The Interaction of Light and Matter
Screen
Metal
plate
Time
Wire
Learning Objectives

Problems with Bohr’s Semiclassical Model of the Atom

Wave-Particle Duality:
Everything exhibits its wave properties in its propagation, and manifests its
particle nature in its interactions
de Broglie’s wavelength for electrons in an atom

Wave-like Description of the Physical World
Probability waves in quantum mechanics
Heisenberg’s uncertainty principle

Some applications of Heisenberg’s Uncertainty Principle in Science/Astrophysics:
Natural widths of spectral lines
Quantum mechanical tunneling and nuclear fusion in stars
Measurement precision

Schrödinger’s Wave Equation
Solutions describe possible quantum states of a physical system
Quantum mechanical atom
Learning Objectives

Pauli’s Exclusion Principle
No two electrons (particles) can share the same quantum state

Relativistic Wave Equation
Solutions to relativistic wave equation for an atom

Complex Spectra of Atoms
Multielectron atoms with different ionization states
Permitted and Non-Permitted Transitions
Learning Objectives

Problems with Bohr’s Semiclassical Model of the Atom

Wave-Particle Duality:
Everything exhibits its wave properties in its propagation, and manifests its
particle nature in its interactions
de Broglie’s wavelength for electrons in an atom

Wave-like Description of the Physical World
Probability waves in quantum mechanics
Heisenberg’s uncertainty principle

Some applications of Heisenberg’s Uncertainty Principle in Science/Astrophysics:
Natural widths of spectral lines
Quantum mechanical tunneling and nuclear fusion in stars
Measurement precision

Schrödinger’s Wave Equation
Solutions describe possible quantum states of a physical system
Quantum mechanical atom
Bohr’s Semiclassical Atom

Recall that Bohr’s model of the atom combined both classical physics and
quantum mechanics, and has as its two central ingredients:
- electrons in circular orbits around the the nucleus (classical physics)
- the orbital angular momenta of electrons are quantized (quantum
mechanics). Bohr also assumed that electrons only absorb or emit
electromagnetic waves when they make make a transition from one permitted
orbit to another.

What are the two problems in Bohr’s model of the atom that cannot be explained
using classical physics?
why is the angular momentum
Bohr’s Semiclassical Atom

Recall that Bohr’s model of the atom combined both classical physics and
quantum mechanics, and has as its two central ingredients:
- electrons in circular orbits around the the nucleus (classical physics)
- the orbital angular momenta of electrons are quantized (quantum
mechanics). Bohr also assumed that electrons only absorb or emit
electromagnetic waves when they make make a transition from one permitted
orbit to another.

What are the two problems in Bohr’s model of the atom that cannot be explained
using classical physics?
- why is the angular momentum of the electron
quantized?
- why does an electron in a permitted orbit not
emit electromagnetic waves as demanded by
Maxwell’s equations?
Bohr’s Semiclassical Atom

Recall that Bohr’s model of the atom combined both classical physics and
quantum mechanics, and has as its two central ingredients:
- electrons in circular orbits around the the nucleus (classical physics)
- the orbital angular momenta of electrons are quantized (quantum
mechanics). Bohr also assumed that electrons only absorb or emit
electromagnetic waves when they make make a transition from one permitted
orbit to another.

What are the two problems in Bohr’s model of the atom that cannot be explained
using classical physics?
- why is the angular momentum of the electron
quantized?
- why does an electron in a permitted orbit not
emit electromagnetic waves as demanded by
Maxwell’s equations?
Learning Objectives

Problems with Bohr’s Semiclassical Model of the Atom

Wave-Particle Duality:
Everything exhibits its wave properties in its propagation, and manifests its
particle nature in its interactions
de Broglie’s wavelength for electrons in an atom

Wave-like Description of the Physical World
Probability waves in quantum mechanics
Heisenberg’s uncertainty principle

Some applications of Heisenberg’s Uncertainty Principle in Science/Astrophysics:
Natural widths of spectral lines
Quantum mechanical tunneling and nuclear fusion in stars
Measurement precision

Schrödinger’s Wave Equation
Solutions describe possible quantum states of a physical system
Quantum mechanical atom
Wave-Particle Duality


Recall the wave-particle duality of light:
Light – in the form of electromagnetic waves – shows its wave-like properties as
it propagates through space.
Light – in the form of photons – shows its particle-like properties as it interacts
with matter.
If electromagnetic waves –
light – can exhibit particlelike properties, can particles
(e.g., protons, electrons,
neutrons, atoms, molecules,
bacteria, plants, animals,
humans, planets, stars,
galaxies) exhibit wave-like
properties?
Wave-Particle Duality

This revolutionary idea for the behavior of matter was first
proposed in 1924 by the French physicist Louis de Broglie in
his PhD thesis.

Recall that, in 1905, Einstein used Planck’s idea that the energy
of an electromagnetic wave is quantized in the manner
whereby a quantum of energy is carried by a photon, to explain
the photoelectric effect.

Louis de Broglie,
1892-1987
Recall that from the Theory of Special Relativity, the energy and momentum of a
photon are related by
as had been verified in 1922 by Compton through the Compton effect.
Wave-Particle Duality

Purely from symmetry arguments, de Broglie proposed that all matter (from
elementary particles to people, planets, stars, and entire galaxies) also exhibit
wave-like properties such that their frequency and wavelength are given by
The
wavelength given by Eq. (5.17) is known as the de Broglie wavelength.

