Transcript Slide 1

Atomic Spectra and Atomic Energy States
13.1.8 – 13.1.13
Observing Atomic Spectra
The diagram below shows some of the energy levels of
the hydrogen atom.
Calculate the frequency associated with the photon
emitted in each of the electron transitions A and B and
identify the part of the EM spectrum where they occur.
The Origin of Energy Levels
The electron is bound to the nucleus by the
Coulomb force and this force will essentially
determine the energy of the electron. If we were
to regard the hydrogen atom for instance as a
miniature Earth-Moon system, the electron’s
energy would fall off with inverse of distance
from the nucleus and could take any value.
However we can the see the origin of the
existence of discrete energy levels within the
atom if we consider the wave nature of the
electron.
Particle in a Box
• To simplify matters we shall consider the
electron to be confined by a one dimensional
box of length L.
• In classical wave
theory, a wave that
is confined is a
standing wave.
Quantization of the Electron’s Energy
The Schrödinger Model of the
Hydrogen Atom
• In 1926 Erwin Schrodinger
proposed a model of the
hydrogen atom based on
the wave nature of the
electron and hence the
de Broglie hypothesis.
This was actually the birth
of Quantum Mechanics.
Quantum mechanics and
General Relativity are
now regarded as the two
principal theories of
physics.
The mathematics of Schrodinger’s socalled wave mechanics is somewhat
complicated so at this level, the best that
can be done is to outline his theory.
Essentially, he proposed that the electron
in the hydrogen atom is described by a
wave function, Y.
This wave function is described by an
equation known as the Schrodinger wave
equation, the solution of which give the
values that the wave function can have.
If the equation is set up for the electron in
the hydrogen atom, it is found that the
equation will only have solutions for which
the energy E of the electron is given by E =
(n + ½ )hf. Hence the concept of
quantization of energy is built into the
equation.
Of course we do need to know what the
wave function is actually describing. The
electron has an undefined position, but
the square of the amplitude of the wave
function gives the probability of finding
the electron at a particular point.
The solution of the equation predicts exactly the line
spectra of the hydrogen atom. If the relativistic motion of the electron is taken into account, the
solution even predicts the fine structure of some of the spectral lines. (For example, the red line on
closer examination, is found to consist of seven lines close together.)
The Schrodinger equation is not an easy equation to solve and to get exact solutions for atoms other
than hydrogen or singly ionised helium, is well-nigh impossible. Nonetheless, Schrodinger’s theory
changed completely the direction of physics and opened whole new vistas- and posed a whole load of
new philosophical problems.
The Heisenberg
Uncertainty Principle
• In 1927 Werner Heisenberg proposed a principle that went along way to
understanding the interpretation of the Schrodinger wave function.
• Suppose the uncertainty
in our knowledge of the
position of a particle is Δx
and the uncertainty in the
momentum is Δp, then
the Uncertainty Principle
states that the product
ΔxΔp is at least the order
of h, the Planck constant.
A more rigorous analysis
shows that
Heisenberg and de Broglie
h
xp 
4
h

p
If a particle has a uniquely defined de Broglie
wavelength, then its momentum is known
precisely but all knowledge of its position is lost.
The Principle also applies to energy and time. If ΔE is the
uncertainty in a particle’s energy and Δt is the uncertainty in the
time for which the particle is observed is Δt, then
h
Et 
4
This is the reason why spectral lines have finite width. For a
spectral line to have a single wavelength, there must be no
uncertainty in the difference of energy between the associated
energy levels. This would imply that the electron must make the
transition between the levels in zero time.
Homework:
• Tsokos
– Page 404
– Questions 1 to 15