Download Chapter 7 - Foothill College

Electronic Structure of Atoms
(Quantum Theory)
Classical Theory:
By the early 1900’s, “classical theory” viewed
light as behaving like a wave, as
demonstrated in 1801 by Thomas Young in his
“double slit” experiment. Waves can be
diffracted and are “spread out”, not localized
in specific place. Electrons were viewed as
behaving like particles. Particles have mass
and momentum and are localized, in a specific
place at a given time. Classical physics also
viewed energy as continuous, meaning that
all values of energy are allowed. However,
certain experimental observations could not
be explained using classical physics. These
included the distribution of light that is given
off from a glowing hot object (like a light bulb
filament or an electric stove), the
photoelectron effect and atomic emission
spectra.
Wave behavior
through a double slit.
Particle behavior
through a double slit.
During the early 1900’s a bold new theory, Quantum Theory was proposed as a way to
explain these phenomena. As we shall see, Quantum Theory defies “common sense”.
It is not something that we have any common, everyday experience with because it
applies to the very small subatomic realm. However, remember that what we view as
common sense evolves, changes, over time!
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Our goals in studying this chapter are to:
1.
develop an understanding of and be able to describe the properties of electromagnetic radiation,
EMR.
2. understand the relationship between frequency, wavelength and energy of EMR; be able to
calculate one from another.
3. understand what is meant by wave-particle duality.
4. understand and be able to describe what the photoelectric effect is.
5. understand and be able to describe the emission of light from atoms.
6. understand and be able to describe the Bohr model of the hydrogen atom and its historical
significance; be able to mathematically relate the wavelength and frequency of emitted light to
energy levels in the hydrogen atom.
7. be able to draw an energy level diagram for an H atom (1 electron system).
8. be able to draw energy level diagrams for atoms with 2 or more electrons; understand in what
ways and why energy diagrams for all other atoms are different compared to the hydrogen atom.
9. be able to define the three quantum numbers: n, l, ml and their relation to atomic orbital energy and
shape.
10. understand and be able to describe what an orbital is.
11. be able to draw the shape of s and p orbitals.
12. understand and be able to sketch the probability density (ψ2) and radial probability (4πr2[ψ2])
functions as a function of distance from the atomic nucleus (r) for s orbitals.
13. learn that hydrogen is the only atom whose electron energy levels depend solely on the principle
quantum number, n.
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Electromagnetic Radiation (Light)
Remember that Rutherford discovered the nuclear nature of the atom by bombarding thin
sheets of metal foil with relatively massive α particles. From his experiment, he concluded
that the electrons in an atom are located in a region surrounding a very tiny, dense nucleus
that contains most of the atom’s mass. However, he could not describe the arrangement of
the electrons, the electronic structure. In order to determine the electronic structure of the
atom we need something less brutal than the relatively massive α particles Rutherford used.
We use electromagnetic radiation to do this.
Electromagnetic radiation consists of oscillating (wavelike) electric and magnetic fields that
can propagate over large distances through empty space. Since it is a wave,
electromagnetic radiation exhibits properties associated with waves. For instance, it can be
diffracted. Electromagnetic radiation originates from the movement of electrons in atoms,
molecules and ions.
Thus, through the study of the electromagnetic radiation absorbed or emitted by
atoms, molecules and ions we can learn something about the arrangement of
electrons in atoms, molecules and ions. Our focus will be on atoms here.
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Electromagnetic Radiation
Electromagnetic radiation, EM, can be described as alternating electric and magnetic
vectors. Each wave has an amplitude, energy, frequency, wavelength, and a propagation
speed.
The relationship between wavelength, λ, frequency, ν, and speed, c is:
λν = c = 2.998x10 8 m/s
λ is measured in m, while ν is measured in hertz (Hz):
1 Hz = 1 cycle/s = 1/s
What is the frequency of light with a wavelength of 435 nm?
Electromagnetic radiation
extends continuously across a
spectrum from the shortest
wavelength (highest frequency)
to the longest wavelength
(lowest frequency). What we
perceive as white light is
electromagnetic radiation
consisting of all the colors of
visible light.
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Electromagnetic Radiation
The reciprocal relationship
of frequency and
wavelength.
