Failures of Classical Physics

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Failures of Classical Physics Some
experimental situations where "classical" physics
fails
Photoelectric effect
Blackbody radiation
Wave properties of electron
Line spectra
The remedies come from some inherently quantum
ideas

Photon energy Photon momentum
Wavelength for particle Uncertainty principle
Wave function Schrödinger equation
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Blackbody Radiation
Blackbody radiation" or "cavity radiation" refers
to an object or system which absorbs all
radiation incident upon it and re-radiates energy
which is characteristic of this radiating system
only, not dependent upon the type of radiation
which is incident upon it. The radiated energy
can be considered to be produced by standing wave
or resonant modes of the cavity which is
radiating.
The amount of radiation emitted in a given
frequency range should be proportional to the
number of modes in that range. The best of
classical physics suggested that all modes had an
equal chance of being produced, and that the
number of modes went up proportional to the
square of the frequency.
But the predicted continual increase in radiated
energy with frequency (dubbed the "ultraviolet
catastrophe") did not happen. Nature knew better.
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The birth of Quantum Mechanics
According to the Planck hypothesis, all
electromagnetic radiation is quantized and occurs
in finite quanta" of energy which we call
photons. The quantum of energy for a photon is
not Planck's constant h itself, but the product
of h and the frequency. The quantization implies
that a photon of blue light of given frequency or
wavelength will always have the same size quantum
of energy. For example, a photon of blue light of
wavelength 450 nm will always have 2.76 eV of
energy. It occurs in quantized chunks of 2.76 eV,
and you can't have half a photon of blue light -
it always occurs in precisely the same sized
energy chunks.
But the frequency available is continuous and has
no upper or lower bound, so there is no finite
lower limit or upper limit on the possible energy
of a photon. On the upper side, there are
practical limits because you have limited
mechanisms for creating really high energy
photons. Low energy photons abound, but when you
get below radio frequencies, the photon energies
are so tiny compared to room temperature thermal
energy that you really never see them as distinct
quantized entities - they are swamped in the
background. Another way to say it is that in the
low frequency limits, things just blend in with
the classical treatment of things and a quantum
treatment is not necessary.
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Blackbody Intensity as a Function of Frequency
Energy per unit volume per unit frequency
k Boltzman constant
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Rayleigh-Jeans vs Planck
Comparison of the classical Rayleigh-Jeans Law
and the quantum Planck radiation formula.
Experiment confirms the Planck relationship
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Radiation Curves
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The Photoelectric Effect
The remarkable aspects of the photoelectric
effect when it was first observed were
1. The electrons were emitted immediately - no
time lag!
2. Increasing the intensity of the light
increased the number of photoelectrons, but not
their maximum kinetic energy!
  
3. Red light will not cause the ejection of
electrons, no matter what the intensity!
4. A weak violet light will eject only a few
electrons, but their maximum kinetic energies are
greater than those for intense light of longer
wavelengths!
The details of the photoelectric effect were in
direct contradiction to the expectations of very
well developed classical physics.
The explanation marked one of the major steps
toward quantum theory.
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The Photoelectric Effect
Analysis of data from the photoelectric
experiment showed that the energy of the ejected
electrons was proportional to the frequency of
the illuminating light. This showed that whatever
was knocking the electrons out had an energy
proportional to light frequency. The remarkable
fact that the ejection energy was independent of
the total energy of illumination showed that the
interaction must be like that of a particle which
gave all of its energy to the electron! This fit
in well with Planck's hypothesis that light in
the blackbody radiation experiment could exist
only in discrete bundles with energy
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Most commonly observed phenomena with light can
be explained by waves. But the photoelectric
effect suggested a particle nature for light.
