Orbitals, Bonds, Interactions
Part 1. Easy Quantum (2.1.07)
Introduction
I am handing out several readings that you can regard as background, or, if you need no
background (ha), as something to skim until you get to the places where I have dropped
questions for you to answer that probably cannot be answered definitively by anyone. In
all cases, these readings should give you a list of the material I expect you to be familiar
with. As I will continue to remind you, students in this class have varied backgrounds, so
if this seems like something you just had in an entire semester-long class at a higher
level, don’t worry about it (unless you really think you understand anything about it).
When I give you handouts on biochemistry and biomacromolecules, the tables will be
turned for those who have had biology but little physics.
These notes on the rudiments of quantum theory, taken from some web tutorials and my
own insertions and deletions, are for students who have not had recent background
experience with waves or blackbody radiation or quantum mechanics, and it is only
meant to remind you or introduce you to these things at an elementary level. For
example, these notes only take you up to the development of the time-independent
Schrodinger equation using the idea of the deBroglie wavelength for particles, although
most of the material is much more elementary. Some (e.g. the example solutions to
Schrodinger’s equation for the particle in the box or related potentials), are deeper than
they appear. If you are a student with background in these areas, you may be surprised
to read through the material anyway, either as a refresher, or to see how differently you
might view these “elementary” topics today. I will try to convert units to cgs (cm instead
of m) but probably missed some spots--the rest of the world is mks/SI. I have also left in
a few links that you can access from this document if you are interested.
We want to work through the topics in Lunine sections 3.2, 3.3, 3.4 by next
week. There are some things that will come up so many times that we might as well go
over them at different levels now. This writeup, which will be very light reading for some
of you and new to others, has headings that tell you what I expect you to become
familiar with:
Waves and light; blackbody radiation; quantization and matter waves; the
hydrogen spectral lines; the Bohr model emerges; orbitals (elementary description--you
will be receiving a reading packet that goes over atomic orbitals in gory detail); electron
spin; standing wave modes in a plucked string; standing waves in the hydrogen atom;
unbound states; physical significance of the wave function; quick derivation of
Schrodinger’s equation; orbitals (again); the quantum numbers; the angular momentum
quantum number (pictures of orbitals here); magnetic quantum number; electron spin
and the exclusion principle.
Following this handout will come one that is a little more challenging, that
reminds you or informs you in more detail how the orbitals of atoms are labelled to
reflect the quantum numbers that are introduced here.
The last section ends with a question about what portions of quantum theory
really are important in determining the chemical characteristics of macromolecules, and
whether the repulsive/attractive nature of the Coulomb force can be derived from first
principles, using quantum theory or anything else, or is stuck in because we observed it
first.
The stars of the show:
Here are the main characters in this play; you may notice that the comic and actor Robin
Williams is a direct descendant of Erwin Schrodinger, although Williams rarely speaks of
it due to his shame for not pursuing a career in physics.
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Lord Kelvin
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Heisenberg
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Maxwell
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Planck
Einstein
deBroglie
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Bohr
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Waves and Light
A particle is a discrete unit of matter having the attributes of mass, momentum (and
thus kinetic energy) and optionally of electric charge. This definition is mostly for
fundamental particles; for “macroscopic” or “mesoscopic” (the realm of interest to us,
inbetween micro and macro) particles the particle can have other attributes like a
measure of its size, shape, composition, etc. But for now we treat everything like a point
particle, since it is the electron we are after. We want to understand how molecular
orbitals come about--that is our main goal (and the topic of Lunine, sec. 3.2) before
learning about chemical bonding (Lunine, sec. 3.3), intermolecular forces (Lunine,
almost nothing), biomolecules (Lunine, Chapter 4) and more. But first we have to
understand the orbitals of electrons in atoms, and for that we need to be comfortable
with certain concepts.
A wave is a periodic variation of some quantity as a function of location or time.
For example, the wave motion of a vibrating guitar string is defined by the displacement
of the string from its center as a function of distance along the string. A sound wave
consists of variations in the pressure with location. Waves are often represented as
trigonometric functions like a sine wave, so you might write a wave as A0 sin(t) in time
or A0 sin(kx) in space, where  is the frequency of the wave (how many waves pass a
point per second) and k is a spatial frequency called the wavenumber, k=2/; it often
takes a while to get used to wavenumber, so think about it now.
Waves are so widespread in nature that they form a fundamental part of just about every
physics and astronomy course, with more types of waves than you have probably ever
heard of. (In fluid dynamics, for example, there are dozens of types of waves; just in
planetary meteorology the discussions are dominated by Rossby waves, gravity waves,
baroclinic waves,...)
The importance of a wave representation has another, practical, twist: Many processes
(only those that don’t involve large amplitudes) can be represented in a convenient way
as a superposition of many waves of different wavelengths, which is called the Fourier
theorem. In this case it is common to see a wave represented as exp(kx) or exp(t); if
you do not understand the relation between exponentials and trig functions, i.e. that they
can both be used to represent waves, find out about it now. Below we will derive the
Schrodinger equation by representing an electron probability wave as an exponential, so
get used to it now.
For any kind of wave in any medium (e.g. variations in the intensities of the local electric
and magnetic fields in space, which constitutes electromagnetic radiation), there is a
characteristic velocity at which the disturbance travels. For now we are mainly
interested only in waves (photons) that travel at the velocity of light, but soon we will be
asking about the wavelengths of particles that move much more slowly.
There are three measurable properties of wave motion: amplitude, wavelength, and
frequency, the number of vibrations per second (unit: Hertz = s-1). The relation between
the wavelength λ (Greek lambda) and frequency of a wave  (Greek nu) is determined
by the propagation velocity v
v=λ
Example. What is the wavelength of the musical note A = 440 hz (=) when it is
propagated through air in which the velocity of sound is 343 m s–1?
λ = v/ = (343 m s–1)/(440 s–1) = 0.80 m = 80 cm.
You should try the same using the velocity of light (c=3E10 cm s-1) at a frequency of 1
MHz or 1 GHz -- what wavelength regions do these correspond to? What is the
frequency of a UV photon? A far infrared 100 micron photon? You should memorize the
speed of light and be able to make these kinds of conversions easily.
If you are not comfortable with the regions of the electromagnetic spectrum, now is the
time to review. Here is a nice chart showing the names given to different wavelength
ranges, their frequency, their energy, and what photons (or absorption of photons) are
associated with different kinds of physical processes. Please make careful note of these
processes and memorize the wavelengths and energies (in eV, or microns if
appropriate) associated with each. If you find that you have to make a small copy of this
illustration and glue it to your hand so that you can frequently study it, so be it.
Two other attributes of waves are the amplitude (the height of the wave crests with
respect to the base line) and the phase, which measures the position of a crest with
respect to some fixed point. The square of the amplitude gives the intensity of the
wave: the energy transmitted per unit time. We will see that Heisenberg took the bold
step of representing a material particle as a wave, with the square of the amplitude (or
wave function) interpreted as the probability of finding an electron in a certain region of
space. A little thought will show you that the intensity (or flux, or irradiance, or whatever
terminology might ring a bell) of light is no different, although it is usually expressed as
an energy per unit something; normalize it to the total energy in the spectrum and it
qualifies as the probability that the light will have a certain wavelength in a certain region
of space. That is what we try to do when we solve the equation of radiative transfer.
Back to waves: A unique property of waves is their ability to combine constructively or
destructively, depending on the relative phases of the combining waves. In the early
19th century, the English scientist Thomas Young carried out the famous two-slit
experiment which demonstrated that a beam of light, when split into two beams and then
recombined, will show interference effects that can only be explained by assuming that
light is a wavelike disturbance. The exact nature of the waves remained unclear until the
1860's when James Clerk Maxwell developed his electromagnetic theory.
