Untitled
11/19/04 6:13 PM
Matter is made of atoms
The atom of each element is characterized by its atomic mass
Atomic mass unit u is (mass of
C)/12 = 931.5 Mev/ .
1 mole contains
grams, and
and atomic number
.
molecules (Avagadro's constant).
An atom is electrically neutral, but
Thomson's and Milliken's experiments showed that part of the atom is an electron, of charge
C and mass 0.511 MeV/
. This mass is only 1/2000 u, so
Rutherford's analysis and Geiger-Marsden experiments showed that most of the mass is
concentrated in a nucleus, which also carries charge
to balance the charge of the electrons.
The principles of these experiments which probe atomic structure include:
``Cathode rays'' (electrons) emitted by heated metals.
Acceleration of electrons by electrical potential
, so that they acquire kinetic energy
Bending of charged particles in magnetic fields, so that
.
.
Scattering of energetic projectile particles from material targets reveals features of the target. e.g., In
Rutherford scattering large-angle alpha particle scatters reflect the force
, which is large when
is small.
Light is ``quantized''.
In the photoelectric effect, light shining on a photocathode creates an electrical current, by kicking
electrons from the cathode.
Although more electrons are emitted as the light intensity increases, the maximum energy of the
electrons depends only on the frequency of the light:
.
Planck's constant
is a small number:
J-s.
Conclusion is that the light's energy is bundled in quanta (photons) equal to
.
Indeed the Compton effect shows that electromagnetic radiation scatters from individual electrons like
massless particles. Energy and momentum conservation lead to
Atoms have quantized energy states. The planetary Bohr model accounts for many features of one-electron
atoms.
file://localhost/Users/noelclark/Desktop/2170%20quantum%20summary3.0.html
Page 1 of 6
Untitled
11/19/04 6:13 PM
The atom is bound by the Coulomb force between electron and nucleus
For circular orbits the centripital acceleration is
Thus for a given radius
.
.
the atom's energy is
The (positive) binding energy
is the energy required to separate the atomic constituents,
.
By classical electromagnetic theory the accelerating electron (
) radiates light.
This is consistent with the observation that atoms which have absorbed extra energy do emit light.
The energy of the light must be accounted for by a change in the energy of the atom.
Eventually the atom would lose so much energy by radiation that it would collapse. ``Eventually''
here means in about a nanosecond!
The observed radiation from atoms occurs in sharp lines of definite wavelength (or frequency or
energy).
The Bohr hypothesis is that the angular momentum of the orbiting electron is quantized:
where
.
Adding the Bohr quantization leads to
Generalizing to one-electron ions of atoms of atomic number
these formulas become
So now the energy of a photon emitted or absorbed as the atom changes its state from
file://localhost/Users/noelclark/Desktop/2170%20quantum%20summary3.0.html
to
is
Page 2 of 6
Untitled
11/19/04 6:13 PM
where the Rydberg constant
The spectral lines for
Those for
.
are in the Lyman series.
are in the Balmer series, of which the first four are in the visible part of
the spectrum.
The spectral lines can be resolved with a prism or a diffraction grating.
Multi-electron atoms are complicated.
Rules to be learned about later limit the number of electrons in any one state. Thus the electrons in an
atom are arranged in concentric shells.
The shells are labelled K, L, M, N, etc., starting from the innermost.
The average arrangement of the charge from the electrons is spherical. Thus Gauss's law applies.
The inner electrons are therefore affected only by the nucleus, except for a small effect from nearby
electrons (screening).
So the energy levels for inner electrons are similar to those for one-electron ions:
where
,
is the screening correction.
X-rays are electromagnetic radiation produced when electrons accelerated to
with a material target. The frequency of X-rays is about
= thousands of eV collide
times that of visible light.
The electrons radiate a continuous ``bremsstrahlung'' spectrum of X-ray frequencies (or energies),
because of classical radiation by accelerating charges.