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The Line Spectrum Problem
In the years before the beginning of the 20th
century, the light emitted from luminous gases
was found to consist not of a continuous range of
wavelengths, but of discrete colours which were
different for different gases. These spectral
"lines" formed regular series and came to be
interpreted as transitions between atomic energy
levels. This presented a considerable problem for
classical physics, because accelerated charges
were known to radiate electromagnetic energy. It
was expected that orbits of electrons about
positive nuclei would be unstable because they
would radiate energy and therefore spiral into
the nucleus. No classical model could be found
which would yield stable electron orbits. The
Bohr model of the atom started the progress
toward a modern theory of the atom with its
postulate that angular momentum is quantized,
giving only specific allowed energies. Then the
development of the quantum theory and the
Schrödinger equation refined the picture of the
energy levels of atomic electrons.
Helium spectrum
Hydrogen spectrum
integers
RH Rydberg constant 1.097 107 m1
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Bohr atomic model - Classical Electron Orbit
In the Bohr theory, this classical result was
combined with the quantization of angular
momentum to get an expression for quantized
energy levels.
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Angular Momentum Quantization
In the Bohr model, the wavelength associated with
the electron is given by the DeBroglie
relationship
and the standing wave condition that
circumference whole number of wavelengths.
These can be combined to get an expression for
the angular momentum of the electron in orbit.

Thus L is not only conserved, but constrained to
discrete values by the quantum number n. This
quantization of angular momentum is a crucial
result and can be used in determining the Bohr
orbit radii and Bohr energies.
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Combining the energy of the classical electron
orbit with the quantization of angular momentum,
the Bohr approach yields expressions for the
electron orbit radii and energies
substitution for r gives the Bohr energies and
radii
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Hydrogen Energy Levels
The basic hydrogen energy level structure is in
agreement with the Bohr model. Common pictures
are those of a shell structure with each main
shell associated with a value of the principal
quantum number n.
This Bohr model picture of the orbits has some
usefulness for visualization so long as it is
realized that the "orbits" and the "orbit radius"
just represent the most probable values of a
considerable range of values.
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The Bohr model for an electron transition in
hydrogen between quantized energy levels with
different quantum numbers n yields a photon by
emission, with quantum energy
This is often expressed in terms of the inverse
wavelength or "wave number" as follows
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Failures of the Bohr Model
While the Bohr model was a major step toward
understanding the quantum theory of the atom, it
is not in fact a correct description of the
nature of electron orbits. Some of the
shortcomings of the model are 1. It fails to
provide any understanding of why certain spectral
lines are brighter than others. There is no
mechanism for the calculation of transition
probabilities. 2. The Bohr model treats the
electron as if it were a miniature planet, with
definite radius and momentum. This is in direct
violation of the uncertainty principle which
dictates that position and momentum cannot be
simultaneously determined. The Bohr model gives
us a basic conceptual model of electrons orbits
and energies. The precise details of spectra and
charge distribution must be left to quantum
mechanical calculations, as with the Schrödinger
equation.
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Particle nature of light - Compton Scattering
Arthur H. Compton observed the scattering of
x-rays from electrons in a carbon target and
found scattered x-rays with a longer wavelength
than those incident upon the target. The shift of
the wavelength increased with scattering angle
according to the Compton formula
Compton explained and modeled the data by
assuming a particle (photon) nature for light and
applying conservation of energy and conservation
of momentum to the collision between the photon
and the electron. The scattered photon has lower
energy and therefore a longer wavelength
according to the Planck relationship.
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The above expression for ?? can be obtained by
exploting energy-momentum conservation
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Wave Nature of Electron
As a young student at the University of Paris,
Louis DeBroglie had been impacted by relativity
and the photoelectric effect, both of which had
been introduced in his lifetime. The
photoelectric effect pointed to the particle
properties of light, which had been considered to
be a wave phenomenon. He wondered if electons and
other "particles" might exhibit wave properties.
The application of these two new ideas to light
pointed to an interesting possibility
Confirmation of the DeBroglie hypothesis came in
the Davisson- Germer experiment which showed
interference patterns in agreement with
DeBroglie wavelength for the scattering of
electrons on nickel crystals.