It was known that a moving electric charge gives rise to a magnetic field, and that a
changing magnetic field can induce electric charges to move. Maxwell showed
theoretically that when an electric charge is accelerated (by being made to oscillate
within a piece of wire, for example), electrical energy will be lost, and an equivalent
amount of energy is radiated into space, spreading out as a series of waves extending in
all directions.
What is "waving" in electromagnetic
radiation? According to Maxwell, it is
the strengths of the electric and
magnetic fields as they travel through
space. The two fields are oriented at
right angles to each other and to the
direction of travel.
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As the electric field changes, it
induces a magnetic field, which then
induces a new electric field, etc.,
allowing the wave to propagate itself
through space
These waves consist of periodic variations in the electrostatic and electromagnetic field
strengths. These variations occur at right angles to each other. Each electrostatic
component of the wave induces a magnetic component, which then creates a new
electrostatic component, so that the wave, once formed, continues to propagate through
space, essentially feeding on itself. In one of the most brilliant mathematical
developments in the history of science, Maxwell expounded a detailed theory, and even
showed that these waves should travel at about 3E10 cm s–1, a value which
experimental observations had shown corresponded to the speed of light. In 1887, the
German physicist demonstrated that an oscillating electric charge (in what was in
essence the world's first radio transmitting antenna) actually does produce
electromagnetic radiation just as Maxwell had predicted, and that these waves behave
exactly like light.
It is now understood that light is electromagnetic radiation that falls within a range of
wavelengths that can be perceived by the eye. The entire electromagnetic spectrum
runs from radio waves at the long-wavelength end, through heat, light, X-rays, and to
gamma radiation.
It is important that you be familiar with the names of wavelength regions (remembering
that they are arbitrary--products of how we detect them or specific physical effects that
they cause: you should think about this fact--in case somebody someday says “Why is
the boundary between X-rays and UV where it is?”). Here is a smaller version of the plot
given earlier:
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Quantum theory of light
Certain aspects of the interaction between light and matter that were observed during
the decade after 1890 proved rather troublesome. The relation between the temperature
of an object and the peak wavelength emitted by it was established empirically by
Wilhelm Wein in 1893. This put on a quantitative basis what everyone knows: the hotter
the object, the "bluer" the light it emits. How can this be understood?
Black body radiation
All objects above the temperature of absolute zero emit electromagnetic radiation
consisting of a broad range of wavelengths described by a distribution curve, its
spectrum. An idealized model, a body that absorbs and emits perfectly, was finally
solved by Planck, by making the radical assumption that light consists of particles called
photons, each with energy h. This allowed him to avoid the problem with the classical
Rayleigh-Jeans law for black body radiation which gives infinite amounts of energy (“the
UV catastrophe”). The details of the derivation will not be given here-you just have to
know it is a model for the continuum radiation and is pretty good for a lot of objects,
although not good enough for quantitative calculations; a useful idea for a blackbody is
an over with a small window by which some radiation can escape and be observed. And
know that it was for the description of this problem that Planck invented Planck’s
constant!
This curve of intensity of light as a function of wavelength (or frequency) and
temperature is known as the Planck, or blackbody, spectrum. Here is the full formula:
If we look at Planck's law for small frequencies h<<kT, we find an expression that
contains no factors h (Taylor series expansion of exponent)
This is the Rayleigh-Jeans law as derived from classical physics. If we integrate these
laws over all frequencies we find
for Planck's law, and infinity for the Rayleigh-Jeans law. The result has been
experimentally confirmed, and predates Planck's law.
At ordinary temperatures this radiation is almost entirely in the infrared region of the
spectrum, but as the temperature rises above about 2000K, more energy is emitted in
the visible wavelength region and the object begins to glow, first with red light, and then
shifting toward the blue as the temperature is increased. The sun is yellow for a good
reason! (See below.)
The way the peak of the Planck curve changes with temperature is quantified by Wien's
Law:
T (K) = 0.29 / max(cm),
max(cm) = 0.29/ T(K)
This is extremely useful and informative, and simple! These relations are useful not
just for planets or stars, but even for rocks that are emitting at some temperature, and
even microscopic dust grains (with some qualifications). On the other hand, they have
nothing to do with radiation that is produced nonthermally (e.g. synchrotron radiation).
Example: for the Sun we can measure the spectrum and find
max = 5500 Å = 5.5 x 10-5 cm
From this we can estimate the surface temperature of the Sun using Wien's Law (this will
only give the approximately correct answer if the solar spectrum does indeed look like a
Planck spectrum).
T = 0.29/5.5 x 10-5 = 5200 K.
Not bad (sun’s effective temperature is more like 5800 K), considering that the sun’s
spectrum is not a Planck spectrum (i.e. not a black body).
Most of the objects we’ll discuss in this class are roughly at room temperature
T ~ 300 K
In this case
 max(cm) = 0.29/T = 9.8 x 10-4 cm
= 9.8 µm  infrared
 This is why the continuum radiation from grains, aerosols, planets, protostellar
disks, and much of the interstellar medium, is mostly in the infrared part of the
spectrum. Cooler objects (like the outer planets and moons, or the outer parts of
protoplanetary disks) will radiate at even longer, far infrared or submillimeter
wavelengths. This is why the advent of infrared satellite missions like IRAS (1984), up
to today’s Spitzer mission, are so important for star and planet formation. At
submillimeter wavelengths (say 1000 microns), different types of instruments must be
used, but at least they can be used from the ground. Submillimeter and infrared
continuum and spectral line astronomy dominate the study of planetary science,
exoplanets, and the search for extraterrestrial life.
In the case of life, the reason is two-fold: 1. The basic problem is detecting the faint
planet in the glare of its parent star, and this is optimized by observing where the planet
puts out most of its energy, the infrared for a planet at the earth’s temperature (which is
of greatest interest--temperatures where liquid water can exist most of the time). 2.
There are several atmospheric gases (CO2, H2O, O2, and especially O3) that have strong
infrared vibrational bands (we’ll go over vibrational spectroscopy later). Some (O3, CH4,
N2O, ...) are considered possible biomarkers--we will discuss in full depth towards the
end of the course.
The other important relation to remember is the Stefan-Boltzmann law, which says that
the energy emitted per unit time by any unit area (say a cm2) of a blackbody, call it F (for
flux) is proportional to T4:
F =  T4
The constant of proportionality  is called the Stefan-Boltzmann constant.
(Think about how you can get the radius of an object from this relation if you know its
absolute luminosity, using F = L/4R2.)
These relations are derived by differentiating (Wien) and integrating (Stefan-Boltzmann)
the complete Planck distribution given earlier, and displayed below.
This type of radiation has two important characteristics. First, the spectrum is a
continuous one, meaning that all wavelengths are emitted, although with intensities that
vary smoothly with wavelength. The other curious property of black body radiation is that
it is independent of the composition of the object; all that is important is the temperature.
Black body radiation, like all electromagnetic radiation, must originate from oscillations of
electric charges which in this case were assumed to be the electrons within the atoms of
an object acting somewhat as miniature Hertzian oscillators. It was presumed that since
all wavelengths seemed to be present in the continuous spectrum of a glowing body,
these tiny oscillators could send or receive any portion of their total energy. However, all
attempts to predict the actual shape of the emission spectrum of a glowing object on the
basis of classical physical theory proved futile.