Differing amplitude
(brightness, or intensity) of
a wave
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Planck and Quantized Energy (The Birth of Quantum Theory)
When heated, solids emit electromagnetic radiation. Tungsten filament light bulbs and electric stoves
are examples of this phenomenon. The wavelength and intensity distribution of the radiation depends
upon the temperature. This distribution cannot be explained using classical physics where energy is
viewed as being “continuous”, all values “allowed”. A new proposal was made by Max Planck: energy
can be either released or absorbed by atoms only in discrete “chunks” of some minimum size. Thus, the
energy of the light emitted from a hot, glowing object is fixed to certain “allowed” quantities.
Quantized energy
works!
Max Planck:
Electric heating element
Classical theory
fails!
The “Ultraviolet Catastrophe” refers to the inability of classical
physics theory to correctly predict the spectrum of light emitted by
an ideal black-body, an object that only emits electromagnetic
radiation, does not reflect it. This is much more pronounced for
short wavelengths, as shown by the difference between the black
curve (the wrong curve predicted by classical physics) and the blue
curve (the correct curve predicted by Planck's Law).
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A conservative thinker who did not
“like” chemistry because he did not
like the idea of atoms and molecules.
His proposal that energy is quantized
“bothered” him, it was a radical idea
that revolutionized physics.
The Nobel Prize in Physics
in 1918 was awarded to
Max Planck "in recognition
of the services he rendered
to the advancement of
Physics by his discovery of
energy quanta".
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Planck’s Proposed Explanation
(Summary in “simple language”.)
In 1900, the German physicist Max Planck (1918 Noble Prize in Physics) was able to explain
the distribution of light emitted by a hot, glowing object by assuming that energy comes in
tiny packets; Planck gave the name quantum, meaning “fixed amount”, to the tiny packets
of energy that are emitted or absorbed as electromagnetic radiation. He proposed that the
energy of a single quantum is proportional to the frequency of the radiation emitted:
E = hν
h = 6.626 x 10–34 J•s
where the proportionality constant (h) is called “Planck’s constant”. Planck further proposed
that electromagnetic energy can be absorbed or emitted only in quantized form as discrete
“chunks” of energy corresponding to whole-number multiples of hν, such as hν, 2hν, 3hν,
and so on.
∆ E = (∆ n)hv
Where ∆E is equal to the energy of the light emitted (or absorbed)
n is in an integer (quantum number)
h is Plank’s constant = 6.626×10–34 J/s
v is the frequency of the light
WOW! Planck’s proposal means that energy is not continuous as classical theory
assumed, energy is quantized, meaning that it is restricted to certain quantities.
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Planck’s Equation
Quantized Energy and What We Now Understand
Planck’s equation indicates that energy of emitted light from a heated source is “quantized”.
We now understand that the quantization comes from the allowed frequencies of vibration
(jiggling) of the heated atoms, unknown to Planck at the time!
•
•
Each “allowed” vibrational frequency emits quantized light; light of a specific energy.
At any given temperature, a distribution of allowed vibrational frequencies (a distribution
of jiggling frequencies) is possible, hence a distribution of emitted light energy.
•
As temperature increases, the distribution shifts to higher energy vibrations, thus higher
frequencies, meaning lower wavelengths. (The atoms in cooler objects jiggle more
slowly than the atoms in hotter objects.)
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Einstein and the Photoelectric Effect
Albert Einstein used Planck’s ideas to explain the photoelectric effect.
The photoelectric effect occurs when light strikes
the surface of a metal causing electrons to be
ejected from the metal. Experiments have shown
that electrons are ejected only if the frequency of
light is high enough. If lower frequency light is
used, no electrons are ejected, regardless of the
light intensity.
The Nobel Prize in Physics
in 1921 was awarded to
Albert Einstein "for his
services to Theoretical
Physics, and especially for
his discovery of the law of
the photoelectric effect".
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Einstein’s Proposed Explanation
The photoelectric effect was studied extensively by Phillip Lenard. Lenard published his
observations, but to his frustration he was unable to explain them. In 1905, Einstein
published a very short, simple paper that explained the effect. Einstein reasoned that the
photoelectric effect is consistent with the idea that light can be thought of as being
composed of tiny particles or packets of energy. Each particle has a given energy, E = hν,
associated with it. One of these particles can ”bump” an electron from the surface of the
metal only if the particle has sufficient energy (a certain minimum frequency) required to
knock the electron off of the surface.
Einstein proposed that light has “particle-like” (momentum) properties. We now call these
particles photons. Each photon has a specific energy depending upon its frequency
(wavelength):
E = hν
In order for an electron to be ejected from the surface of a substance by a photon, the
photon must have a certain minimum amount of energy or greater.