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When x-rays are scattered from a crystal lattice,
peaks of scattered intensity are observed which
correspond to the following conditions The angle
of incidence angle of scattering. The
pathlength difference is equal to an integer
number of wavelengths. The condition for maximum
intensity contained in Bragg's law above allow us
to calculate details about the crystal structure,
or if the crystal structure is known, to
determine the wavelength of the x-rays incident
upon the crystal.
The Davisson-Germer experiment showed that
electrons exhibit the DeBroglie wavelength given
by
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DeBroglie Wavelengths

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Wave-Particle Duality Light
Does light consist of particles or waves? When
one focuses upon the different types of phenomena
observed with light, a strong case can be built
for a wave picture
Phenomenon Can be explained in terms of waves. Can be explained in terms of particles.
Reflection
Refraction
Interference
Diffraction
Polarization
Photoelectric effect
Compton scattering
Most commonly observed phenomena with light can
be explained by waves. But the photoelectric
effect and the Compton scatering suggested a
particle nature for light. Then electrons too
were found to exhibit dual natures.
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Wavefunction Properties
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Schrödinger Equation
The Schrödinger equation plays the role of
Newton's laws and conservation of energy in
classical mechanics - i.e., it predicts the
future behavior of a dynamic system. It is a wave
equation in terms of the wavefunction which
predicts analytically and precisely the
probability of events or outcome. The detailed
outcome is not strictly determined, but given a
large number of events, the Schrödinger equation
will predict the distribution of results.
The kinetic and potential energies are
transformed into the Hamiltonian which acts upon
the wavefunction to generate the evolution of the
wavefunction in time and space. The Schrödinger
equation gives the quantized energies of the
system and gives the form of the wavefunction so
that other properties may be calculated.
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Time-independent Schrödinger Equation
For a generic potential energy U the
1-dimensional time-independent Schrodinger
equation is
In three dimensions, it takes the form
for cartesian coordinates. This can be written in
a more compact form by making use of the
Laplacian operator
The Schrodinger equation can then be written
H? E?
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Time Dependent Schrödinger Equation
The time dependent Schrödinger equation for one
spatial dimension is of the form
For a free particle where U(x) 0 the
wavefunction solution can be put in the form of a
plane wave
For other problems, the potential U(x) serves to
set boundary conditions on the spatial part of
the wavefunction and it is helpful to separate
the equation into the time-independent
Schrödinger equation and the relationship for
time evolution of the wavefunction
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The Postulates of Quantum Mechanics
1. The Wavefunction Postulate
Associated with any particle moving in a
conservative field of force is a wave function
which determines everything that can be known
about the system.
It is one of the postulates of quantum mechanics
that for a physical system consisting of a
particle there is an associated wavefunction.
This wavefunction determines everything that can
be known about the system. The wavefunction may
be a complex function, since it is its product
with its complex conjugate which specifies the
real physical probability of finding the particle
in a particular state.
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Probability in Quantum Mechanics
The wavefunction represents the probability
amplitude for finding a particle at a given point
in space at a given time. The actual probability
of finding the particle is given by the product
of the wavefunction with it's complex conjugate
(like the square of the amplitude for a complex
function).
Since the probability must be 1 for finding the
particle somewhere, the wavefunction must be
normalized. That is, the sum of the probabilities
for all of space must be equal to one. This is
expressed by the integral

infinitesimal volume
Part of a working solution to the Schrodinger
equation is the normalization of the solution to
obtain the physically applicable probability
amplitudes
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2. The Operator Postulate
With every physical observable q there is
associated an operator Q, which when operating
upon the wavefunction associated with a definite
value of that observable will yield that value
times the wavefunction.
With every physical observable there is
associated a mathematical operator which is used
in conjunction with the wavefunction. Suppose the
wavefunction associated with a definite quantized
value (eigenvalue) of the observable is denoted
by ?n (an eigenfunction) and the operator is
denoted by Q. The action of the operator is given
by
The mathematical operator Q extracts the
observable value qn by operating upon the
wavefunction which represents that particular
state of the system. This process has
implications about the nature of measurement in a
quantum mechanical system. Any wavefunction for
the system can be represented as a linear
combination of the eigenfunctions ?n (see basis
set postulate), so the operator Q can be used to
extract a linear combination of eigenvalues
multiplied by coefficients related to the
probability of their being observed (see
expectation value postulate).