In this view, the wave-particle duality applies to everything in the physical world:
- everything exhibits its wave properties in its propagation
- everything manifests its particle nature in its interactions
Wave-Particle Duality

If everything – including us – propagate as waves, should we worry about being
diffracted when walking through the door?
Wave-Particle Duality

If everything – including us – propagate as waves, should we worry about being
diffracted when walking through the door?
Wave-Particle Duality

If everything – including us – propagate as waves, should we worry about being
diffracted when walking through the door?
Wave-Particle Duality

So, our wave-like properties have very short wavelengths (depending on our mass
and speed). Why then do we not have to worry about diffracting?

Hint: Conditions for destructive interference
for m = 1, 2, 3, …
Notice also that the bright fringes in the
interference pattern become dimmer for larger θ.
L»D
Wave-Particle Duality

So, our wave-like properties have very short wavelengths (depending on our mass
and speed). Why then do we not have to worry about diffracting? The maxima in
the interference pattern is concentrated at θ = 0o; i.e., we walk through and appear
behind the door in the direction of motion at the door.
L»D
Wave-Particle Duality


In 1927, Clinton J. Davisson and Lester H. Germer directed a beam of electrons
on a highly polished single crystal of nickel. The lines of atoms on the surface
form a sort of diffraction grating for the electron waves with a spacing equal to the
lattice spacing of the nickel crystal of 0.215 nm, about the size of an atom.
Clinton J Davisson (1881-1958; left)
and
Lester H. Germer(1896-1971; right)
The electron beam they used had an energy of 54 eV,
corresponding to a de Broglie wavelength of 0.167 nm.
The first-order (m = 1) maximum should therefore occur
at  = sin-1 (/d) = 51o, in agreement with their measurements.
Wave-Particle Duality


In 1927, Clinton J. Davisson and Lester H. Germer directed a beam of electrons
on a highly polished single crystal of nickel. The lines of atoms on the surface
form a sort of diffraction grating for the electron waves with a spacing equal to the
lattice spacing of the nickel crystal of 0.215 nm, about the size of an atom.
Clinton J Davisson (1881-1958; left)
and
Lester H. Germer(1896-1971; right)
The electron beam they used had an energy of 54 eV,
corresponding to a de Broglie wavelength of 0.167 nm.
The first-order (m = 1) maximum should therefore occur
at  = sin-1 (/d) = 51o, in agreement with their measurements.
Wave-Particle Duality

Modern experiments such as that performed in 1989 by A. Tonomura and his
colleagues at the Hitachi Advanced Laboratory and Gakushuin University in
Tokyo (based on Clauss Jönsson’s experiment in 1961) is equivalent to the
double-slit experiment for light. About 10 electrons are emitted a second, each
accelerated to ~0.4 c.
Metal
plate
Wire
Wave-Particle Duality

Modern experiments such as that performed in 1989 by A. Tonomura and his
colleagues at the Hitachi Advanced Laboratory and Gakushuin University in
Tokyo (based on Clauss Jönsson’s experiment in 1961) is equivalent to the
double-slit experiment for light. About 10 electrons are emitted a second, each
accelerated to ~0.4 c.

How do electrons
interact with the screen?
Metal
plate
Wire
Screen
Wave-Particle Duality

Modern experiments such as that performed in 1989 by A. Tonomura and his
colleagues at the Hitachi Advanced Laboratory and Gakushuin University in
Tokyo (based on Clauss Jönsson’s experiment in 1961) is equivalent to the
double-slit experiment for light. About 10 electrons are emitted a second, each
accelerated to ~0.4 c.

How do electrons
interact with the screen?
particles, which is why
points appear on
(where electrons
interact with
screen).
Metal
plate
Wire
ScreenAs
individual
the screen
with strike –
– the
Wave-Particle Duality

Modern experiments such as that performed in 1989 by A. Tonomura and his
colleagues at the Hitachi Advanced Laboratory and Gakushuin University in
Tokyo (based on Clauss Jönsson’s experiment in 1961) is equivalent to the
double-slit experiment for light. About 10 electrons are emitted a second, each
accelerated to ~0.4 c.

What image would you see if
electrons behave purely like
Metal
plate
Wire
Screen particles?
Wave-Particle Duality

Modern experiments such as that performed in 1989 by A. Tonomura and his
colleagues at the Hitachi Advanced Laboratory and Gakushuin University in
Tokyo (based on Clauss Jönsson’s experiment in 1961) is equivalent to the
double-slit experiment for light. About 10 electrons are emitted a second, each
accelerated to ~0.4 c.

What image would you see if
electrons behave purely like
Metalstripes.
Two
plate
Wire
Screen particles?
Wave-Particle Duality

Modern experiments such as that performed in 1989 by A. Tonomura and his
colleagues at the Hitachi Advanced Laboratory and Gakushuin University in
Tokyo (based on Clauss Jönsson’s experiment in 1961) is equivalent to the
double-slit experiment for light. About 10 electrons are emitted a second, each
accelerated to ~0.4 c.

What image would you see if
electrons propagate with wave-like
properties?
Metal
plate
Wire
Screen
Wave-Particle Duality

Modern experiments such as that performed in 1989 by A. Tonomura and his
colleagues at the Hitachi Advanced Laboratory and Gakushuin University in
Tokyo (based on Clauss Jönsson’s experiment in 1961) is equivalent to the
double-slit experiment for light. About 10 electrons are emitted a second, each
accelerated to ~0.4 c.