Quantization and matter waves
In 1900, the great German physicist Max Planck (who earlier in the same year had
worked out an empirical formula giving the detailed shape of the black body emission
spectrum) showed that the shape of the observed spectrum could be exactly predicted if
the energies emitted or absorbed by each oscillator were restricted to integral values of
h, where  ("nu") is the frequency and h is a constant 6.626E–34 J s which we now
know as Planck's Constant. (Planck’s headstone at his gravesite purportedly has nothing
written on it besides his name and this number; an amusing game and effective way to
see physics in a new perspective is to rank physicists according to whether they have
named after them fundamental constants, classes of elementary particles (e.g. bosons),
laws, effects, etc.) The allowable energies of each oscillator are quantized, but the
emission spectrum of the body remains continuous because of differences in frequency
among the uncountable numbers of oscillators it contains.This modification of classical
theory, the first use of the quantum concept, was as unprecedented as it was simple,
and it set the stage for the development of modern quantum physics.
The photoelectric effect was the turning point in understanding the quantization of
energy. (I’m assuming you know what that effect is--go to Wikipedia if you don’t.)
Although the number of electrons ejected from the metal surface per second depends on
the intensity of the light, as expected, the kinetic energies of these electrons (as
determined by measuring the retarding potential needed to stop them) does not, and this
was definitely not expected. Just as a more intense physical disturbance will produce
higher energy waves on the surface of the ocean, it was supposed that a more intense
light beam would confer greater energy on the photoelectrons. But what was found, to
everyone's surprise, is that the photoelectron energy is controlled by the wavelength of
the light, and that there is a critical wavelength below which no photoelectrons are
emitted at all.
Albert Einstein quickly saw that if the kinetic energy of the photoelectrons depends on
the wavelength of the light, then so must its energy. Further, if Planck was correct in
supposing that energy must be exchanged in packets restricted to certain values, then
light must similarly be organized into energy packets. But a light ray consists of electric
and magnetic fields that spread out in a uniform, continuous manner; how can a
continuously-varying wave front exchange energy in discrete amounts? Einstein's
answer was that the energy contained in each packet of the light must be concentrated
into a tiny region of the wave front. This is tantamount to saying that light has the nature
of a quantized particle whose energy is given by the product of Planck's constant and
the frequency:
E = h  = hc/
where c is the speed of light. Einstein's publication of this explanation in 1905 led to the
rapid acceptance of Planck's idea of energy quantization, which had not previously
attracted much support from the physics community of the time. It is interesting to note,
however, that this did not make Planck happy at all. Planck, ever the conservative, had
been reluctant to accept that his own quantized-energy hypothesis was much more than
an artifice to explain black-body radiation; to extend it to light seemed an absurdity that
would negate the well-established electromagnetic theory and would set science back to
the time before Maxwell.
Einstein's special theory of relativity arose from his attempt to understand why the laws
of physics that describe the current induced in a fixed conductor when a magnet moves
past it are not formulated in the same way as the ones that describe the magnetic field
produced by a moving conductor. The details of this development are not relevant to our
immediate purpose, but some of the conclusions that this line of thinking led to very
definitely are. Einstein showed that the velocity of light, unlike that of a material body,
has the same value no matter what velocity the observer has. Further, the mass of any
material object, which had previously been regarded as an absolute, is itself a function of
the velocity of the body relative to that of the observer (hence "relativity"). As you
probably know, according to Einstein, the mass of an object increases without limit as
the velocity approaches that of light. Where does the increased mass come from?
Einstein's answer was that the increased mass is that of the kinetic energy of the object;
that is, energy itself has mass, so that mass and energy are equivalent according to the
famous formula E = mc2.
The only particle that can move at the velocity of light is the photon itself, due to its zero
rest mass.
In 1924, the French physicist Louis deBroglie proposed that, although the photon has no
rest mass, its energy, given by E=h, confers upon it an effective mass of (since E=mc2)
“m” = E/c2 = h/c2 = h/c 
and a momentum of
“mv” = h/c2 x c =  /c = h /
or
p = h/ = hk/2.
DeBroglie (in his doctoral thesis) proposed that just as light possesses particle-like
properties, so should particles of matter exhibit a wave-like character. Within two years
this hypothesis had been confirmed experimentally by observing the diffraction (a wave
interference effect) produced by a beam of electrons as they were scattered by the row
of atoms at the surface of a metal. (Needless to say, deBroglie got a good postdoctoral
position.)
deBroglie showed that the wavelength of a particle is inversely proportional to its
momentum:
 = h/m v
where v is the velocity of the particle.
Notice that the wavelength of a stationary particle is infinitely large, while that of a
particle of large mass approaches zero. For most practical purposes, the only particle of
interest to chemistry that is sufficiently small to exhibit wavelike behavior is the electron
(mass 9.11E-28 g). I will some day ask you a question like: What is the deBroglie
wavelength of a baseball travelling at 100 miles/hr? What is the deBroglie wavelength of
an electron moving around an H atom in the ground state? What velocity should you
use?
A wave is a change that varies with location in a periodic, repeating way. What sort of a
change do the crests and troughs of a "matter wave" trace out? The answer is that the
wave amplitude at some position x represents the value of a quantity whose square is a
measure of the probability of finding the particle at position x. In other words, what is
"waving" is the value of a mathematical probability function. The evolution of this
function is described by the Schrodinger equation.
In 1927, Werner Heisenberg proposed that certain pairs of properties of a particle cannot
simultaneously have exact values. In particular, the position and the momentum of a
particle have associated with them uncertainties x and p given by
(x) (p) ≥ h/2
As with the de Broglie particle wavelength, this has practical consequences only for
electrons and other particles of very small mass. It is very important to understand that
these "uncertainties" are not merely limitations related to experimental error or
observational technique, but instead they express an underlying fact that Nature does
not allow a particle to possess definite values of position and momentum at the same
time. This principle (which would be better described by the term "indeterminacy" than
"uncertainty") has been thoroughly verified and has far-reaching practical consequences
which extend to chemical bonding and molecular structure. We will examine these later.
The uncertainty principle is consistent with particle waves, because either one
really implies the other. Consider the following two limiting cases: · A particle whose
velocity is known to within a very small uncertainty will have a sharply-defined energy
(because its kinetic energy is known) which can be represented by a probability wave
having a single, sharply-defined frequency. A "monochromatic" wave of this kind must
extend infinitely in space:
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But if the peaks of the wave represent locations at which the particle is most likely to
manifest itself, we are forced to the conclusion that it can "be" virtually anywhere, since
the number of such peaks is infinite! Now think of the opposite extreme: a particle whose
location is closely known. Such a particle would be described by a short wavetrain
having only a single peak, the smaller the uncertainty in position, the more narrow the
peak.
To help you see how waveforms of different wavelength combine, two such
combinations are shown below:
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It is apparent that as more waves of different frequency are mixed, the regions in which
they add constructively diminish in extent. The extreme case would be a wavetrain in
which destructive interference occurs at all locations except one, resulting in a single
pulse:
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Is such a wave possible, and if so, what is its wavelength? Such a wave is possible, but
only as the sum (interference) of other waves whose wavelengths are all slightly
different. Each component wave possesses its own energy (momentum), and adds that
value to the range of momenta carried by the particle, thus increasing the uncertainty p.
In the extreme case of a quantum particle whose location is known exactly, the
probability wavelet would have zero width which could be achieved only by combining
waves of all wavelengths-- an infinite number of wavelengths, and thus an infinite range
of momentum dp and thus kinetic energy.
As Young’s double slit experiment and other evidence soon showed, photons have to be
considered as both particles and waves, or one or the other depending on the
experimental situation. This seems like a conundrum, leading to 100s of books, many of
which you have probably seen at your local bookstore. This is unfortunate, because
there is no “answer” to the question of wave-particle duality, and all the fun thought
experiments (e.g. Schrodinger’s cat) are just that: fun. There is no problem here if you
realize that the idea of something in terms of properties or attributes is simply a model of
our interpretation of perceptions or experiments, and bears an extremely uncertain and
even tenuous relation to what is “really” “out there,” if anything. There are some “things”
that we will call electrons and photons and they can be given attributes for the purpose
of calculations using our little models, but their “real” nature may have little to do with
these models.