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Operators in Quantum Mechanics
Associated with each measurable parameter in a
physical system is a quantum mechanical operator.
Such operators arise because in quantum mechanics
you are describing nature with waves (the
wavefunction) rather than with discrete particles
whose motion and dynamics can be described with
the deterministic equations of Newtonian physics.
Part of the development of quantum mechanics is
the establishment of the operators associated
with the parameters needed to describe the
system. Some of those operators are listed below.
It is part of the basic structure of quantum
mechanics that functions of position are
unchanged in the Schrödinger equation, while
momenta take the form of spatial derivatives. The
Hamiltonian operator contains both time and space
derivative.
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3. Hermitian Property Postulate
Any operator Q associated with a physically
measurable property q will be Hermitian.
The quantum mechanical operator Q associated with
a measurable property q must be Hermitian.
Mathematically this property is defined by
where ?a and ?b are arbitrary normalizable
functions and the integration is over all of
space. Physically, the Hermitian property is
necessary in order for the measured values
(eigenvalues) to be constrained to real numbers.
if Q is Hermitian, then all qi are real numbers
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4. Basis Set Postulate
The set of eigenfunctions of Hermitian operators
Q will form a complete set of linearly
independent functions.
The set of functions ?j which are eigenfunctions
of the eigenvalue equation
form a complete set of linearly independent
functions. They can be said to form a basis set
in terms of which any wavefunction representing
the system can be expressed
This implies that any wavefunction ? representing
a physical system can be expressed as a linear
combination of the eigenfunctions of any physical
observable of the system.
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5. Expectation Value Postulate
For a system described by a given wavefunction,
the expectation value of any property q can be
found by performing the expectation value
integral with respect to that wavefunction.
For a physical system described by a wavefunction
?, the expectation value of any physical
observable q can be expressed in terms of the
corresponding operator Q as follows
It is presumed here that the wavefunction is
normalized and that the integration is over all
of space. This postulate follows along the lines
of the operator postulate and the basis set
postulate. The function can be represented as a
linear combination of eigenfunctions of Q, and
the results of the operation gives the physical
values times a probability coefficient. Since the
wavefunction is normalized, the integral gives a
weighted average of the possible observable
values.
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A physical system is described by the wave
function ?, which can always be written as a
linear combination of the eigenfunctions of a
Hermitian operator Q
A measure of Q for the state ? will give as a
result any of its eigenvalues qn, each with a
probability cn2, so that
The normalization condition of the wavefunction
implies that
A measurement of Q forces the system to be in one
of the eigenstates, ?n, of Q any subsequent
measure of Q will give the result qn
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6. Time Evolution Postulate
The time evolution of the wavefunction is given
by the time dependent Schrödinger equation
If ?(x,y,z t) is the wavefunction for a physical
system at an initial time and the system is free
of external interactions, then the evolution in
time of the wavefunction is given by
where H is the Hamiltonian operator formed from
the classical Hamiltonian by substituting for the
classical observables their corresponding quantum
mechanical operators. The role of the Hamiltonian
in both space and time is contained in the
Schrödinger equation.
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1. Associated with any particle moving in a
conservative field of force is a wave function
which determines everything that can be known
about the system.
2. With every physical observable q there is
associated an operator Q, which when operating
upon the wavefunction associated with a definite
value of that observable will yield that value
times the wavefunction.
3. Any operator Q associated with a physically
measurable property q will be Hermitian
4. The set of eigenfunctions of each Hermitian
operator Q will form a complete set of linearly
independent functions.
5. For a system described by a given
wavefunction, the expectation value of any
property q can be found by performing the
expectation value integral with respect to that
wavefunction
6. The time evolution of the wavefunction is
given by the time dependent Schrödinger equation.