What image would you see if
electrons propagate with wave-like
properties?
Interference pattern.

Each electron passes
through both slits
and interferes
with itself, thus revealing
it’s wave-like properties.
(Obviously, a particle can
only go through one slit.)
Metal
plate
Wire
Screen
Wave-Particle Duality
Wave-Particle Duality

Similar experiments have been performed on other elementary particles as well as
atoms.
Wave-Particle Duality

Instead of double slits, experiments using just one slit also have been performed.
Wave-Particle Duality

Instead of double slits, experiments using just one slit also have been performed.
Wave-Particle Duality

Interference pattern reported by Zeilinger and his collaborators for an experiment
where they directed a beam of neutrons through a single slit.
Wave-Particle Duality


Recall that Bohr’s model of the atom combined both classical physics and
quantum mechanics, and has as its two central ingredients:
- electrons in circular orbits around the the nucleus (classical physics)
- the orbital angular momenta of electrons are quantized (quantum
mechanics). Bohr also assumed that electrons only absorb or emit
electromagnetic waves when they make make a transition from one permitted
orbit to another.
If electrons behave as waves, can you explain
why the angular momentum of electrons are
quantized (can only have certain permitted
orbits)?
Wave-Particle Duality

de Broglie reasoned that the quantization of orbital angular momentum in Bohr’s
model of the atom is simply a manifestation of the wave-like nature of the
electron. The circumference of an electron’s orbit must be equal to an integral
number of wavelengths (i.e., integral number of de Broglie wavelengths) for the
electron to undergo constructive interference. Otherwise, the electron will find
itself out of phase and suffer destructive interference.

Based on this consideration, one can show that
electron can only have angular momenta given
.
Assignment
question

In this description, we no longer envisage
electrons as particles orbiting at different
permitted radii from the nucleus in an atom,
but as standing waves surrounding the
nucleus in an atom.
by
Wave-Particle Duality
Wave-Particle Duality

Now, using the principles for the wave-particle duality of the physical world
- everything exhibits its wave properties in its propagation
- everything manifests its particle nature in its interactions
Can you now explain why electrons can orbit an atom without emitting
electromagnetic radiation?
Wave-Particle Duality

Now, using the principles for the wave-particle duality of the physical world
- everything exhibits its wave properties in its propagation
- everything manifests its particle nature in its interactions
Can you now explain why electrons can orbit an atom without emitting
electromagnetic radiation? Electrons are not orbiting as particles interacting
through Coulomb forces with their nuclei, but propagate as waves.
Wave-Particle Duality

Now, using the principles for the wave-particle duality of the physical world
- everything exhibits its wave properties in its propagation
- everything manifests its particle nature in its interactions
Can you now explain why electrons in atoms can absorb electromagnetic radiation
but only at certain wavelengths?
Wave-Particle Duality

Now, using the principles for the wave-particle duality of the physical world
- everything exhibits its wave properties in its propagation
- everything manifests its particle nature in its interactions
When absorbing electromagnetic radiation, an electron, a photon, and the atomic
nucleus interact as particles. Electrons can only absorb photons at wavelengths
that change their de Broglie wavelengths by certain fixed amounts.
Wave-Particle Duality

Now, using the principles for the wave-particle duality of the physical world
- everything exhibits its wave properties in its propagation
- everything manifests its particle nature in its interactions
Can you now explain why electrons in atoms can emit electromagnetic radiation
but only at certain wavelengths?
Wave-Particle Duality

Now, using the principles for the wave-particle duality of the physical world
- everything exhibits its wave properties in its propagation
- everything manifests its particle nature in its interactions
When emitting electromagnetic radiation, an electron, a photon, and the atomic
nucleus interact as particles. Electrons can only emit photons at wavelengths that
change their de Broglie wavelengths by certain fixed amounts.
Learning Objectives

Problems with Bohr’s Semiclassical Model of the Atom

Wave-Particle Duality:
Everything exhibits its wave properties in its propagation, and manifests its
particle nature in its interactions
de Broglie’s wavelength for electrons in an atom

Wave-like Description of the Physical World
Probability waves in quantum mechanics
Heisenberg’s uncertainty principle

Some applications of Heisenberg’s Uncertainty Principle in Science/Astrophysics:
Natural widths of spectral lines
Quantum mechanical tunneling and nuclear fusion in stars
Measurement precision

Schrödinger’s Wave Equation
Solutions describe possible quantum states of a physical system
Quantum mechanical atom
Probability Waves

Quantum mechanics describe particles in terms of probability waves.

Consider a particle that comprises the following probability wave, Ψ, a sine wave
with a precise wavelength  propagating along the x-direction
Probability wave Ψ :
x
(x,t) = 0 ei(kx-t)
where k = 2 / 
=2ν
The momentum, p = h / , of a particle described by such a wave is known
precisely (as the wavelength is known precisely).

The probability of finding the particle at a given location x is given by
P(x) =  *
[0 ei(kx-t)] [0 e-i(kx-t)]
= |02|
which is a constant independent of x or t. Thus, the particle can be found with
equal probability at any point along the x-direction: its position is perfectly
uncertain; i.e., a sinusoidal wave has no beginning or end.
=
Probability Waves

Consider now a particle that has a probability wave, Ψ, that is equal to the
addition of several sine waves with different wavelengths
Probability wave Ψ :
x
Probability Waves

Consider now a particle that has a probability wave, Ψ, that is equal to the
addition of several sine waves with different wavelengths
Probability wave Ψ :
x
The position of such a particle can be determined with a greater certainty because
P(x) =  * is large only for a narrow range of locations.

