Download Chapter 2 (Lecture 2-3) Old Quantum Theory The Postulates of Bohr

Chapter 2 (Lecture 2-3)
Old Quantum Theory
The Postulates of Bohr and Bohr Atom
Wilson-Summerfeld rules of quantization
Selection rules and correcpondence principles
Quantization of Simple Systems
Harmonic oscillator
Rigit rotator
Particle in a box
Rutherford Atom
In the 1880's physicists discovered the existence of the electron, and knew that it somehow lived inside the atom. J.J.Thompson
theorized that the atom was made of a positive substance with the little negative electrons mixed in, like negative plums mixed in
with positive pudding. It was called the "plum pudding model" of the atom (plum pudding was a popular dessert at that time).
In the 1900's Ernest Rutherford performed his famous "gold foil" experiments, and discovered that the atom has a positive
concentrated nucleus, and that the electrons orbit the nucleus almost as the planets orbit the sun. Most of the atom is empty space.
Rutherford shot alpha particles at a thin sheet of gold foil (like aluminum foil, but made of gold). An alpha particle is exactly the
same thing as the nucleus of a helium atom--two protons and two neutrons all bound together. Obviously it has a net positive
charge.
When shot at the thin gold foil most of the alphas passed through, but were deflected somewhat. Getting close to the gold atoms
made them change their direction--a force was exerted.
Now if the positively-charges alphas were passing through plum pudding atoms, with the negative and positive charges pretty
much equally distributed throughout the atom, you would expect a certain amount of deflection, but small. Instead, a much
o
greater amount of deflection was observed. In fact, some of the alpha particles were deflected back 180 , back to the direction
from which they came. This could only happen if they came close to a very concentrated positively charged object (positive and
positive repel). Rutherford's interpretation was that the positive charge in an atom is concentrated in a dense nucleus at the center
of the atom, and that the negative electrons must be in orbit around this nucleus. Most of the atom is empty space.
Bohr Atom
In 1911, Rutherford introduced a new model of the atom in which cloud of negatively charged electrons surrounding a small,
dense, positively charged nucleus. This model is result of experimental data and Rutherford naturally considered a planetarymodel atom. The laws of classical mechanics (i.e. the Larmor formula, power radiated by a charged particle as it accelerates.),
predict that the electron will release electromagnetic radiation while orbiting a nucleus. Because the electron would lose energy, it
would gradually spiral inwards, collapsing into the nucleus. This atom model is disastrous, because it predicts that all atoms are
unstable.
To overcome this difficulty, Niels Bohr proposed, in 1913, what is now called the Bohr model of the atom. He suggested that
electrons could only have certain classical motions (Bohr postulates):
1.
The electrons can only travel in special orbits: at a certain discrete set of distances from the nucleus with specific
energies.
2.
The electrons of an atom revolve around the nucleus in orbits. These orbits are associated with definite energies and are
also called energy shells or energy levels. Thus, the electrons do not continuously lose energy as they travel in a
particular orbit. They can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting
electromagnetic radiation with a frequency ν determined by the energy difference of the levels according to the Planck
relation:
3.
Kinetic energy of the electron in the orbit is related to the frequency of the motion of the electron:
4.
For a circular orbit the angular momentum L is restricted to be an integer multiple of a fixed unit:
where n = 1, 2, 3, ... is called the principal quantum number. The lowest value of n is 1; this gives a smallest possible orbital
radius of 0.0529 nm known as the Bohr radius.
Bohr's condition, that the angular momentum is an integer multiple of ħ was later reinterpreted by de Broglie as a standing wave
condition: the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the
electron's orbit:
The Bohr model gives almost exact results only for a system where two charged points orbit each other at speeds much less than
that of light.
To calculate the orbits requires two assumptions:
1. (Classical Rule)The electron is held in a circular orbit by electrostatic attraction. The centripetal force is equal to the Coulomb
force.
It also determines the total energy at any radius:
The total energy is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from
the proton. For infinite values of r, the energy is zero, corresponding to a motionless electron infinitely far from the proton.
2.
(Quantum rule) The angular momentum
so that the allowed orbit radius at any n is:
The energy of the n-th level is determined by the radius:
An electron in the lowest energy level of hydrogen (n = 1) therefore has 13.6 eV less energy than a motionless electron infinitely
far from the nucleus.
The combination of natural constants in the energy formula is called the Rydberg energy (RE):
This expression is clarified by interpreting it in combinations which form more natural units. We define
of the electron (511 keV) and
is rest mass energy
is the fine structure constant then
Bohr Atom and Rydberg formula
The Rydberg formula, which was known empirically before Bohr's formula, is now in Bohr's theory seen as describing the
energies of transitions or quantum jumps between one orbital energy level, and another. When the electron moves from one
energy level to another, a photon is emitted. Using the derived formula for the different 'energy' levels of hydrogen one may
determine the 'wavelengths' of light that a hydrogen atom can emit.
The energy of a photon emitted by a hydrogen atom is given by the difference of two hydrogen
energy levels:
where nf is the final energy level, and ni is the initial energy level.