Only a slight digression: The conflation or confusion of a model for the thing it represents
is called “reification” (one of my favorite words), or “hypostatization.” The old Mr.
Magoo cartoons were all about reification due to a cognitive organism walking around
with extremely nearsighted vision, and we are probably doing it constantly in daily life, so
we won’t worry about it further (although it is certainly a more important subject than
quantum mechanics, contrary to what most physicists must say in order to rescue the
objective nature of reality, without which physics would be a sham; you will see
examples of this as we go through the course). If you are interested in evidence and
arguments that infants produce conceptual models in the same way that science does,
or that non-human organisms have a conceptual apparatus, or that human conceptual
organization and categorization is highly variable (but with some interesting uniformities
also), see me outside of class. For now, we resume with the reification of the world in
order to have problems to solve and homework to give.
The hydrogen spectral lines
That atoms have electron energies with well-defined values was a result not of theory,
but of the work of Balmer (a Swiss school teacher in 1885) and others to find a formula
that relates the wavelengths of the known spectral lines of hydrogen in a simple way.
Balmer’s formula was not based on theory, but worked, and he predicted a fifth line; later
35 lines were discovered, all having wavelengths fitting his formula, which we write today
as
1/
= R [(1/m2) - (1/n2) ]
wherer R is the “Rydberg constant” = 1.096...E5 cm-1 and for Balmer m=2. Soon it was
found that replacing m with 1 (Lyman) or 3 (Paschen), other series of H lines could be
accounted for. These series’, called Lyman, Balmer, Paschen, Brackett, and Pfund (up
to m=5) cover the UV to the IR. (There are embarrassing 8mm home movies of Lyman
trying to climb mountains to detect “his series” in the solar spectrum, requiring high
altitude to reduce the UV attenuation; he was unfamiliar with the molecular absorbers in
our atmosphere so did not know that he would need a mountain as high as our
stratosphere for this purpose.) Nearly all astronomers are extremely familiar with the
first two of these series’. (Lyman’s lines have come in mighty handy too, since the
Hubble Space Telescope UV spectrometer was decommissioned; very distant galaxies
at redshift around 3 can be observed by the Lyman break being redshifted into the
visible, and so are called Lyman break galaxies).
There is no limit to how large n can be: values in the hundreds have been observed,
although doing so is very difficult because of the increasingly close spacing of
successive levels as n becomes large. Atoms excited to very high values of n are said to
be in Rydberg states. (See http://www.phys.ttu.edu/~gglab/rydberg_state.html)
Attempts to adapt Balmer's formula to describe the spectra of atoms other than
hydrogen generally failed, although certain lines of some of the spectra seemed to fit this
same scheme, with the same value of R. Today we know those were the hydrogenic
atoms and that quantum mechanics does not fare a lot better for non-hydrogenic atoms,
although a few cases must be counted as successes.
As n becomes larger, the spacing between neighboring levels diminishes and the
discrete lines merge into a continuum. This can mean only one thing: the energy levels
converge as n approaches infinity. This convergence limit corresponds to the energy
required to completely remove the electron from the atom; it is the ionization energy.
At energies in excess of this, the electron is no longer bound to the rest of the atom,
which is now of course a positive ion. But an unbound system is not quantized; the
kinetic energy of the ion and electron can now have any value in excess of the ionization
energy. When such an ion and electron pair recombine to form a new atom, the light
emitted will have a wavelength that falls in the continuum region of the spectrum.
Spectroscopic observation of the convergence limit is an important method of measuring
the ionization energies of atoms.
The Bohr model emerges
Rutherford's demonstration that the mass and the positive charge of the atom is mostly
concentrated in a very tiny region called the nucleus forced the question of just how the
electrons are disposed outside the nucleus. By analogy with the solar system, a
planetary model was suggested: if the electrons were orbiting the nucleus, there would
be a centrifugal force that could oppose the electrostatic attraction and thus keep the
electrons from falling into the nucleus. This of course is similar to the way in which the
centrifugal force produced by an orbiting planet exactly balances the force due to its
gravitational attraction to the sun.
Niels Bohr was born in the same year (1885) that Balmer published his formula for the
line spectrum of hydrogen. Beginning in 1913, the brilliant Danish physicist published a
series of papers that would ultimately derive Balmer's formula from first principles.
Bohr's first task was to explain why the orbiting electron does not radiate energy as it
moves around the nucleus. This energy loss, if it were to occur at all, would do so
gradually and smoothly. But Planck had shown that black body radiation could only be
explained if energy changes were limited to jumps instead of gradual changes. If this
were a universal characteristic of energy- that is, if all energy changes were quantized,
then very small changes in energy would be impossible, so that the electron would in
effect be "locked in" to its orbit.
From this, Bohr went on to propose that there are certain stable orbits in which the
electron can exist without radiating and thus without falling into a "death spiral". This
supposition was a daring one at the time because it was inconsistent with classical
physics, and the theory which would eventually lend it support would not come along
until the work of de Broglie and Heisenberg more than ten years later.
Since Planck's quanta came in multiples of h, Bohr restricted his allowed orbits to those
in which the product of the radius r and the momentum of the electron mv (which has the
same units as h, J-s) are integral multiples of h:
2rmv = nh
(n = 1,2,3, . .)
Each orbit corresponds to a different energy, with the electron normally occupying the
one having the lowest energy, which would be the innermost orbit of the hydrogen atom.
Taking the lead from Einstein's explanation of the photoelectric effect, Bohr assumed
that each spectral line emitted by an atom that has been excited by absorption of energy
from an electrical discharge or a flame represents a change in energy given by
E = h = hc/, the energy lost when the electron falls from a higher orbit (value of n) into
a lower one.
Finally, as a crowning triumph, Bohr derived an expression giving the radius of the nth
orbit for the electron in hydrogen as
rn = h2 n2/(4  me e2)
Substitution of the observed values of the electron mass and electron charge into this
equation yielded a value of 0.529E-8 cm for the radius of the first orbit, a value that
corresponds to the radius of the hydrogen atom obtained experimentally from the kinetic
theory of gases. Bohr was also able to derive a formula giving the value of the Rydberg
constant, and thus in effect predict the entire emission spectrum of the hydrogen atom.
His formula for the energy of level n in hydrogen was
where 0 is the permittivity of free space (from Maxwell’s equations), and the appearance
of h shows clearly that the formula is of quantum origin. We view the model as primitive
and even ridiculous today, even though it does incorporate the idea of quantization of
energies, but Bohr’s success gained him what few of us will ever have, our pictures on
stamps:
Here are some more pictures to remind you of the Bohr model and what it explained.
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Problems, problems.
There were two kinds of difficulties. First, there was the practical limitation that it only
works for atoms that have one electron-- that is, for H, He+, Li2+, etc. The second
problem was that Bohr was unable to provide any theoretical justification for his
assumption that electrons in orbits described by the preceding equation would not lose
energy by radiation (all accelerating charges must radiate--says who?). This reflects the
fundamental underlying difficulty: because de Broglie's picture of matter waves would not
come until a decade later, Bohr had to regard the electron as a classical particle
traversing a definite orbital path.