Wording we now use: When a photon of sufficient energy is used, it transfers all of its
energy to an electron enabling the electron to overcome the surface potential or what we
call the “work function” of the metal.
Work function (Θ ) refers to the minimum energy needed to remove an electron from a
solid to a point immediately outside the solid surface. The work function is a characteristic
property for any solid surface of a substance with a conduction band. (In simpler words,
the work function is the minimum amount of energy needed in order for an electron in a
metallic atom to escape to the surface of the metal.)
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Wave-Particle Duality
Other experiments followed that showed light behaving as a particle is not limited to the
photoelectric effect: The Compton Experiment (published in 1923) showed that when electrons
are irradiated with x-rays, the x-rays lose energy (frequency is lowered) and the electrons gain
kinetic energy. Momentum is thus transferred from the x-rays to the electrons! The 1927 Nobel
Prize in physics was awarded for this experiment.
WOW, NEAT! .........and a bit strange.
Electromagnetic radiation is now said to have a wave-particle duality. It behaves
like a wave (can be diffracted for example) as well as a particle (the Photoelectric
Effect, the Compton Effect), but never the two at the same time.
The overall equation related energy, wavelength and frequency of light is:
Ephoton = hν = hc/λ
h = 6.626 x 10–34 J•s
c = 2.998 x 108 m/s
Example Problem:
Calculate the frequency in megahertz and wavelength in meters of electromagnetic
radiation with energy 6.527 x 10–26 J/photon.
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Using the Photoelectric Effect
Molybdenum metal must absorb radiation with a minimum frequency of 1.09 x 1015 s−1 to
emit an electron from its surface via the photoelectric effect.
(a)
What is the minimum energy needed to produce this effect? (This energy is called the “work
function, Θ, of the metal surface.)
(b)
What wavelength of radiation will provide a photon of this energy?
(c)
If molybdenum is irradiated with 120 nm radiation, what is the velocity of the emitted electrons?
Note: Ephoton = hν = Θ + Ek
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Example Problem
The light-sensitive substance in black-and-white photographic film is AgBr. Photons provide
the energy necessary to transfer an electron from Br- to Ag+ to produce Ag and Br and
thereby darken the film.
(a)
If a minimum energy of 2.00 × 105 J/mol is needed for this process, what is the minimum energy
needed by each photon?
(b)
Calculate the wavelength of the light necessary to provide photons of this energy.
(c)
Explain why this film can be handled in a darkroom under red light.
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Emission of Light from Atoms
Emission Spectrum: A figure showing the distribution of the wavelengths of electromagnetic
radiation emitted by an object.
White Light: visible (continuous, all
wavelengths “allowed”) spectrum
The light emitted by excited atoms IN THE GAS PHASE was found not to
be continuous in energy like like a rainbow. Atomic emission spectra
show a series of sharp lines at specific wavelengths. Each element has a
unique emission spectrum. This phenomenon cannot be explained
using classical physics and it puzzled scientists in the early 1900s.
The above figure illustrates a
“continuous emission spectrum”.
The electromagnetic radiation
emitted from a light bulb or the sun
are examples. Note that a “rainbow”
or all wavelengths (energies) of light
are observed, an observation
consistent with “classical physics”.
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Interpretation of the Line Spectra of Hydrogen
Johannes Rydberg was the first to propose a mathematical relationship that would describe
the wavelengths of light emitted by heated hydrogen gas.
⎛ 1
1
1⎞
= R ⎜ 2 − 2 ⎟ , where R=1.09677x10 7 m −1
λ
⎝ n1 n2 ⎠
•
n1 and n2 are positive integers with n2 > n1.
•
The visible emission lines of hydrogen shown above correspond to n1 = 2.
•
This relationship was empirically derived; that is it was derived based on observation
without regard to theory. Notice that only certain wavelengths of light are allowed as
determined by the values of n. This is another example of “quantization” of energy.
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Line Spectra and the Bohr Model of the HYDROGEN Atom
The Nobel Prize in Physics in 1922
was awarded to Niels Bohr "for his
services in the investigation of the
structure of atoms and of the
radiation emanating from them".
En = −
ℜhc
2.179x10 -18 J
=−
2
n
n2
Potential Energy →
Bohr proposed the following formula for the potential energy of the
electron in the nth energy level of a hydrogen atom:
electron ionization, n = ∞
En = 0 J
ground state, n = 1
En = -2.179x10-18 J
Where n is any integer ≥ 1. According to the Bohr model, a hydrogen
atom contains cetain allowed energy states for the electron. These are
QUANTIZED energy states (Bohr called these stationary states).