Since the energy of a photon is
the wavelength of the photon given off is given by
This is known as the Rydberg formula, and the Rydberg constant R is RE / hc. This formula was
known in the nineteenth century to scientists studying spectroscopy, but there was no theoretical
explanation for this form or a theoretical prediction for the value of R, until Bohr. In fact, Bohr's
derivation of the Rydberg constant, as well as the concomitant agreement of Bohr's formula with
experimentally observed spectral lines of the Lyman (nf = 1), Balmer (nf = 2), and Paschen (nf =
3) series, and successful theoretical prediction of other lines not yet observed, was one reason
that his model was immediately accepted.
Improvement of Bohr Model: Wilson- Sommerfeld quantization
Several enhancements to the Bohr model were proposed; most notably the Sommerfeld model or Bohr-Sommerfeld model,
which suggested that electrons travel in elliptical orbits around a nucleus instead of the Bohr model's circular orbits. This model
supplemented the quantized angular momentum condition of the Bohr model with an additional radial quantization condition, the
Sommerfeld-Wilson quantization condition
where pr is the radial momentum canonically conjugate to the coordinate q which is the radial position and T is one full orbital
period. The Bohr-Sommerfeld model was fundamentally inconsistent and led to many paradoxes. The Sommerfeld quantization
can be performed in different canonical coordinates, and sometimes gives answers which are different. In the end, the model was
replaced by the modern quantum mechanical treatment of the hydrogen atom, which was first given by Wolfgang Pauli in 1925,
using Heisenberg's matrix mechanics. The current picture of the hydrogen atom is based on the atomic orbitals of wave mechanics
which Erwin Schrödinger developed in 1926.
However, this is not to say that the Bohr model was without its successes. Calculations based on the Bohr-Sommerfeld model
were able to accurately explain a number of more complex atomic spectral effects.
Quantization of some simple systems
Harmonic oscillator degenerate energy levels
For a system consisting of a particle of mass m bound to the equilibrium position
by a restoring force
and
constrained to move along the axis the classical motion consists in a harmonic oscillation with frequency as described by the
equation
The momentum
. Using Wilson Summerfeld quantization rule, integrating the equation over a period, we obtain
CW (Class work) Make explicit calculation in the class.
Thus we see that the energy levels allowed by the old quantum theory are integral multiples of , as indicated in Figure (given in
the class). The selection rule
permits the emission and absorption of light of frequency only.
A particle bound to an equilibrium position in a plane by restoring forces with different force constants in the x and y directions,
corresponding to the potential function
is similarly found to carry out independent harmonic oscillations along the two axes. The quantization restricts the energy to the
values
determined by the two quantum numbers
In case that
.
, the oscillator is said to be isotropic. The energy levels are then given by the equation
CW: Table energy levels and explain degenerate levels.
CW: Energy of 3 D oscillator
CW: Discrete and continuous levels.
Different states of motion, corresponding to different sets of values of the two quantum numbers
, may then correspond
to the same energy level. Such an energy level is said to be degenerate, the degree of degeneracy being given by the number of
independent sets of quantum numbers. In this case the nth level shows (n + l)-fold degeneracy. The nth level of the threedimensional isotropic harmonic oscillator shows
The Particle in a Box
Let us consider a particle of mass m in a box in the shape of a rectangular parallelepiped with edges a, b, and c, the particle being
under the influence of no forces except during collision with the walls of the box, from
which it rebounds elastically. The linear momenta
will then be constants of the motion, except that they will change
sign on collision of the particle with the corresponding walls. Their values are restricted by the rule for quantization as follows:
Consequently the total energy is restricted to the values
The Rigid Rotator
The configuration of the system of a rigid rotator restricted to a plane is determined by a single angular coordinate, say
The
canonically conjugate angular momentum,
where I is the moment of inertia, is a constant of the motion. Hence the
quantum rule is
Thus the angular momentum is an integral multiple of h, as originally assumed by Bohr. The allowed energy values are
The decline of the old quantum theory began with the introduction of half-integral values for quantum numbers in place of integral
values for certain systems, in order to obtain agreement with experiment. It was discovered that the pure rotation spectra of the
hydrogen halide molecules are not in accordance
with K = 0, 1, 2, , but instead require
.
Correspondence Principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or
by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says that for
large orbits and for large energies, quantum calculations must agree with classical calculations.
Successes and Limitations of old quantum theory
We shall not pursue the discussion of old quantum theory any further. The treatment of H atom can easily be extended to H like
atoms (He+, Li++) and to the alkali atoms. The theory also applies to the vibration and rotation spectra of molecules.
Nevertheless it is incomplete theory. The Bohr-Sommerfeld rule applied only periodic or multiple periodic systems. There is no
method of quantization for aperiodic systems. As a general rule all collision phenomena and chaotic systems lie outside the scope
of the theory. The old quantum theory provides no means to calculate the intensities of the spectral lines. It fails to explain the
anomalous Zeeman effect (that is, where the spin of the electron cannot be neglected).
Its limitations are still under investigation.
References
Pauling Linus and Wilson Bright. Introduction to Quantum Mechanics with applications. ISBN-10: 0486648710 | ISBN-13: 9780486648712.
Quantum Mechanics, David McMahon
Introduction To Quantum Mechanics, Harald J W Müller-Kristen