On the other hand, because Ψ is now a combination of waves of various
wavelengths, the particle’s momentum, p = h / , is less certain.

This is nature’s intrinsic tradeoff: the uncertainty in a particle’s position, Δx, and
the uncertainty in its momentum, Δp, is inversely related. As one decreases, the
other must increase. This fundamental inability of a particle to simultaneously
have a well-defined position and a well-defined momentum is a direct result of the
wave-particle duality of nature.
Probability Waves and the Two-Slit Experiment

As an illustration of nature’s intrinsic tradeoff
between the uncertainty in a particle’s position, Δx,
and the uncertainty in its momentum, Δp, consider
the following.

How do you prevent the electron beam from
producing an interference pattern?
Probability Waves and the Two-Slit Experiment

As an illustration of nature’s intrinsic tradeoff
between the uncertainty in a particle’s position, Δx,
and the uncertainty in its momentum, Δp, consider
the following.

How do you prevent the electron beam from
producing an interference pattern?
By precisely controlling the position of the electrons
so that they only enter one slit. If we can be sure
that an electron only passes through one slit, we no
longer see an (double-slit) interference pattern.

How do you explain this if electrons behave like
waves? Hint: how can you precisely control the
the position of an electron, and what are the consequences of doing this?
Probability Waves and the Two-Slit Experiment

As an illustration of nature’s intrinsic tradeoff
between the uncertainty in a particle’s position, Δx,
and the uncertainty in its momentum, Δp, consider
the following.

How do you prevent the electron beam from
producing an interference pattern?
By precisely controlling the position of the electrons
so that they only enter one slit. If we can be sure
that an electron only passes through one slit, we no
longer see an (double-slit) interference pattern.

How can you precisely control the
the position of an electron, and what are the consequences of doing this?
By using electrical or magnetic deflecting plates. But these deflecting plates
change the momentum of the electrons (by accelerating electrons to the desired
direction), and so you lost accurate control of the momentum and hence
wavelength of these electrons. Electron waves at difference wavelengths produce
interference patterns with different locations for their maxima and minima, hence
an (double-slit) interference pattern is lost.
Probability Waves and the Two-Slit Experiment

As an illustration of nature’s intrinsic tradeoff
between the uncertainty in a particle’s position, Δx,
and the uncertainty in its momentum, Δp, consider
the following.

Conversely, if you try to precisely control the
momentum of an electron, you will lose good
control of its position. As a consequence, an
electron can go through both slits (obviously, an
electron can go through both slits only if it behaves
like a wave), and interfere with itself to produce an
(double-slit) interference pattern. (I leave it up to
you to imagine how you could try to precisely
control the momentum of an electron. It is not
trivial!)
Probability Waves and the Two-Slit Experiment

In practice, it is impossible to control the momentum of the electrons to an
arbitrary accuracy, or focus the electrons to an arbitrarily narrow beam.

In the double-slit experiment for electrons, why is the minima in the interference
pattern not perfectly dark?
Metal
plate
Wire
Screen
Probability Waves and the Two-Slit Experiment

In the double-slit experiment for neutrons, why is the minima in the interference
pattern not perfectly dark?
Probability Waves and the Single-Slit Experiment

In the single-slit experiment for neutrons, why is the minima in the interference
pattern not perfectly dark?
Probability Waves and the Two-Slit Experiment

In the double-slit experiment for electrons, why is the minima in the interference
pattern not perfectly dark?
Because the momentum, and hence wavelength, of the electrons cannot be
perfectly controlled.
Metal
plate
Wire
Screen
Heisenberg’s Uncertainty Principle

In 1927, the German physicist Werner Heisenberg presented the
theoretical framework for the inherent “fuzziness” of nature,
showing that the uncertainty in a particle’s position, Δx, and the
uncertainty in its momentum, Δp, is related by

He also showed that the uncertainty of energy measurement, ΔE,
and the time interval over which this measurement is taken, Δt, is
related by
Recall that
=
= 1.054571596(82) x 10-34 J s.
Werner Heisenberg,
1901-1976
Heisenberg’s Uncertainty Principle

To explain the relationship between the inherent uncertainty in a particle’s
position, Δx, and its momentum, Δp, Heisenberg appealed to a thought experiment
known as Heisenberg’s microscope.

Suppose that an electron, behaving like a classical particle,
moves in the x direction along a line below a microscope.
Suppose that a photon, also moving in the x direction, strikes
the electron and enters the microscope. The photon transfers
the least momentum to the electron if it enters along path 1,
and the most momentum if it enters along path 2.

We do not know along which path the photon entered the
microscope, and hence how much the photon changed the
momentum of the electron. We only know that the momentum
of the electron has changed by an amount that spans the range
2
in the x-direction.
1
Heisenberg’s Uncertainty Principle

The angular resolution of the microscope is limited by the
observing wavelength and its aperture.
(sin)

It can be shown that the microscope can only determine the
position of the electron to within a range
Note that sin ε is related to the aperture of the microscope.