Each stationary state corresponds to a certain allowed circular “orbit”.
Only certain orbits are permitted! His proposal offered a reason why
electrons do not “fall” into the nucleus. They are “not allowed” to do
so.
All these energy levels are between 0 and -2.18x10-18 J. The energy
levels become “bunched” near 0 J as n increases.
When n = 1, the electron is said to be in the “ground” state of energy.
This is the lowest possible energy. For n >1, the electron is said to be in
an “excited” state. As n increases, the radius and energy of the “orbit”
increases.
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Bohr Model Explains the Line Spectrum of the Hydrogen Atom
Bohr postulated that an electron can move between allowed energy levels by absorbing or
emitting a photon of light with energy equal to the difference in energy between the two
states. The energy of the absorbed or emitted light would exactly match the difference in
energy levels between between which the electron moves.
Bohr postulated that absorption of a photon occurs when an
electron moves from a lower energy state to a higher energy
state. The electron gains energy in the process. Emission of a
photon occurs when an electron moves from a higher energy
state (higher n) to a lower energy state (lower n). The electron
loses energy in the process and this energy is converted into a
photon of light. His theory thus explained the observation of
atomic “line” spectra.
Using the Bohr Model for the hydrogen atom, the difference
between allowed energy levels can be calculated as follows:
⎛ 1
1⎞
∆ E = E f − Ei = −2.179x10 -18 J ⎜ 2 − 2 ⎟
⎝ n f ni ⎠
hc
E photon = ∆ E = hv =
λ
Note: Absolute values are used for ∆E when calculating the corresponding
frequency (ν) or wavelength (λ) of the light. Why?
The infrared, visible and
ultraviolet series of spectral lines
emitted by the H atom.
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Bohr Model of Visible Emission Spectrum of H Atom
(Analogy of a ball on a staircase.)
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Calculations Using the Bohr Model of the Hydrogen Atom
⎛ 1
1⎞
∆ E = E f − Ei = −2.179x10 -18 J ⎜ 2 − 2 ⎟
⎝ n f ni ⎠
1.
E photon = ∆ E = hv =
hc
λ
What energy of photon is emitted when an electron transitions from the n4 to n1 level?
What is the wavelength of this photon?
In what area of the EM spectrum is this photon?
2.
What is the shortest wavelength of light in nm that can be emitted from a hydrogen
atom with its electron in the n= 5 state?
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Bohr, Planck and Einstein
So light posses a “wave-particle duality”:
Wave-like properties of diffraction, refraction etc.
Particle-like properties of fixed energy and momentum.
Through Planck’s and Einstein’s work we now see energy as having properties that were
thought to be reserved for matter only: fixed quanta (Planck) and discrete packets (Einstein).
While Bohr’s model worked well for hydrogen and other one-electron systems such as He+,
it could not explain the emission spectra of any of the other elements. Even so, Bohr’s work
marks an important step in the evolution of Quantum Theory; he showed that quantum ideas
are applicable to the atom. However, the problem needed further thought!
Einstein wrote in his notes that when he first read Bohr’s theory of the atom he laughed.
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The “Wave” Properties of Matter
Around 1925 Louis de Broglie, who had a degree in history and literature before studying
physics, pondered the question:
“If light can be thought of as having both a wave and a particle (photon) behavior, can matter, such as
electrons, also possess wave properties?” (This was presented as part of his pH.D dissertation.)
de Broglie proposed that any moving particle has a corresponding “wavelength”. Using
Einstein’s equation E = mc2 for the relationship between mass and energy, and E = hν for the
energy of a photon, he derived the de Broglie equation from which the wavelength (λ) of any
particle with a mass (m in kg) and velocity (u in m/s) can be calculated:
λ=
h
mu
From physics the term mu is the momentum of the particle. Consider an electron traveling
at 1/100 the speed of light:
λ=
(
6.626x10 −34 J • s
≈ 2.4 x10 −10 m = 0.24 nm = 2.4 Å = 240 pm
⎛
−31
6 m⎞
9.109x10 kg ⎜ 2.998x10 ⎟
⎝
s⎠
)
This is roughly on the order of atomic distances, thus predicting that electrons have “wave” properties that are
observable at very short, atomic distances. This is consistent with the Bohr model and, in fact, seems to make the
Bohr model more reasonable. Realistically, the idea of the electron existing in only certain allowed orbits/energies is
ridiculous if the electron is a particle. However, if the electron is a wave, then each “orbit” could correspond to a
certain, even number of wavelengths, so that the wave will not cancel itself out (destructively interfere). Only in the
allowed “orbits” will waves of certain energies be able to exist. De Broglie received the Noble prize in physics for his
work.
