lecture 20 spherical harmonics – not examined remember phils problems and your notes = everything...

Lecture 20Lecture 20

Spherical Harmonics – not examinedSpherical Harmonics – not examined

http://www.hep.shef.ac.uk/Phil/PHY226.htmRemember Phils Problems and your notes = everything

last lecture !!!!!!

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

We can all imagine the ground state of a particle in an infinite quantum well in 1D

IntroductionIntroduction

Or the 2D representation of 2 harmonics of a wave distribution in x and y interacting on a plate

Visualisation of the spherical harmonics in a 3D spherical potential well is more tricky !!!!!

XY

ψ

X

ψ

www.falstad.com/mathphysics.html

http://www.falstad.com/mathphysics.html

e.g.

IntroductionIntroduction

In 3D Cartesian coordinates we write:

In spherical polar coordinates last lecture we stated that: 2

2 22 2 2 2 2

1 1 1sin

sin sinr

r r r r r

Let’s think about the Laplace equation in 3D

02 ),,( zyx

and so the Laplace equation in spherical polar coordinates is:

02 ),,( r

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.

)()()(),,( zZyYxXzyx )()()(),,( FPrRr

zkykxkAzyx zyx sinsinsin),,( iar eArer sin),,( 02

Let’s look at electron orbitals for the Hydrogen atomLet’s 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: 2

613

n

eVEn

.

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 ),,( r ),,( mlnare


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

),,( r ),,( mlnare


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 (n=3, l=0), or (n=3, l=1) or (n=3, l=2) 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 . | |m l

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 …..


)()()(),,( FPrRr


)()()(),,( FPrRr 0

230

2 area

2

1

21

0

230

1 area

00 230

2

230

2 11 arar ea

ea

r

),,(

Once we have the solution to the wave equation in 3D spherical polar coordinates we can deduce the probability function.

So

For example the probability density function in 3D for ground state (1,0,0) is …..

drrea

dP ar 2230

41

0

drea

rdrre

aP arar

0

230

22

0

230

004

41

1422

4

0

30

20

202

30

0

ararae

aP ar

The radial probability density for the hydrogen ground state is obtained by multiplying the square of the wavefunction by a spherical shell volume element.

If we integrate over all space

we can show that the total probability is 1.


drrea

dP ar 2230

41

0

Probability density function in 3D for ground state (1,0,0) is

Let’s look at electron orbitals for the Hydrogen atomLet’s look at electron orbitals for the Hydrogen atomIt 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 θ and φ as well as of radius r making them exceedingly difficult to plot.


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.


Spherical Harmonics

The solution of a PDE in spherical polar coordinates is )()()(),,( FPrRr

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.

)(rR

)()( FP

),( mlY

),( mlY

( , )mlY )(P )(FThe spherical harmonics can be directly compared with the and

solutions for the wave function describing the electron orbitals of 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 θ and φ 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).

http://www.lifesmith.com/spharmin.html

Summing harmonics can produce some really pretty shapes



Oil droplets or soap bubbles oscillatingOil droplets or soap bubbles oscillatingSpherical Harmonics also describe the wobbling deformations of an

oscillating, elastic sphere.

( , )mlY

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. ( , )m

lY

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.

What sine and cosine are for a one-dimensional, linear string, the spherical harmonics are for the surface of a sphere.

Oil droplets or soap bubbles oscillatingOil droplets or soap bubbles oscillating

lecture 20 spherical harmonics – not examined remember phils problems and your notes = everything...

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