Combining the relations for Δx and Δp, we have
Learning Objectives

Problems with Bohr’s Semiclassical Model of the Atom

Wave-Particle Duality:
Everything exhibits its wave properties in its propagation, and manifests its
particle nature in its interactions
de Broglie’s wavelength for electrons in an atom

Wave-like Description of the Physical World
Probability waves in quantum mechanics
Heisenberg’s uncertainty principle

Some applications of Heisenberg’s Uncertainty Principle in Science/Astrophysics:
Natural widths of spectral lines
Quantum mechanical tunneling and nuclear fusion in stars
Measurement precision

Schrödinger’s Wave Equation
Solutions describe possible quantum states of a physical system
Quantum mechanical atom
Electronic Orbitals in Atoms

Note that spectral lines are not arbitrarily narrow, but have a certain width. What
factors contribute to the widths of spectral lines?
Electronic Orbitals in Atoms

Note that spectral lines are not arbitrarily narrow, but have a certain width. What
factors contribute to the widths of spectral lines? Spectral resolution of measuring
instrument, random motion of atoms, disturbance of electronic orbitals by
neighboring atoms, and inherent fuzziness of electronic orbitals.
Electronic Orbitals in Atoms

Heisenberg’s uncertainty principle implies that electrons cannot have precisely
defined angular momenta and orbital radii as prescribed in Bohr’s model of the
atom. Rather, the electron orbits must be imagined as fuzzy clouds of probability,
with the clouds being mode “dense” in regions where the electron is more likely
to be found.
Bohr’s model
for the hydrogen atom
Modified Bohr’s model
for the hydrogen atom
Electronic Orbitals in Atoms

How do we explain the concept of “fuzzy orbits” in terms of the de Broglie
wavelengths for electrons in an atom?
Modified Bohr’s model
for the hydrogen atom
Electronic Orbitals in Atoms

How do we explain the concept of “fuzzy orbits” in terms of the de Broglie
wavelengths for electrons in an atom? Wavelengths closer to the de Broglie
wavelengths interfere more constructively, and so there is a higher probability of
finding electrons closer to the de Broglie wavelengths.
Modified Bohr’s model
for the hydrogen atom
Electronic Orbitals in Atoms

How do we explain the concept of “fuzzy orbits” in terms of the de Broglie
wavelengths for electrons in an atom? Conversely, wavelengths further from the
de Broglie wavelengths interfere more destructively, and so there is a lower
probability of finding electrons further from the de Broglie wavelengths.
Modified Bohr’s model
for the hydrogen atom
Electronic Orbitals in Atoms

As a consequence, spectral lines cannot be infinitely narrow but must have a
certain width (natural linewidth), as is indeed observed.

Natural line profile is a Lorentzian profile.
Modified Bohr’s model
for the hydrogen atom
Quantum Mechanical Tunneling

Because of the inherent uncertainty between a particle’s momentum and position,
a particle can penetrate a barrier even though the particle’s energy is lower than
the barrier potential.

If the energy of the particle is known precisely (and even if smaller than the
barrier potential), the position of the particle is not known precisely and so it can
be on either side of the barrier (provided the barrier width is sufficiently small).

So, a (small) fraction of particles on one side of a barrier can tunnel through to the
other side, an effect called quantum mechanical tunneling.
Probability wave Ψ :
x
Quantum Mechanical Tunneling

A corresponding effect is seen in a classical wave such as water or light waves.
When a wave enters a medium with different wave motion properties, its
amplitude can decay with distance in this medium: the wave becomes evanescent
(fades away).

If the barrier width is sufficiently small, the amplitude of the wave may not decay
away completely before reaching the other side of the barrier, where the wave can
once again propagate freely.
Quantum Mechanical Tunneling

A corresponding effect is seen in a classical wave such as water or light waves.

For example, when light waves undergoes a total internal reflection at the
boundary of a medium, the light waves are observed to decay exponentially away
from this boundary (evanescent waves). The existence of light waves beyond this
boundary is required
because electric and
magnetic fields cannot
be discontinuous at a
boundary.

In this example, the
light ray undergoes total
internal reflection (i.e.,
no emerging refracted
ray) if it strikes the back
surface of the prism at a
sufficiently large angle
to the surface normal.
Quantum Mechanical Tunneling
Incident waves
Refracted waves
Boundary between two media
Incident waves
Evanescent waves
Boundary between two media
Quantum Mechanical Tunneling

The picture below shows a light ray undergoing total internal reflection in a
triangular prism. By placing a lenticular prism close to the triangular prism, the
evanescent wave from the triangular prism can be made to propagate in the
lenticular prism (a phenomenon known as frustrated total internal reflection).
Quantum Mechanical Tunneling and Stellar Nuclear Fusion

Quantum mechanical tunneling plays an
essential role in stellar nuclear fusion.

The first step in converting hydrogen to
helium is the fusion of two protons to
produce deuterium (proton + neutron nucleus)
along with a positron and a γ-ray photon.

For two protons to overcome their Coulomb
repulsion and approach each other
sufficiently close for their strong nuclear
force to bind them together, their required
kinetic energy corresponds to a gas
temperature of ~1010 K.
Quantum Mechanical Tunneling and Stellar Nuclear Fusion

For comparison, the central temperature of
the Sun is only 1.57 × 107 K. Even taking
into consideration the fact that a significant
number of particles have speeds well in
excess of the average speed (i.e., particle
speeds are distributed according to the
Maxwell-Boltzmann distribution), not
enough protons can overcome their Coulomb
repulsion to produce the Sun’s observed
luminosity.