You can find some derivations and elaborations about the Bohr model here:
http://www.nat.vu.nl/~wimu/Bohrtheory.html
Once it became apparent that the electron must have a wavelike character, things began
to fall into place. The possible states of an electron confined to a fixed space are in
many ways analogous to the allowed states of a vibrating guitar string. These states are
described as standing waves that must possess integral numbers of nodes. The states
of vibration of the string are described by a series of integral numbers n = 1,2,... which
we call the fundamental, first overtone, second overtone, etc. The energy of vibration is
proportional to n2. Each mode of vibration contains one more complete wave than the
one below it.
In exactly the same way, the mathematical function that defines the probability of finding
the electron at any given location within a confined space possesses n peaks and
corresponds to states in which the energy is proportional to n2.
The electron in a hydrogen atom is bound to the nucleus by its spherically symmetrical
electrostatic charge, and should therefore exhibit a similar kind of wave behavior. This is
most easily visualized in a two-dimensional cross section that corresponds to the
conventional electron orbit. But if the particle picture is replaced by de Broglie's
probability wave, this wave must follow a circular path, and- most important of all- its
wavelength (and consequently its energy) is restricted to integral multiples n = 1,2,.. of
the circumference
2r=n
(n = 1, 2, 3, ...)
for otherwise the wave would collapse owing to self-interference. That is, the energy of
the electron must be quantized; what Bohr had taken as a daring but arbitrary
assumption was now seen as a fundamental requirement. Indeed the above equation
can be derived very simply by combining Bohr's quantum condition 2rmv = nh with the
expression mv = h/ for the deBroglie wavelength of a particle.
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Viewing the electron as a standing-wave pattern also explains its failure to lose energy
by radiating. Classical theory predicts that an accelerating electric charge will act as a
radio transmitter; an electron traveling around a circular wire would certainly act in this
way, and so would one rotating in an orbit around the nucleus. In a standing wave,
however, the charge is distributed over space in a regular and unchanging way; there is
no motion of the charge itself, and thus no radiation.
Orbitals
Because the classical view of an electron as a localizable particle is now seen to be
untenable, so is the concept of a definite trajectory, or orbit. Instead, we now use the
word orbital to describe the state of existence of an electron. An orbital is really no more
than a mathematical function describing the standing wave that gives the probability of
the electron manifesting itself at any given location in space. More commonly (and
loosely) we use the word to describe the region of space occupied by an electron. Each
kind of orbital is characterized by a set of quantum numbers n, l, and m. These relate,
respectively, to the average distance of the electron from the nucleus, to the shape of
the orbital, and to its orientation in space. There is also a spin quantum number. We
will learn about these in some gory detail soon!
Are electrons in orbitals moving?
In its lowest state in the hydrogen atom (in which l=0) the electron has zero angular
momentum, so electrons in s orbitals are not in motion. In orbitals for which l>0 the
electron does have an effective angular momentum, and since the electron also has a
definite rest mass me = 9.11E-28 g, it must possess an effective velocity. Its value can
be estimated from the Uncertainty Principle; if the volume in which the electron is
confined is about 10–10 m, then the uncertainty in its momentum is at least
h/(1010) = 6.6E–19 g cm s–1, which implies a velocity of around 107 m s–1, or almost onetenth the velocity of light.
The stronger the electrostatic force of attraction by the nucleus, the faster the effective
electron velocity. In fact, the innermost electrons of the heavier elements have effective
velocities so high that relativistic effects set in; that is, the effective mass of the electron
significantly exceeds its rest mass. This has direct chemical effects; it is the cause, for
example, of the low melting point of metallic mercury and of the
Why does the electron not fall into the nucleus? The negatively-charged electron is
attracted to the positive charge of the nucleus. What prevents it from falling in? This
question can be answered in various ways at various levels. All start with the statement
that the electron, being a quantum particle, has a dual character and cannot be treated
solely by the laws of Newtonian mechanics.
We saw above that in its wavelike guise, the electron exists as a standing wave which
must circle the nucleus at a sufficient distance to allow at least one wavelength to fit on
its circumference. This means that the smaller the radius of the circle, the shorter must
be the wavelength of the electron, and thus the higher the energy. Thus it ends up
"costing" the electron energy if it gets too close to the nucleus. The normal orbital radius
represents the balance between the electrostatic force trying to pull the electron in, and
what we might call the "confinement energy" that opposes the electrostatic energy. This
confinement energy can be related to both the particle and wave character of the
electron.
If the electron as a particle were to approach the nucleus, the uncertainty in its position
would become so small (owing to the very small volume of space close to the nucleus)
that the momentum, and therefore the energy, would have to become very large. The
electron would, in effect, be "kicked out" of the nuclear region by the confinement
energy.
The standing-wave patterns of an electron in a box can be calculated quite easily. For a
spherical enclosure of diameter d, the energy is given by
E = n2 h2/8med2 where n = 1,2,3,.
Electron Spin
Each electron in an atom has associated with it a magnetic field whose direction is
quantized; there are only two possible values that point in opposite directions. We
usually refer to these as "up" and "down", but the actual directions are parallel and
antiparallel to the local magnetic field associated with the orbital motion of the electron.
The term spin implies that this magnetic moment is produced by the electron charge as
the electron rotates about its own axis. Although this conveys a vivid mental picture of
the source of the magnetism, the electron is not an extended body and its rotation is
meaningless. Electron spin has no classical counterpart; the magnetic moment is a
consequence of relativistic shifts in local space and time due to the high effective
velocity of the electron in the atom. This effect was predicted theoretically by
P.A.M. Dirac in 1928.
Bohr's theory worked; it completely explained the observed spectrum of the hydrogen
atom, and this triumph would later win him a Nobel prize. The main weakness of the
theory, as Bohr himself was the first to admit, is that it could offer no good explanation of
why these special orbits immunized the electron from radiating its energy away. The only
justification for the proposal, other than that it seems to work, comes from its analogy to
certain aspects of the behavior of vibrating mechanical systems.
Standing wave modes in a plucked string
In order to produce a tone when plucked, a guitar string must be fixed at each end (that
is, it must be a bound system) and must be under some tension. Only under these
conditions will a transverse disturbance be countered by a restoring force (the string's
tension) so as to set up a sustained vibration. Having the string tied down at both ends
places a very important boundary condition on the motion: the only allowed modes of
vibration are those whose wavelengths produce zero displacements at the bound ends
of the string; if the string breaks or becomes unattached at one end, it becomes silent.
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In its lowest-energy mode of vibration there is a single wave whose point of maximum
displacement is placed at the center of the string. In musical terms, this corresponds to
the fundamental note to which the string is tuned; in terms of the theory of vibrations, it
corresponds to a "quantum number" of 1. Higher modes, known as overtones (and in
music, as octaves), contain 2, 3, 4 and more points of maximum displacement
(antinodes) spaced evenly along the string, separated by points of zero displacement
(nodes). These correspond to successively higher quantum numbers and higher
energies.
The vibrational states of the string are quantized in the sense that an integral number of
antinodes must be present. Note again that this condition is imposed by the boundary
condition that the ends of the string, being fixed in place, must be nodes. Because the
locations of the nodes and antinodes do not change as the string vibrates, the vibrational
patterns are known as standing waves.
A similar kind of quantization occurs in other musical instruments; in each case the
vibrations, whether of a stretched string, a column of air, or of a stretched membrane.
Standing waves in the hydrogen atom
The analogy with the atom can be seen by imagining a guitar string that has been closed
into a circle. The circle is the electron orbit, and the boundary condition is that the waves
must not interfere with themselves along the circle. This condition can only be met if the
circumference of an orbit can exactly accommodate an integral number of wavelengths.
Thus only certain discrete orbital radii and energies are allowed, as depicted in the two
diagrams below.
