron, so there
is an inherent uncertainty in its position
There is some probability for finding the electron at
virtually any spot in the box
The uncertainty in position, Δx, is approximately the
size of the box. Smaller box: smaller Δx and
shorter wavelengths. Meaning higher frequencies.
Guess what has happened to the energy levels!
Section 28.5
Quiz
• The uncertainty in position, Δx, is approximately the
size of the box. Smaller box: smaller Δx and
shorter wavelengths. Meaning higher frequencies.
• Compared to a larger box, the energy levels in the
smaller box are:
• A) the same
• B) smaller
• C) larger
• D) not determinable
Section 28.5
Uncertainty, an Example
• Electrons are incident
on a narrow slit
• The electron wave is
diffracted as it passes
through the slit
• The interference pattern
gives a measure of how
the wave function of the
electron is distributed
throughout space after it
passes through the slit
Section 28.5
Uncertainty, Example, cont.
• The initial momentum is
in the y direction
• The diffracted electron
acquires a nonzero
momentum along x
• The width of the slit
affects the interference
pattern
• The narrower the slit,
the broader the
distribution pattern
• Meaning more spread
(uncertainty Δpx ) of the
sideways momentum
Heisenberg Uncertainty Principle
• The Heisenberg Uncertainty Principle gives the lower
limit on the product of Δx and Δp
h
Dx Dp ³
4p
• The uncertainties Δx and Δp are limits set by
quantum theory
• As usual, Planck’s constant sets the (tiny) scale of
this quantum effect
• The relationship holds for any quantum situation and
for any wave-particle
Section 28.5
Heisenberg Time-Energy Uncertainty
• You can also derive a relation between the uncertainties
in the energy ΔE of a particle and the time interval Δt
over which this energy is measured or generated
• The Heisenberg energy-time uncertainty principle is
h
DE Dt ³
4p
• Again the scale is so small that we can observe it
classically. But it’s very real.
• You can’t exactly determine the frequency of a wave if
you observe it for a finite length of time – this happens
even for sound waves or radio waves, classically. It’s a
mathematical property of any sine or cos wave that
doesn’t repeat forever.
Section 28.5
Heisenberg Uncertainty Principle, final
• Quantum theory and the uncertainty principle mean
that there is always a trade-off between the
uncertainties
• It is not possible, even in principle, to have perfect
knowledge of both x and px
• This suggests that there is always some inherent
uncertainty in our knowledge of the physical universe
• Quantum theory says that the world is inherently
unpredictable
• For any macroscale object, the uncertainties in the
actual measurement will always be much larger than
the inherent uncertainties due to the Heisenberg
uncertainty relation
Section 28.5
Heisenberg Uncertainty and Relativity
• Relativity deals with 3-D space and one time
dimension
vector
x, y, z, ct
we call this set a four-
• Similarly, px, py, pz and E/c form a four-vector
• So it’s entirely natural to have FOUR Heisenberg
uncertainty relations, one each for the x, y, and z
components, and one for the time component
• There are a lot of beautiful symmetries in nature that
guide our construction of theories of the universe
Section 28.5