Quantum mechanical tunneling helps to
overcome the Coulomb repulsion between
protons for nuclear fusion to proceed.
Measurement Precision

The figure below shows a spectrum (solid line) observed towards a star. The
fluctuations in the spectrum (from a relative intensity of 1.0) is caused by the
inherent shape of the stellar continuum, absorption in the interstellar medium, and
uncertainties in the measurement of light energy at a given wavelength.
Measurement Precision

What can you do the decrease the uncertainty in the measurement of light energy
at a given wavelength (assume you cannot change your observing equipment)?
Measurement Precision

What can you do the decrease the uncertainty in the measurement of light energy
at a given wavelength (assume you cannot change your observing equipment)?
Integrate for a longer time interval, since ΔE Δt = ħ.
Measurement Precision

This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to
make an improved image of Betelgeuse:
We target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the bandwidth
of each of eight images that we plan to make over the frequency range 40–48
GHz) with robust weighting, about two orders of magnitude higher than that
previously attained by Lim et al. (1998) with the VLA as shown in Fig. 1. The
required integration time is about 3 hrs. With an estimated observing efficiency of
60% (including overheads for absolute flux calibration and pointing checks), we
will require a total observing time of 5 hrs.
Measurement Precision

This is the last paragraph of a proposal I submitted to the EVLA in Feb 2012 to
make an improved image of Betelgeuse:
We target an rms sensitivity of 15 μJy for a bandwidth of 1 GHz (the bandwidth
of each of eight images that we plan to make over the frequency range 40–48
GHz) with robust weighting, about two orders of magnitude higher than that
previously attained by Lim et al. (1998) with the VLA as shown in Fig. 1. The
required integration time is about 3 hrs. With an estimated observing efficiency of
60% (including overheads for absolute flux calibration and pointing checks), we
will require a total observing time of 5 hrs.

Eq. (5.20) explain why observers sometimes integrate for hours when observing
astronomical objects, so as to reduce the uncertainty in the measurement of the
light intensity (in the case of images, from different directions of the sky).
Learning Objectives

Problems with Bohr’s Semiclassical Model of the Atom

Wave-Particle Duality:
Everything exhibits its wave properties in its propagation, and manifests its
particle nature in its interactions
de Broglie’s wavelength for electrons in an atom

Wave-like Description of the Physical World
Probability waves in quantum mechanics
Heisenberg’s uncertainty principle

Some applications of Heisenberg’s Uncertainty Principle in Science/Astrophysics:
Natural widths of spectral lines
Quantum mechanical tunneling and nuclear fusion in stars
Measurement precision

Schrödinger’s Wave Equation
Solutions describe possible quantum states of a physical system
Quantum mechanical atom
Wave-Particle Duality

Although, to be able to relate to the familiar, we imagine an atom as electrons
orbiting a nucleus, a proper description of an atom requires describing electrons as
waves.
Schrödinger’s Equation

Motivated to find a proper wave equation for the electron, in
1926 the Austrian physicist Erwin Schrödinger formulated the
wave equation
now known as Schrödinger’s equation that describes how the
quantum state of a physical system (elementary particles, atoms,
molecules, etc.) changes in time.
Erwin Schrödinger,
1887-1961

Wavefunctions Ψ(x, t) that satisfy Schrödinger’s equation describe possible
quantum states of a physical system. E.g., wavefunctions that satisfy
Schrödinger’s equation for a single particle describe the allowed values of the
particle’s energy, momentum, etc., as well as its propagation through space.

Schrödinger’s equation is to quantum mechanics what Newton’s equations are to
classical physics (mechanics).
Schrödinger’s Equation

Wavefunctions that satisfy Schrödinger’s equation for a single particle describe
the allowed values of a particle’s energy, momentum, etc., as well as its
propagation through space.
Schrödinger’s Equation

Wavefunctions that satisfy Schrödinger’s equation for a single particle describe
the allowed values of a particle’s energy, momentum, etc., as well as its
propagation through space.

Plot of the real part of a possible wavefunction for a particle moving at a constant
velocity.
Quantum Mechanical Atom

Schrödinger’s wave equation
can be solved analytically for the hydrogen atom, giving exactly
the same set of allowed energies as those obtained by Bohr. In
addition to the principal quantum number n, Schrödinger found
that two other quantum numbers, and , are required for a
complete description of the electron orbitals such that the orbital
angular momentum of the electron has a magnitude
where
an is the angular momentum
quantum number and n is the principal quantum number. For
historical reasons related to how spectral lines were first
designated, = 0, 1, 2, 3, 4, 5, etc. are referred to as s, p, d, g, f,
h, etc.
Erwin Schrödinger,
1887-1961
Quantum States of the Hydrogen Atom

Quantum numbers and energies
for the ground (n = 1) and first
excited state (n = 2) of the
hydrogen atom.

For a single-electron atom such
as hydrogen, the different
angular momentum quantum
numbers with the same
principal quantum number n
have (almost exactly) the same
energy and are said to be
degenerate. (Note that this is
not the case for a multi-electron
atom.)
Quantum Mechanical Atom

Cross-section in xz-plane of the probability densities for an electron in different
states (n, ) in a hydrogen atom.