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Unbound states
If a guitar string is plucked so harshly that it breaks, the restoring force and boundary
conditions that restricted its motions to a few discrete harmonically related frequencies
are suddenly absent; with no constraint on its movement, the string's mechanical energy
is dissipated in a random way without musical effect. In the same way, if an atom
absorbs so much energy that the electron is no longer bound to the nucleus, then the
energy states of the atom are no longer quantized; instead of the line spectrum
associated with discrete energy jumps, the spectrum degenerates into a continuum in
which all possible electron energies are allowed. The energy at which the ionization
continuum of an atom begins is easily observed spectroscopically, and serves as a
simple method of experimentally measuring the energy with which the electron is bound
to the atom.
Physical significance of the wave function
How can such a simple-looking expression contain within it the quantum-mechanical
description of an electron in an atom— and thus, by extension, of all matter? The catch,
as you may well suspect, lies in discovering the correct form of Ψ , which is known as
the wave function. As this names suggests, the value of Ψ is a function of location in
space relative to that of the proton which is the source of the binding force acting on the
electron. As in any system composed of standing waves, certain boundary conditions
must be applied, and these are also contained in Ψ; the major ones are that the value of
must approach zero as the distance from the nucleus approaches infinity, and that the
function be continuous.
When the functional form of has been worked out, the Schrödinger equation is said to
have been solved for a particular atomic system. The details of how this is done are only
outlined in this course, but the consequences of doing so are extremely important to us.
Once the form of is known, the allowed energies E of an atom can be predicted from the
above equation. Soon after Schrödinger's proposal, his equation was solved for several
atoms, and in each case the predicted energy levels agreed exactly with the observed
spectra.
Quick derivation of Schrodinger’s equation
We give an “easy” derivation of Schrodinger’s equation here. It is the steady state (timeindependent) case that we want to solve, but I’ll include the time dependence at first and
then take the limit where ∂/∂t = 0. You should try to do the derivation from scratch with
no time dependence.
Given the uncertainty principle, it is clear that all we can derive is an equation for the
probability that the electron will have a certain position at a certain time. The probability
density (pdf) of its position is P(x) such that P(x) gives the probability that the particle will
be between x and x+dx; this is just the standard definition of probability density function.
Schrodinger’s equation will be for a wave function whose square is this probability
density, a function of position, and, in 3D spherical coordinates, a function of r, , . It
will turn out that the solutions are spherical harmonics of various orders and that is why
the orbitals, which are surfaces of constant probability density, have the forms they do.
For example, one solution might have r increasing away from the nucleus, reaching a
peak, then decreasing, and the angular solution, instead of being uniform in the angles
(which would give a spherically symmetric orbital) being peaked at   in  and  (say);
then you can see how the orbital (surface of constant probability) will be a pair of blobshaped objects with a minimum between them.
The standard example of a classical wave is the motion of a string. Typically a string can
move up and down, and the standard solution to the wave equation
can be positive as well as negative. Actually the square of the wave function is a
possible choice for the probability (this is proportional to the intensity for radiation). Now
we try to argue what wave equation describes the quantum analog of classical
mechanics, i.e., quantum mechanics.
The basic idea is to use the standard representation of a propagating plane wave, i.e.
which is a wave propagating in the x direction with wavelength  = 2/k and frequency 
= /(2). But we interpret this wave as a propagating beam of particles.
If we define the probability as the square of the wave function, it is not very sensible to
take the real part of the exponential: the probability would be an oscillating function of x
for given t. If we take the complex function
, however, the
probability, defined as the absolute value squared, is a constant (A2 ) independent of
x and t, which is very sensible for a beam of particles. Thus we conclude that the wave
function (x,t) is complex, and the probability density is (x,t) 2.
Using de Broglie's relation for p and also use E = h.
One of the important goals of quantum mechanics is to generalise classical mechanics.
We shall attempt to generalise the relation between momenta and energy,
to the quantum realm. Notice that (just use the property of the differential of an
exponential)
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Using this we can guess a wave equation of the form
Actually using the definition of energy when the problem includes a potential (for our
case it will be the Coulomb potential)
(when expressed in momenta, this quantity is usually called a "Hamiltonian") we find the
time-dependent Schrödinger equation
We will only spend limited time on this equation. Initially we are interested in the timeindependent Schrödinger equation, where the probability (x,t) 2 is independent of
time. In order to reach this simplification, we find that (x,t) must have the form
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If we substitute this in the time-dependent equation, we get (using the product rule for
differentiation)
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Taking away the common factor
we have an equation for that no longer
contains time, the time-indepndent Schrödinger equation
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The corresponding solution to the time-dependent equation is the standing wave
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Notice that in deriving the wave equation we replaced the number or by a differential
acting on the wave function. The energy (or rather the Hamiltonian) was replaced by an
"operator", which when multiplied with the wave function gives a combination of
derivatives of the wave function and function multiplying the wave function, symbolically
written as
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This appearance of operators (often denoted by hats) where we were used to see
numbers is one of the key features of quantum mechanics.
The boundary conditions on this equation, which we just state here, are that
1. (x) is a continuous function, and is single valued.
2.  (x,t) 2dx must be finite, so that the normalized quantity
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is a probability density.
3. (x)/x is continuous except where V(x) has an infinite discontinuity (we will use
this when solving the particle in the box problem).
OK, now we see that there is an equation to solve but we will just presume we have
the solutions.
There is another very useful kind of information contained in . Recalling that its value
depends on the location in space with respect to the nucleus of the atom, the square of
this function ||2, evaluated at any given point, represents the probability of finding the
electron at that particular point. Although the electron remains a particle having a definite
charge and mass, and the question of "where" it is located is no longer meaningful. Any
single experimental observation will reveal a definite location for the electron, but this will
in itself have little significance; only a large number of such observations (similar to a
series of multiple exposures of a photographic film) will yield meaningful results which
will show that the electron can "be" anywhere with at least some degree of probability.
This does not mean that the electron is "moving around" to all of these places, but that
(in accord with the uncertainty principle) the concept of location has limited meaning for
a particle as small as the electron. If we count only those locations in space at which the
probability of the electron manifesting itself exceeds some arbitrary value, we find that
the Ψ function defines a definite three-dimensional region which we call an orbital.
The region near the nucleus can be thought of as an extremely small funnel-shaped box,
the walls of which correspond to the electrostatic attraction that must be overcome if an
electron confined within this region is to escape. As an electron is drawn toward the
nucleus by electrostatic attraction, the volume to which it is confined diminishes rapidly.
Because its location is now more precisely known, its kinetic energy must become more
uncertain; the electron's kinetic energy rises more rapidly than its potential energy falls,
so that it gets ejected back into one of its allowed values of n.
The electron well. The red circles show the average distance of the electron from the
nucleus for the allowed quantum levels (standing wave patterns) of n=1 through n=3. As
n decreases the potential energy of the system becomes more negative and the electron
becomes more confined in space. According to the uncertainty principle, this increases
the momentum of the electron, and hence its kinetic energy. The latter acts as a kind of
"confinement energy" that restores the electron to one of the allowed levels.
We can also dispose of the question of why the orbiting electron does not radiate its
kinetic energy away as it revolves around the nucleus. The Schrödinger equation
completely discards any concept of a definite path or trajectory of a particle; what was
formerly known as an "orbit" is now an "orbital", defined as the locations in space at
which the probability of finding the electrons exceeds some arbitrary value. It should be
noted that this wavelike character of the electron coexists with its possession of a
momentum, and thus of an effective velocity, even though its motion does not imply the
existence of a definite path or trajectory that we associate with a more massive particle.
