Lecture 20 Spherical Harmonics

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Lecture 20Spherical Harmonics not examined

• last lecture !!!!!!
• There will be 1 revision lecture next
• Come to see me before the end of term
• Ive put more sample questions and answers in
  Phils Problems
• Past exam papers
• Have a look at homework 2 (due in on 15/12/08)

Remember Phils Problems and your notes
everything
http//www.hep.shef.ac.uk/Phil/PHY226.htm
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Introduction
We can all imagine the ground state of a particle
in an infinite quantum well in 1D
?
X
Or the 2D representation of 2 harmonics of a wave
distribution in x and y interacting on a plate
?
X
Y
www.falstad.com/mathphysics.html
Visualisation of the spherical harmonics in a 3D
spherical potential well is more tricky !!!!!
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Introduction
Lets think about the Laplace equation in 3D
In 3D Cartesian coordinates we write
In spherical polar coordinates last lecture we
stated that
and so the Laplace equation in spherical polar
coordinates is
Comparing the Cartesian case with the spherical
polar case, it is not difficult to believe that
the solution will be made up of three separate
functions, each comprising an integer variable to
define the specific harmonic solution.
e.g.
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Lets look at electron orbitals for the Hydrogen
atom
This topic underlies the whole of atomic and
nuclear physics. Next semester in atomic physics
you will cover in more detail the radial
spherical polar solutions of the Schrödinger
equation for the hydrogen atom.
Bohr and Schrodinger predicted the energy levels
of the H atom to be
This means that the energy of an electron in any
excited orbital depends purely on the energy
level in which it resides. From your knowledge of
chemistry, you will know that each energy level
can contain more than one electron. These
electrons must therefore have the same energy.
We say that there exists more than one quantum
state corresponding to each energy level of the H
atom. (Actually there are 2n2 different quantum
states for the nth level).
For the 1D case it was sufficient to define a
quantum state fully using just one quantum
number, e.g. n 2 because our well extended only
along the x axis. In 3D we have to consider
multiple axes within a 3D potential well, and
since the probability density functions
corresponding to the EPCs are mostly not radially
symmetric, we must represent wavefunctions with
the same energy but different eigenfunctions,
using a unique set of quantum numbers.
The quantum numbers for polar coordinates
corresponding to
are
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Lets look at electron orbitals for the Hydrogen
atom
An electron probability cloud (EPC) is a
schematic representation of the likely position
of an electron at any time.
This figure shows the EPCs corresponding to the
ground state and some excited states of the
hydrogen atom.
For each energy level there are several different
EPC distributions corresponding to the different
3D harmonic solutions for that energy level.
The quantum numbers for polar coordinates
corresponding to
are
6
Lets look at electron orbitals for the Hydrogen
atom
n is defined as the principal quantum number (and
sets the value of the energy level of the wave).
For each wave with quantum number n, there exist
quantum states of l from l 0 to l (n - 1)
where l is defined as the orbital quantum number.
So for example an electron in the 3rd excited
state can be in (n3, l0), or (n3, l1) or
(n3, l2) quantum states.
Each one of these states has further states
represented by quantum number m defined as the
magnetic quantum number, a positive or negative
integer where .
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Lets look at electron orbitals for the Hydrogen
atom
The full solution
, for the ground state and first few
excited states corresponding to each specific
combination of quantum numbers is shown below. a0
is the first Bohr radius corresponding to the
ground state of the H atom ..
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Lets look at electron orbitals for the Hydrogen
atom
Once we have the solution to the wave equation in
3D spherical polar coordinates we can deduce the
probability function.
For example the probability density function in
3D for ground state (1,0,0) is ..
The radial probability density for the hydrogen
ground state is obtained by multiplying the
square of the wavefunction by a spherical shell
volume element.
So
If we integrate over all space
we can show that the total probability is 1.
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Lets look at electron orbitals for the Hydrogen
atom
Probability density function in 3D for ground
state (1,0,0) is
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Lets look at electron orbitals for the Hydrogen
atom
It would be very interesting to plot the full 3D
probability density distributions for each
combination of quantum states. Unfortunately,
distributions for non spherically symmetric
solutions (i.e. p and d quantum states) would be
a function of ? and f as well as of radius r
making them exceedingly difficult to plot.
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Lets look at electron orbitals for the Hydrogen
atom
If we were to plot only the probability density
functions for spherically symmetric solutions
(i.e. s quantum states) for each quantum state n
we would find the following distributions
corresponding to the EPCs shown earlier for
hydrogen.
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Lets look at electron orbitals for the Hydrogen
atom
Spherical Harmonics
The solution of a PDE in spherical polar
coordinates is
We can say that the solution is comprised of a
radially dependent function and two
angular dependent terms which
can be grouped together to form specific
spherical harmonic solutions .
Formally the spherical harmonics
are the angular portion of the solution
to Laplace's equation in spherical coordinates
derived in the notes.
The spherical harmonics
can be directly compared with the
and
solutions for the wave function describing the
electron orbitals of the hydrogen atom.
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Lets look at electron orbitals for the Hydrogen
atom
Spherical Harmonics
Spherical harmonics are useful in an enormous
range of applications, not just the solving of
PDEs.
They allow complicated functions of ? and f to be
parameterised in terms of a set of solutions.
For example a summed series of specific harmonics
as a Fourier series can be used to describe the
earth (nearly but not exactly spherical).
Summing harmonics can produce some really pretty
shapes
http//www.lifesmith.com/spharmin.html
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Oil droplets or soap bubbles oscillating
Spherical Harmonics also describe
the wobbling deformations of an oscillating,
elastic sphere.
What sine and cosine are for a one-dimensional,
linear string, the spherical harmonics are for
the surface of a sphere.
A tiny oil droplet is placed on an oil bath which
is set into vertical vibrations to prevent
coalescence of the droplet with the bath. The
droplet, which at rest would have spherical form
due to surface tension, bounces periodically on
the bath.
A movie shows the oscillations of the drop and
the corresponding calculations using spherical
harmonics with l 2, 3, 4 and m
0.
The magnetic quantum number m determines
rotational symmetry of the wobbling around the
vertical axis. For m ? 0, deformations are not
symmetric with respect to the vertical, and in
this case, the droplet starts to move around on
the oil bath. This can be seen in a second movie.
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Oil droplets or soap bubbles oscillating
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Oil droplets or soap bubbles oscillating