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Wave Properties of Electrons
http://www.youtube.com/watch?v=DfPeprQ7oGc
•
•
In 1927, de Broglie’s proposal was verified by C. J.
Davison and L. H. Germer in the U.S. and by George
Thomson in the U.K. who showed that electrons can
be diffracted like waves by a thin sheet of metal foil.
So, electrons do have wave properties under some
circumstances! Wave-particle duality applies to
particles!
Ernest Ruska, a German physicist, constructed the
first electron microscope in 1933 and shared the
Nobel Prize in physics in 1986 for his work. The
electron microscope works on the basis that
electrons can be diffracted. Biological molecules are
now routinely studied using modern electron
microscopes that have a resolving power of about 1
nm.
x-ray diffraction of aluminum foil
electron diffraction of aluminum foil
Example Problem:
Calculate the wavelength of a 145 g fast ball moving at 95 miles per hour.
“Think about it”-Why are we unable to observe wave-like motions for macroscopic objects
such as baseballs and people?
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“Wave” Properties lead to Uncertainty in Position and Momentum
The Heisenberg Uncertainty Principle, proposed by the German physicist Werner
Heisenberg, states that it is impossible to know simultaneously the position and
momentum (energy) of an electron in an atom since the electron has “wave” properties. If
we know one precisely, say the momentum, then the other, the position, becomes uncertain.
Heisenberg’s Uncertainty Principle in equation form is:
∆ x ⋅ ∆(mu) ≥
h
4π
where
∆x is the uncertainty in the position and
∆(mu) is the uncertainty in the momentum (energy)
For example, in a hydrogen atom if we want to measure the momentum
(energy) of the electron to ±1%, then the uncertainty in position becomes
≈ 1 nm about the size of the atom!
On the other hand, if we want to pin-down the momentum of a 95 mph
baseball to ±1%, the the uncertainty in position is only 1x10-35 m. A
distance too small to even consider.
The Nobel Prize in
Physics in 1932 was
awarded to Werner
Heisenberg "for the
creation of quantum
mechanics".
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Heisenberg’s Uncertainty Principle is a limitation built into nature, not due
to limitations in technology or equipment. At the subatomic level, nature
prevents us from knowing everything. There is a built in uncertainty in the
world. This bothered Einstein greatly, and he thought that there must be
some “rules” that we just did not know yet. Bohr argued that “some things
are fundamentally unknowable.” They debated for years about this!
Einstein: “I can’t believe that God would play dice with the universe.”
Through Einstein’s, Planck’s, de Broglie’s and Heisenberg’s work it became
apparent that a new theory was needed to describe the energies and
positions of small particles. Enter Erwin Schrödinger.
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The Birth of Quantum Mechanics-Matter Waves
Quantum Mechanics: Examines the wave motion and
energies of matter on the atomic scale. The mathematical
equations involved are difficult to solve; we will be
concerned only with the results. For the hydrogen atom,
the electron is described as a standing wave (like the
waves on a plucked guitar string). Like the waves on
guitar strings, only certain “wavelengths” are allowed.
These wavelengths are embedded in what are called
“wave-functions”. The wave-functions are “under the
influence” of the local electrostatic potential between the
positive nucleus and the negative electrons.
The Nobel Prize in Physics
in 1933 was awarded
jointly to Erwin Schrödinger
and Paul Adrien Maurice
Dirac "for the discovery of
new productive forms of
atomic theory"
In 1926 Schrödinger proposed his wave equation
describing the energy for electrons in a hydrogen atom
based on the “wave” properties of the electron, thinking
of the electron as a probability wave:
Hψ = Eψ
Where H is the Hamiltonian operator
(defines the electron and proton interactions),
ψ is an allowed wave-function for the electron, this
describes the behavior of the electron in space.
E is the electron energy.
Bottom line:
Each allowed electron wave-function, ψ,
corresponds to an allowed energy, E, for the
electron.
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Wave motion in a restricted (quantized) system.