Orbitals (again)
The modern view of atomic structure dismisses entirely the old but comfortable planetary
view of electrons circling around the nucleus in fixed orbits. As so often happens in
science, however, the old outmoded theory contains some elements of truth that are
retained in the new theory. In particular, the old Bohr orbits still remain, albeit as
spherical shells rather than as two-dimensional circles, but their physical significance is
different: instead of defining the "paths" of the electrons, they merely indicate the
locations in the space around the nucleus at which the probability of finding the electron
has higher values. The electron retains its particle-like mass and momentum, but
because the mass is so small, its wavelike properties dominate. The latter give rise to
patterns of standing waves that define the possible states of the electron in the atom.
The quantum numbers
Modern quantum theory tells us that the various allowed states of existence of the
electron in the hydrogen atom correspond to different standing wave patterns. In the
preceding lesson we showed examples of standing waves that occur on a vibrating
guitar string. The wave patterns of electrons in an atom are different in two important
ways:
1. Instead of indicating displacement of a point on a vibrating string, the electron waves
represent the probability that an electron will manifest itself (appear to be located) at any
particular point in space. (Note carefully that this is not the same as saying that "the
electron is smeared out in space"; at any given instant in time, it is either at a given point
or it is not.)
2. The electron waves occupy all three dimensions of space, whereas guitar strings
vibrate in only two dimensions.
Aside from this, the similarities are striking. Each wave pattern is identified by an integer
number n, which in the case of the atom is known as the principal quantum number.
The value of n tells how many peaks of amplitude (antinodes) exist in that particular
standing wave pattern; the more peaks there are, the higher the energy of the state.
The three simplest orbitals of the hydrogen atom are depicted above in pseudo-3D, in
cross-section, and as plots of probability (of finding the electron) as a function of
distance from the nucleus. The average radius of the electron probability is shown by the
blue circles or plots in the two columns on the right. These radii correspond exactly to
those predicted by the Bohr model.
Physical significance of n
The potential energy of the electron is given by the formula
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in which e is the charge of the electron, m is its mass, h is Planck's constant, and n is the
principal quantum number. The negative sign ensures that the potential energy is always
negative. Notice that this energy in inversely proportional to the square of n, so that the
energy rises toward zero as n becomes very large, but it can never exceed zero.
This formula was actually part of Bohr's original theory (compare with the formula given
above), and is still applicable to the hydrogen atom, although not to atoms containing
two or more electrons. In the Bohr model, each value of n corresponded to an orbit of a
different radius. The larger the orbital radius, the higher the potential energy of the
electron; the inverse square relationship between electrostatic potential energy and
distance is reflected in the inverse square relation between the energy and n in the
above formula. Although the concept of a definite trajectory or orbit of the electron is no
longer tenable, the same orbital radii that relate to the different values of n in Bohr's
theory now have a new significance: they give the average distance of the electron from
the nucleus. As you can see from the figure, the averaging process must encompass
several probability peaks in the case of higher values of n. The spatial distribution of
these probability maxima defines the particular orbital.
This physical interpretation of the principal quantum number as an index of the
average distance of the electron from the nucleus turns out to be extremely useful
from a chemical standpoint, because it relates directly to the tendency of an atom
to lose or gain electrons in chemical reactions.
The angular momentum quantum number
The electron wave functions that are derived from Schrödinger's theory are
characterized by several quantum numbers. The first one, n, describes the nodal
behavior of the probability distribution of the electron, and correlates with its potential
energy and average distance from the nucleus as we have just described.
The theory also predicts that orbitals having the same value of n can differ in shape and
in their orientation in space. The quantum number l, known as the angular momentum
quantum number, determines the shape of the orbital. (More precisely, l determines the
number of angular nodes, that is, the number of regions of zero probability encountered
in a 360° rotation around the center.)
When l = 0, the orbital is spherical in shape. If l = 1, the orbital is elongated into
something resembling a figure-8 shape, and higher values of l correspond to still more
complicated shapes— but note that the number of peaks in the radial probability
distributions (below) decreases with increasing l. The possible values that l can take are
strictly limited by the value of the principal quantum number; l can be no greater than n –
1. This means that for n = 1, l can only have the single value zero which corresponds to
a spherical orbital. For historical reasons, the orbitals corresponding to different values
of l are designated by letters, starting with s for l = 0, p for l = 1, d for l = 2, and f for l = 3.
The shapes and radial distributions of the orbitals corresponding to the three allowed
values of l for the n = 3 level of hydrogen are shown above. Notice that the average
orbital radius r decreases somewhat at higher values of l. The function relationship is
given by
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in which z is the nuclear charge of the atom, which of course is unity for hydrogen.
The magnetic quantum number
An s-orbital, corresponding to l = 0, is spherical in shape and therefore has no special
directional properties. The probability cloud of a p orbital is aligned principally along an
axis extending along any of the three directions of space. The additional quantum
number m is required to specify the particular direction along which the orbital is aligned.
"Direction in space" has no meaning in the absence of a force field that serves to
establish a reference direction. For an isolated atom there is no such external field, and
for this reason there is no distinction between the orbitals having different values of m. If
the atom is placed in an external magnetic or electrostatic field, a coordinate system is
established, and the orbitals having different values of m will split into slightly different
energy levels. This effect was first seen in the case of a magnetic field, and this is the
origin of the term magnetic quantum number. In chemistry, however, electrostatic
fields are much more important for defining directions at the atomic level because
it is through such fields that nearby atoms in a molecule interact with each other.
The electrostatic field created when other atoms or ions come close to an atom
can cause the energies of orbitals having different direction properties to split up
into different energy levels; this is the origin of the colors seen in many inorganic salts
of transition elements, such as the blue color of copper sulfate.
The quantum number m can assume 2l + 1 values for each value of l, from –l through 0
to +l. When l = 0 the only possible value of m will also be zero, and for the p orbital
(l = 1), m can be –1, 0, and +1. Higher values of l introduce more complicated orbital
shapes which give rise to more possible orientations in space, and thus to more values
of m.
Electron spin and the exclusion principle
Certain fundamental particles have associated with them a magnetic moment that can
align itself in either of two directions with respect to an external magnetic field. The
electron is one such particle, and the direction of its magnetic moment is called its spin.
The mechanical analogy implied by the term spin is easy to visualize, but should not be
taken literally. Physical rotation of an electron is meaningless. However, the coordinates
of the electron's wave function can be rotated mathematically. Electron spin is basically
a relativistic effect in which the electron's momentum distorts local space and time. It has
no classical counterpart and thus cannot be visualized other than through mathematics.
Today it is believed that for particles such as electrons that possess half-integral values
of spin, no two of them can be in identical quantum states within the same system. The
quantum state of a particle is defined by the values of its quantum numbers, so what this
means is that no two electrons in the same atom can have the same set of quantum
numbers. This is known as the Pauli exclusion principle, named after the German
physicist Wolfgang Pauli (1900-1958, Nobel Prize 1945). In the last section of this
reading I will argue that there would probably no complex chemistry without the
exclusion principle.
The best non-mathematical explanation of the exclusion principle that I have come
across is Phil Fraundorf's Candle Dances and Atoms page at U. Missouri-St. Louis,
which appears to be a version of an explanation given by Richard Feynmann.
The exclusion principle was discovered empirically and was placed on a firm theoretical
foundation by Pauli in 1925. A complete explanation requires some familiarity with
quantum mechanics, so all we will say here is that if two electrons possess the same
quantum numbers n, l, m and s (defined elsewhere, in case you are not familiar with
them), the wave function that describes the state of existence of the two electrons
together becomes zero, which means that this is an "impossible" situation.
A given orbital is characterized by a fixed set of the quantum numbers n, l, and m. The
electron spin itself constitutes a fourth quantum number s, which can take the two values
+1 and –1. Thus a given orbital can contain two electrons having opposite spins,
which "cancel out" to produce zero magnetic moment. Two such electrons in a single
orbital are often referred to as an electron pair.