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Free particle approach to the Schrödinger equation
Though the Schrodinger equation cannot be
derived, it can be shown to be consistent with
experiment. The most valid test of a model is
whether it faithfully describes the real world.
The wave nature of the electron has been clearly
shown in experiments like the Davisson-Germer
experiment. This raises the question "What is the
nature of the wave?". We reply, in retrospect,
that the wave is the wavefunction for the
electron. Starting with the expression for a
traveling wave in one dimension, the connection
can be made to the Schrödinger equation. This
process makes use of the deBroglie relationship
between wavelength and momentum and the Planck
relationship between frequency and energy.
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It is easier to show the relationship to the
Schrödinger equation by generalizing this
wavefunction to a complex exponential form using
the Euler relationship. This is the standard form
for the free particle wavefunction.
One can check that ? is eigenunction of momentum
and energy operators
The connection to the Schrodinger equation can be
made by examining wave and particle expressions
for energy
Asserting the equivalence of these two
expressions for energy and putting in the quantum
mechanical operators for both brings us to the
Shrödinger equation
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The Uncertainty Principle
The position and momentum of a particle cannot be
simultaneously measured with arbitrarily high
precision. There is a minimum for the product of
the uncertainties of these two measurements.
There is likewise a minimum for the product of
the uncertainties of the energy and time
This is not a statement about the inaccuracy of
measurement instruments, nor a reflection on the
quality of experimental methods it arises from
the wave properties inherent in the quantum
mechanical description of nature. Even with
perfect instruments and technique, the
uncertainty is inherent in the nature of things.
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Uncertainty Principle
Important steps on the way to understanding the
uncertainty principle are wave-particle duality
and the DeBroglie hypothesis. As you proceed
downward in size to atomic dimensions, it is no
longer valid to consider a particle like a hard
sphere, because the smaller the dimension, the
more wave-like it becomes. It no longer makes
sense to say that you have precisely determined
both the position and momentum of such a particle.
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The exact definition of ?x and ?p is
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Particle Confinement
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Confinement Calculation
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The Hydrogen Atom
The solution of the Schrodinger equation for the
hydrogen atom is better achieved by using
spherical polar coordinates and by separating the
variables so that the wavefunction is represented
by the product
The separation leads to three equations for the
three spatial variables, and their solutions give
rise to three quantum numbers associated with the
hydrogen energy levels.
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Quantum Numbers from Hydrogen Equations
The hydrogen atom solution requires finding
solutions to the separated equations which obey
the constraints on the wavefunction. The solution
to these equations can exist only when a few
constants which arise in the solution are
restricted to integer values. This gives the
hydrogen atom quantum numbers
n principal quantum number
l orbital quantum number
ml magnetic quantum number
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Vector Model for Orbital Angular Momentum
The orbital angular momentum for an atomic
electron can be visualized in terms of a vector
model where the angular momentum vector is seen
as precessing about a direction in space. While
the angular momentum vector has the magnitude
shown, only a maximum of l units of h can be
measured along a given direction, where l is the
orbital quantum number.
  
While called a "vector", the orbital angular
momentum in Quantum Mechanics is a special kind
of vector because its projection along a
direction in space is quantized to values one
unit of angular momentum (h) apart. The diagram
shows that the possible values for the "magnetic
quantum number" ml (for l 2), can take the values

   -2, -1, 0, 1, 2
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Electron Spin
An electron spin s 1/2 is an intrinsic property
of electrons. In addition to orbital angular
momentum electrons have intrinsic angular
momentum characterized by quantum number 1/2. In
the pattern of other quantized angular momenta
ms ½ spin up ms ½ spin down
Spin "up" and "down" allows two electrons for
each set of spatial quantum numbers
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Pauli Exclusion Principle
No two electrons in an atom can have identical
quantum numbers. This is an example of a general
principle which applies not only to electrons but
also to other particles of half-integer spin
(fermions). It does not apply to particles of
integer spin (bosons).