Each orbital has a
characteristic shape
reflecting the motion of the
electron in that particular
orbital, this motion being
characterised by an angular
momentum that reflects the
angular velocity of the
electron moving in its
orbital.
f
g
ml = 0
Quantum States of the Hydrogen Atom

The energy diagram for the hydrogen atom plotted in columns of constant orbital
angular momentum quantum number .
Quantum States of the Helium Atom

The energy diagram for the helium atom plotted in columns of constant orbital
angular momentum quantum number . For a multi-electron atom like helium,
interactions between electrons
result in atoms with the
outermost electron having the
same principal quantum
number but different angular
momentum quantum numbers
having different energies.
Quantum Mechanical Atom

The projection of the orbital angular momentum in a specified direction (z-axis),
the angular momentum vector component, is also referred to as the magnetic
quantum number,
. The z-component of the angular momentum vector, Lz, can
only have values
, with
equal to any of the
integers
between
and
inclusive. Thus, the angular momentum vector can point in
different directions.

E.g., for n = 1, = 0
and
for n = 2, = 0, 1 and
for n = 3, = 0, 1, 2 and
= 0.
= −1, 0, +1
= −2, −1, 0, +1, +2
Quantum States of the Hydrogen Atom

Quantum numbers and energies
for the ground (n = 1) and first
excited state (n = 2) of the
hydrogen atom.

In the absence of any preferred
direction in space (e.g., as
defined by an electric or
magnetic field), different
orbitals with the same principal
quantum number n have the
same energy and are said to be
degenerate.
Quantum Mechanical Atom

Cross-section in xz-plane of the probability densities for an electron in different
excited states (n, ,
) in a hydrogen atom.
The Zeeman Effect

An electron in an atom will feel the effect
of a magnetic field: the magnitude of this
effect depends on the electron’s orbital
motion (i.e., magnitude and orientation of
the electron’s orbital angular momentum
through the magnetic quantum number
) and magnetic field strength B.

Electron orbitals with the same n and
but different values
therefore have
(slightly) different energies. The splitting
of spectral lines in the presence of a
magnetic field is called the Zeeman
effect after the Dutch physicist Pieter
Zeeman.

In the example shown, the three
frequencies of the split line are given by
The Zeeman Effect

Zeeman splitting of the Iron line as observed for a sunspot.

The Zeeman effect provides the only direct measure of magnetic field strengths in
astrophysics. (There are several indirect methods to estimate magnetic field
strengths).
spatial dimension
along slit
slit
λ
Anomalous Zeeman Effect

More complicated splitting patterns of spectral lines by magnetic fields are
sometimes seen, usually involving even number of unequally spaced spectral
lines. This effect is called the anomalous Zeeman effect.
Learning Objectives

Pauli’s Exclusion Principle
No two electrons (particles) can share the same quantum state

Relativistic Wave Equation
Solutions to relativistic wave equation for an atom

Complex Spectra of Atoms
Multielectron atoms with different ionization states
Permitted and Non-Permitted Transitions
Pauli Exclusion Principle


Based on the empirical knowledge of the properties of atoms
(e.g., from their spectra), in 1925 the Austrian theoretical
physicist Wolfgang Pauli proposed that electrons in an atom
cannot share the same quantum state provided that each
electron state is defined by four quantum numbers. This rule,
which at the time did not have a theoretical basis, is now
known as the Pauli exclusion principle.
Recall that three quantum numbers were known at the time: the
principal quantum number n, the angular momentum quantum
number , and the magnetic quantum number . The new
quantum number that Pauli introduced could take on two
possible values.
Wolfgang Pauli, 18691955
Quantum Mechanical Atom

In 1925, George Uhlenbeck, Samuel Goudsmit, and Ralph Kronig associated the
new quantum number introduced by Paul with the spin of the electron. The
electron spin is not a classical top-like rotation (although this is often drawn for
visualization purposes) but a purely quantum effect.

The spin angular momentum S is a vector of constant magnitude
component
are +½ or −½.
with a z. The only values for the electron spin quantum number ms
Electron
Proton
Quantum States of the Helium Atom

Parahelium/orthohelium corresponds to helium atoms with their two electrons
having antiparallel/parallel spins.

In orthohelium, one electron is
in the 1s state. That state is not
shown because the second
electron cannot decay to the
1s state.
Quantum Mechanical Atom

According to Maxwell’s equation, a moving charge generates a magnetic field.
Quantum Mechanical Atom

According to Maxwell’s equation, a moving charge generates a magnetic field.
An orbiting electron therefore generates a magnetic field.

A spinning charged sphere is an electrical current, which according to Maxwell’s
equations generates a magnetic field. By analogy, a “spinning” electron generates
a magnetic field.

As a consequence, the
magnetic field generated
by the electron due to its
spin interacts with the
magnetic field generated
by the electron due to its
orbital motion. This
effect is called spin-orbit
coupling.
Quantum Mechanical Atom

When the spectral lines of hydrogen are examined at very high spectral resolution,
they are found to be closely-spaced doublets. The lines are split due to spin-orbit
coupling of the electron, and are known as fine structure lines.

(The reason why the = 1 but not = 0 angular momentum quantum number is
split is beyond the scope of this course.)
ms = +½
ms = −½
0.016 nm
Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line
in the Sun’s spectrum corresponds to the wavelength of yellow light emitted when
salt is sprinkled in a flame.

The sodium (D) line is actually a doublet (two closely-spaced lines).
Quantum States of the Sodium Atom

Recall that Fraunhofer determined that the wavelength of one prominent dark line
in the Sun’s spectrum corresponds to the wavelength of yellow light emitted when
salt is sprinkled in a flame.