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Quantum Numbers
The results of the math give the following:
1.
Each wave-function, ψ , returns a single allowed energy value, E, for the electron
(quantized energy).
2.
Each wave-function, ψ , can be labeled by using three (3) integers (quantum numbers).
ψ2 gives the probability of an electron, with a certain energy, to be at a given location in
the atom.
The three quantum numbers are symbolized, n, l and ml. The quantum numbers are not
randomly chosen, but are required components for the solutions to the wave equation.
They are a property of Schrödinger’s wave equation, not some derived evil scheme made
up by physicists! The three quantum numbers come in a sequence, have names, and we
give them a “physical” interpretation:
3.
n, the Principle Quantum Number = 1, 2, 3, …∞
Designates the primary POTENTIAL energy (shell) for a electron in the nth level. This is analogous
to Bohr’s value of n. As n increases, the potential energy of the electron also increases (becomes
less negative).
l, the Angular Momentum Quantum Number = 0, 1, 2, … n-1
Designates the subshell type (orbital shape) for the electron. Each different l value corresponds to
a different subshell (different orbital shape). The number of allowed subshells in a given shell is
equal to n.
ml, the Magnetic Quantum Number = 0, ±1, ±2, … ±l
Designates the orientation of the orbital in space with respect to the nucleus and the other
orbitals. Each value of ml corresponds to a different orbital.
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Allowed Quantum Numbers and Orbitals
The subshells as given by quantum number l are replaced by single letter designations for
simplicity:
l = 0 designates an s subshell
l = 1 designates a p subshell
l = 2 designates a d subshell
l = 3 designates an f subshell
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Shapes of Orbitals — Quantum Number l
Each value of l describes a different orbital shape. The value of l in the mathematical solution
to Schrödinger’s wave equation comes from a unique wave function ψ.
These shapes are really probability clouds (called orbitals) for the electrons. An orbital is the
region in space where an electron can be found with 90% probability. The other 10% of the
time the electron will be outside this region.
Why to we have to talk in probabilities? Remember Heisenberg’s Uncertainty Principle: you
cannot simultaneously determine both the energy and position of an electron around a
nucleus.
If the energy is known precisely then the electron position has high uncertainty.
If the electron position is known precisely then the energy has high uncertainty.
s orbital shape
The 2p orbital shapes
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The 3d Orbitals
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[ Ψ(r)]
2
Radial Electron Density Functions — s Orbitals
= Probability Density Function
(Probability of finding an electron with a specific potential energy at a point with
distance r from the nucleus.)
90% probability contours
P(r) = 4π r 2 [ Ψ(r) ] = Radial Probability Distribution
2
(Total Probability of finding an electron on
the surface of a sphere at distance r from
the nucleus.)
1s
2s
3s
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Energy Levels of the H-atom
(A special case.)
Hydrogen is a rather simple system, one electron and one
proton. Due to the simple nature of hydrogen, hydrogen is
the ONLY atom whose energy state depends solely on the
principle quantum number, n. Details about this will be
covered in the next chapter.
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Problems
Consider the probability density and radial probability distribution functions shown on slide
30 of the notes.
(a) What is the difference between the probability density as a function of r and the radial
probability distribution as a function of r?
(b) Based on the figures shown, make sketches of what you think the probability density and
the radial probability distribution would look like for the 4s orbital of the hydrogen atom as a
function of r.
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Problems
In an experiment to study the photoelectric effect, a scientist measures the kinetic energy of ejected
electrons as a function of the frequency of radiation hitting a metal surface. She obtains the following
plot:
The point labeled “v0” corresponds to light with a wavelength of 680 nm.
(a) What is the value of v0 in s-1?
(b) What is the value of the work function of the metal in units of kJ/mol of ejected electrons?
(c) What happens when the metal is irradiated with light of frequency less than v0?
(d) Note that when the frequency of the light is greater than v0, the plot shows a straight line with a
nonzero slope. Why is this the case?
(e) Can you determine the slope of the line segment discussed in part (d)? Explain.
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Amusing Questions
If human height were quantized in one-foot increments, what would happen to the height of
a child as s/he grows up?
In the television series Star Trek, the transporter beam is a device used to “beam down”
people from the Starship Enterprise to another location, such as the surface of a planet. The
writers of the show put in a “Heisenberg compensator” into the transporter beam
mechanism. Explain why such a compensator (that is entirely fictional) would be necessary
to deal with Heisenberg’s uncertainty principle.
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