The nature of the Pauli exclusion principle can
be illustrated by supposing that electrons 1 and
2 are in states a and b respectively. The
wavefunction for the two electron system would be
but this wavefunction is unacceptable because the
electrons are identical and indistinguishable. To
account for this we must use a linear combination
of the two possibilities since the determination
of which electron is in which state is not
possible to determine.
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The wavefunction for the state in which both
states "a" and "b" are occupied by the electrons
can be written
The Pauli exclusion principle is part of one of
our most basic observations of nature particles
of half-integer spin must have antisymmetric
wavefunctions, and particles of integer spin must
have symmetric wavefunctions. The minus sign in
the above relationship forces the wavefunction to
vanish identically if both states are "a" or "b",
implying that it is impossible for both electrons
to occupy the same state.
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Pauli Exclusion Principle Applications
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Periodic Table of the Elements
The quantum numbers associated with the atomic
electrons along with the Pauli exclusion
principle provide insight into the building up of
atomic structures and the periodic properties
observed.
For a given principal number n there are n2
different possible states.
The order of filling of electron energy states is
dictated by energy, with the lowest available
state consistent with the Pauli principle being
the next to be filled. The labeling of the levels
follows the scheme of the spectroscopic notation
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Spectroscopic Notation
Before the nature of atomic electron states was
clarified by the application of quantum
mechanics, spectroscopists saw evidence of
distinctive series in the spectra of atoms and
assigned letters to the characteristic spectra.
In terms of the quantum number designations of
electron states, the notation is as follows
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Order of Filling of Electron States
As the periodic table of the elements is built up
by adding the necessary electrons to match the
atomic number, the electrons will take the lowest
energy consistent with the Pauli exclusion
principle. The maximum population of each shell
is determined by the quantum numbers and the
diagram at left is one way to illustrate the
order of filling of the electron energy states.
For a single electron, the energy is determined
by the principal quantum number n and that
quantum number is used to indicate the "shell" in
which the electrons reside. For a given shell in
multi-electron atoms, those electrons with lower
orbital quantum number l will be lower in energy
because of greater penetration of the shielding
cloud of electrons in inner shells. These energy
levels are specified by the principal and orbital
quantum numbers using the spectroscopic notation.
When you reach the 4s level, the dependence upon
orbital quantum number is so large that the 4s is
lower than the 3d. Although there are minor
exceptions, the level crossing follows the scheme
indicated in the diagram, with the arrows
indicating the points at which one moves to the
next shell rather than proceeding to higher
orbital quantum number in the same shell
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The division into main shells encourages a kind
of "planetary model" for the electrons, and while
this is not at all accurate as a description of
the electrons, it has a certain mnemonic value
for keeping track of the buildup of heavier
elements.
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The electron orbital configurations provide a
structure for understanding chemical reactions,
which are guided by the principle of finding the
lowest energy (most stable) configuration of
electrons. We say that sodium has a valence of 1
since it tends to lose one electron, and chlorine
has a valence of -1 since it has a tendency to
gain one electron. Both of these atoms are very
active chemically, and their combination is the
classic case of an ionic bond.
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Bose-Einstein Condensation
In 1924 Einstein pointed out that bosons could
"condense" in unlimited numbers into a single
ground state since they are governed by
Bose-Einstein statistics and not constrained by
the Pauli exclusion principle. Little notice was
taken of this curious possibility until the
anomalous behavior of liquid helium at low
temperatures was studied carefully. When helium
is cooled to a critical temperature of 2.17 K, a
remarkable discontinuity in heat capacity occurs,
the liquid density drops, and a fraction of the
liquid becomes a zero viscosity "superfluid".
Superfluidity arises from the fraction of helium
atoms which has condensed to the lowest possible
energy. A condensation effect is also credited
with producing superconductivity. In the BCS
Theory, pairs of electrons are coupled by lattice
interactions, and the pairs (called Cooper pairs)
act like bosons and can condense into a state of
zero electrical resistance