The sodium (D) line is actually a doublet (two closely-spaced lines), and is caused
by splitting of a single spectral line into two by spin-orbit coupling.
ms = +½
ms = −½
Quantum Mechanical Atom

The anomalous Zeeman effect, usually involving the splitting of a spectral line
into an even number of unequally spaced spectral lines in the presence of
magnetic field, is the same as the Zeeman effect but acting on lines which are split
due to spin-orbit coupling.
Quantum Mechanical Atom

The anomalous Zeeman effect is the same as the Zeeman effect but acting on
spectral lines that are split by spin-orbit coupling. The number of energy levels
that results from the application of a magnetic field is beyond the scope of this
course.
Learning Objectives

Pauli’s Exclusion Principle
No two electrons (particles) can share the same quantum state

Relativistic Wave Equation
Solutions to relativistic wave equation for an atom

Complex Spectra of Atoms
Multielectron atoms with different ionization states
Permitted and Non-Permitted Transitions
Relativistic Schrödinger Equation

In 1928, the English physicist Paul Dirac combined Schrödinger’s equation with
Einstein’s theory of special relativity to produce a relativistic wave equation for
the electron.

Dirac’s solution
- naturally included the spin of the electron
- naturally explained Pauli’s exclusion principle as being applicable to all
particles with spin of an odd integer times
(such as electrons, protons, and
neutrons) known collectively as fermions
- particles (such as photons) that have an integral spin do not obey Pauli’s
exclusion principle and are known as bosons
- predicted the existence of antiparticles (identical to their corresponding
particles except for their opposite electrical charges and magnetic moments)
Quantum Mechanical Atom

In 1933, Otto Stern and Walther Gerlach measured the effect of nuclear spin.

For hydrogen atoms, the nuclear spin quantum number can only have values of
+½ or −½. Hydrogen atoms with parallel proton and electron spins have higher
energies than those with antiparallel proton and electron spins (spin-spin
coupling). The transition between these two states emits a photon with a
wavelength of 21 cm.
Quantum Mechanical Atom

In 1933, Otto Stern and Walther Gerlach measured the effect of nuclear spin.

For hydrogen atoms, the nuclear spin quantum number can only have values of
+½ or −½. Hydrogen atoms with parallel proton and electron spins have higher
energies than those with antiparallel proton and electron spins. The transition
between these two states in the ground (n = 1) state emits a photon at a
wavelength of 21 cm.
spin-orbit
coupling
spin-spin coupling
Hydrogen 21-cm Line

The 21-cm line is one of the most important spectral lines in astronomy, and
permits astronomers to map the distribution and bulk motion of neutral (unionized)
atomic hydrogen gas in galaxies.
Hydrogen 21-cm Line

The 21-cm line is one of the most important spectral lines in astronomy, and
permits astronomers to map the distribution and bulk motion of neutral (unionized)
atomic hydrogen gas in galaxies.
Intensity of 21-cm line
Velocity of 21-cm line
Learning Objectives

Pauli’s Exclusion Principle
No two electrons (particles) can share the same quantum state

Relativistic Wave Equation
Solutions to relativistic wave equation for an atom

Complex Spectra of Atoms
Multielectron atoms with different ionization states
Permitted and Non-Permitted Transitions
Complex Spectra of Atoms

In summary, an electron in an atom is described by four quantum numbers
- principal quantum number, n
- orbital quantum number,
- magnetic quantum number,
=
…
- spin quantum number, ms = ±½
The electron can interact with itself through spin-orbit coupling.

The nucleus also has a spin quantum number. The electron can interact with the
nucleus through spin-spin coupling.

In an atom/ion with a single electron, there is no other electron to interact with.
The spectrum of such an atom/ion is hydrogen-like.

In a multielectron atom, electrons interact not only with themselves and the
nucleus, but also with each other (described as orbit-orbit and spin-spin coupling).
The spectrum of multi-electron atoms therefore increases rapidly in complexity
with the total number of electrons.

Furthermore, multielectron atoms with different ionization states (e.g., O I, O II,
etc.) have different spectra.
Complex Spectra of Atoms

Finally, different transitions have different likelihoods of occurring.

Transitions that have a high
likelihood of occurring are
known as permitted
transitions, and the resulting
spectral lines known as
permitted lines. The
lifetime of an electron at a
permitted transition is ≪1 s.

Transitions that have a low
likelihood of occurring are
known as non-permitted or
forbidden transitions, and
the resulting spectral lines
known as forbidden lines.
The lifetime of an electron
at a forbidden transition is
>1 s.
Quantum Mechanical Atom

An example of a forbidden transition is the transition of the hydrogen atom that
produces the 21-cm line.
Complex Spectra of Atoms

At optical wavelengths, forbidden transitions are indicated by enclosing square
brackets; e.g., [O III]. These lines are not seen from gas even under the best
vacuum conditions on the Earth (hence their designations as forbidden lines), but
are only seen from gas in space. Why?
Complex Spectra of Atoms

At optical wavelengths, forbidden transitions are indicated by enclosing square
brackets; e.g., [O III]. These lines are not seen from gas even under the best
vacuum conditions on the Earth (hence their designations as forbidden lines), but
are only seen from gas in space. Why? Forbidden transitions have a relatively
low probability of occurring, or equivalently relatively long lifetimes. At
relatively high densities, collisions between atoms (or ions) occur on timescales
much shorter than the lifetimes of forbidden transitions: the de-excitation energy
is transferred to the kinetic energy of the colliding atom (or ion). At relatively low
densities (as in interstellar space), collisions between atoms (or ions) occur on
timescales much longer than the lifetimes of forbidden transitions, thus permitting
forbidden transitions that produce photons to occur.