You can find out more a little more detail on the Pauli exclusion principle from
Wikipdedia or Hyperphysics.com. [Physics students: Have you ever seen it derived from
first principles? Have you ever seen anything derived from “first principles”?] For a more
rigorous look at quantum mechanics, there are many options, a book still probably being
the best. Cheaper, and good, online accounts can be found at several sites. One is the
2005 course by Niels Walet, Univ. Manchester at
http://walet.phy.umist.ac.uk/QM/LectureNotes/ which works from the beginning up to the
easy potentials to scattering to the hydrogen atom.
Concluding Discussion
Given the basic features of quantum mechanics: What part of the hierarchy of
chemical complexity really does depend on the existence of the quantum nature
of the microscopic world?
To what extent do the properties of mesoscopic objects (that’s what I’m calling
everything from protoplanetary grains to biomacomolecules and supramolecular
assemblies) depend on quantum effects? In particular, it is the hierarchical layering of
structure from monomers, to polymers, through the stages of macromolecular structure,
to supramolecular assemblies, that constitute not only biology but the entire macroscopic
world (if you think about it a while).
Here are some simple-minded ways to answer this, both problematic. There are others-if you have one, please let me know.
First: I think it is clear that if energy levels and other microscopic properties (that we refer
to as their mechanical analogues, like angular momentum) were not quantized, that is,
discrete, the world would be a mess. If electrons could be at any energy possible, even
with larger probability for some than others, there would be no identifiable units of
chemical complexity, from atoms, molecules, polymers, ... This progression of
hierarchical structure, perhaps most amazingly clear in the structure of proteins, but
everywhere else too, owes the discreteness of its levels of complexity to the
discreteness of the quantum world. I would like to know if you have come across other
speculations about this question. You should think about how a lot of physics and
macroscopic phenomena does not depend on this discreteness at all, but other
mysterious features of the microscopic world that are beyond our understanding.
As an example: The equations of fluid mechanics, also called hydrodynamics, or of
magnetohydrodynamics (MHD), can be derived an easy way (conserve this in a volume,
conserve that in a volume) that makes it clear that the momentum equation is only a
form of Newton’s second law, etc., but the rigorous derivation (the one that really justifies
why the pressure tensor can be written as an isotropic part and an anisotropic part that
must satisfy certain criteria...) shows that it really depends entirely on the existence of
the conservation properties of the Boltzmann equation, which itself is a conundrum
because it is where an arrow of time suddenly appears in physics (Boltzmann committed
suicide, partly over this problem). I would claim that nowhere in that rigorous derivation
can you find a trace of quantum theory, even if you derive higher-order corrections to the
equations of fluid flow (they are not exact).
So not everything in the world depends on the discreteness of the microscopic
world. But discreteness of chemical levels of organization seems to.
A problem with this is whether or not physics actually derives the discreteness of
the quantum world, or whether, frustrated by observations of hydrogen lines, Bohr stuck
out his neck and made an outlandish suggestion only to fit the data; in fact there is some
question whether he thought it was “real.” We will see the same thing below, concerning
the exclusion principle. Perhaps you have studied special relativity and noticed the
same sort of thing (Einstein wondering, “What if...” and then everything falling in place;
or was it his wife that thought of that while she was correcting his math errors?)
SECOND: I would say, and probably others would agree, that
If it were not for the exclusion principle, the atoms of all elements would behave in
the same way, and there would be no chemistry!
This is of such direct importance for us that you should understand it clearly. As we
have seen, the lowest-energy standing wave pattern the electron can assume in an atom
corresponds to n=1, which describes the state of the single electron in hydrogen, and of
the two electrons in helium. Since the quantum numbers m and l are zero for n=1, the
pair of electrons in the helium orbital have the values (n, l, m, s) = (1,0,0,+1) and (1,0,0,–
1)— that is, they differ only in spin. These two sets of quantum numbers are the only
ones that are possible for a n=1 orbital. The additional electrons in atoms beyond helium
must go into higher-energy (n>1) orbitals. Electron wave patterns corresponding to these
greater values of n are concentrated farther from the nucleus, with the result that these
electrons are less tightly bound to the atom and are more accessible to interaction with
the electrons of neighboring atoms, thus influencing their chemical behavior.
If it were not for the Pauli exclusion principle, all the electrons of every element
would be in the lowest-energy n=1 state, and the differences in the chemical
behavior the different elements would be minimal. Chemistry would certainly be a
simpler subject, but it would not be very interesting!
In particular, there is virtually no chance that the kinds of bonds between different atomic
orbitals that give rise to organic chemistry and life would exist without the Pauli principle.
So at this level we can say specifically how the origin of life depends on quantum
mechanics.
But there is are less severe constraints imposed by quantum mechanics, and what we
want to know is, given the Pauli principle and the existence of molecules, what is the
importance of, say, the deBroglie wavelength by the time we get to something the size of
a protein or nucleic acid or dust particle? See if you can guess the deBroglie
wavelength of a macromolecule just by thinking about their speeds. Where does
quantum mechanics play a role then, besides the two possibly profound instances of
discrete states and the exclusion principle? Certainly in allowing only certain vibrational
states and transitions in the monomers that make up the macromolecules, but this is at
the molecular level where we expect such quantum effects.
I will give you one example, which has to do with the mobility of electrons on linear
biopolymers, in our next problem set (I will be outlining it in class--it is pretty elementary).
See if you can think of any others yourself, for any kind of mesoscopic particle. If they
are minimal, then can we conclude that all we have to do is treat the macromolecule like
some complex collection of dipoles and induced dipoles, throw in whatever other
intermolecular forces are at work, add up all the forces between all the molecular
building blocks, and minimize the total energy to find, say, the lowest-energy folding
conformation of a protein? People have been trying this for many years and now
basically admit that it is not soluble, but does this have anything to do with quantum
mechanics, or is it simply that it is a combinatorial nightmare, with so many possible
ways to fold and rotate at each amino acid “hinge” that no computer in the near future
will have the computational power to solve such a problem? We will look at this later in
the course.
If the complex properties and behaviors of organic macromolecules are not due
fundamentally to quantum effects, yet the particles are too small to be part of the
“macroscopic” world, then to what should we attribute these properties?
To partially answer this, think about the properties of gravitational systems. The twobody problem can be solved, the three- and higher- body problems can’t (except with
severe approximations) because they are so nonlinear that they introduce chaotic
behavior and also complex resonance effects, but at least we can integrate the
equations of motion on a computer and get an accurate answer. For a system with a
billion gravitating particles, we can appeal to theories stolen from the atomic world, like
statistical mechanics, kinetic theory, fluid mechanics. But this is at the “macroscopic”
level. The case that is most like ours might be a star cluster, and indeed, the evolution
of a globular cluster remains challenging even for the largest specialized computers.
However a little thought will show you why globular clusters are not going to show any
complex behavior like biomolecules do: The gravitational field has no dipoles, because it
has only one “charge,” attractive.
Think about what a huge difference the + - nature of the electrostatic interaction makes,
from the structure of the atoms and molecules to the weak forces that dominate
supramolecular aggregates.
[You physics majors: Does the existence of two signs for the Coulomb force emerge in
any way from quantum theory? At a fundamental level, from what does it derive? Is
there something conserved for electromagnetic fields that is not conserved for
gravitational fields? Do they exist in our theories only because it was obvious from
experiments that there were two signs, so of course it had to emerge from the theory? At
what point?]
Yet we still have not really answered our question: So the polarity of the electric charge
is a major difference, and that gives rise to molecular binding and the rest. Why do
some mesoscopic entities behave like DNA or even lipids, while others behave like
microscopic rocks--and are we sure about the latter? More to come.