The maximum frequency of the X-rays obeys
, the Duane-Hunt law, because one electron
can emit at most all of its kinetic energy into one quantum.
The projectile electron also disturbs atoms in its path, randomly ionizing them.
Because X-ray energies are high, deeply-bound electrons from the inner orbits of a multi-electron atom
can be dislodged.
Characteristic X-rays are those emitted by an atom after it has lost an inner electron.
Transitions to the K shell from L, M, N, etc. emit X-Ray lines called
,
,
, etc.
The wavelengths of these lines can be calculated from the Bohr model, with the screening correction
Moseley measured K frequencies for many elements, unambiguously establishing their atomic
.
numbers in terms of their nuclear charge.
Particles are also waves.
De Broglie proposed that a particle of momentum
.
With this assumption Bohr's quantization rule
file://localhost/Users/noelclark/Desktop/2170%20quantum%20summary3.0.html
has associated with it a wave with wavelength
is equivalent to requiring an integral
Page 3 of 6
Untitled
11/19/04 6:13 PM
number of wavelenths around the orbit's circumference.
We also have
, and
, where
is the phase velocity of the wave.
The Davisson-Germer experiment confirms that a beam of electrons exhibits Bragg refraction from
crystals, just like X-rays.
The wave function
(More generally
is the probability amplitude.
, defined for 3-dimensional space.)
The time dependence factors away, leaving
The probability
to find the particle at
the probability for the particle to be between
The function is normalized to unity, i.e.,
.
is given by the absolute square of
and
, or of
:
.
(the particle has to be somewhere).
A travelling wave has the form
There's a trade-off (uncertainty principle) between momentum and localization of a particle.
The De Broglie wave with
represents a particle of definite momentum, but with probability
spread over all space.
A wave function that describes a particle localized in a small region has a sharply peaked shape. What
would it's momentum (wavelength) be?
Fourier analysis shows that a localized particle wave packet can be describedas a Fourier superposition
of sin/cosine components:
file://localhost/Users/noelclark/Desktop/2170%20quantum%20summary3.0.html
Page 4 of 6
Untitled
11/19/04 6:13 PM
One can show, and we demonstrated, that the narrower the wave packet is, the more frequencies k we
have to include.
From this property we deduce the uncertainty principles for particle waves:
Schrodinger Equation
For a confined particle (bound state)
One dimensional problems
particle in a box, energy eigenvalues
where n = 1,2,3...
harmonic oscillator , energy eigenvalues
where n = 0,1,2,3....
Three dimensional problems
In each space dimension we get boundary conditions that produce a quantum number associated
with that dimension.
e.g., cubical box
file://localhost/Users/noelclark/Desktop/2170%20quantum%20summary3.0.html
.
Page 5 of 6
Untitled
11/19/04 6:13 PM
Schrödinger equation is solved by assuming a trial ``separated'' solution, e.g.,
, leading to separate ordinary differential equations for
An atom is bound by the attractive potential
.
, a ``central force''.
Spherical symmetry here implies the choice
and a messy looking
Schrödinger equation.
; boundary condition in
(single-valued as
increases by
) gives
.
Comparison with
leads to
.
); boundary condition in
(finite at
) leads to
.
From the radial equation, we identify the rotational kinetic energy
angular momentum is given by
, showing that the
.
Advanced treatment shows that
is limited to
.
Only one angular momentum component is a constant of the motion, with a well defined quantum
mechanical value. If that component is
(the usual convention) then
are not
determined). Think of a precessing top.
Again, these statements about angular momentum apply to any central force.
.
defines forbidden regions.
By the Schrödinger equation,
bends away from the axis in forbidden regions, oscillates in allowed ones.
Addition of the boundary conditions (
finite everywhere) leads to a discrete set of allowed
energies.
...for example the rigid box
(integral number of half wavelengths must fit in the
box).
file://localhost/Users/noelclark/Desktop/2170%20quantum%20summary3.0.html
Page 6 of 6