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Hydrogen Atom

- We will examine the simplest atom and describe

what the wave function tells us about the

behavior - When we solve the Schrodinger Equation in three

dimensions we find we can only get acceptable

solutions if some measureable quantities take on

a series of precise values - Other quantities are not precise and have

undertainties

Hydrogen Atom

- These precisely observable quantities are called

eigenvalues (proper values) - An eigenstate is a set of particular values for

these observable quantities - Each eigenstate is characterized by integer

numbers quantifying the observable quantities

Principle Quantum Number

- The principle quantum number defines the energy

of the electron just as in the Bohr theory

Orbital Quantum Number

- The orbital quantum number is related to the

magnitude of the angular momentum of the

electron. - Classically, Lmvr
- In quantum mechanics the result is

Magnetic Quantum Number

- It turns out that just as charge is quantized, so

is space!!! Directions in space are not allowed

to be in just any old direction - The direction of the angular momentum vector

(think of an orbiting electron as going in a

circle and the plane of the circle is at right

angles to the angular momentum) can only take on

particular values

Magnetic Quantum Number

- These values are described by the magnetic

quantum number, ml which takes on values from -l

to l in integer steps

Heres an example for l2.

Magnetic Quantum Number

- This quantum number has the name because when

atoms are placed in an external magnetic field,

the spectra shows that the energy levels are

modified slightly and depend on this orientation - This is called the Zeeman Effect
- Since an electron with angular momentum must have

some circular component it will be a rotating

current and will generate a magnetic field which

interacts with the external field

Magnetic Quantum Number

In a magnetic field, the electrons energy

depends on the value of ml. The result is that a

single spectral line is now becomes many lines

depending on the value of the magnetic quantum

number!

Spin Quantum Number

- The solution to the Schrodinger Equation doesnt

make any predictions about spin - However, if you calculate the speed of the

electron in the ground state (n0) you find that

it is about 2 x 106 m/s and special relativity

needs to be considered

Spin Quantum Number

- P.A.M. Dirac had to rewrite the Schrodinger

Equation (which cannot work for relativity) to

make it agree with Einstein - Miraculously, the solutions turned out to predict

some additional angular momentum for the electron

which could take on only two values,

Spin Quantum Number

- Careful examination of spectra indeed showed this

fine structure splitting of spectral lines which

exactly agreed with Diracs prediction - It is called electron spin, which implies a

particle spinning on its axis - But we know that a particle picture is silly, so

we just deal with the angular momentum - The quantum number is or - 1/2 (spin up or spin

down)

The Whole Picture

- So the electron in a hydrogen atom is described

by four quantum numbers - These are n, l, ml, and ms
- They are related in that some are limited by the

size of others - Best displayed in a table

The Whole Picture

The Whole Picture

- Energy determined by n
- Orientation and angular momentum determined by

the other three - In more complex atoms, some dependence of energy

on the others due to magnetic interactions

between the various magnetic fields

Spectra and Selection Rules

- It turns out that a photon also has angular

momentum of h-bar - That means that to conserve angular momentum,

when an electron changes states, it must change

its angular momentum by one unit of h-bar so - This is called a selection rule

Pauli Esclusion Principle

- Look at more complex atoms with multiple

electrons - The Bohr model failed badly
- Solutions to the Schrodinger equation (must be

done numerically) predict the correct energy

levels for the same quantum numbers as for

hydrogen - The electrons interact with each other which

complicates things enormously

Pauli Exclusion Principle

- It quickly became apparent that a new principle

was needed to explain the states of electrons in

multi-electron atoms - No two electrons ever had the same quantum

numbers at the same time in any individual atom - This is the Pauli Exclusion Principle
- It turns out to have a deeper meaning
- This provides for the chemical properties which

appear in the periodic table

Periodic Table

- States are occupied by electrons in increasing

order of energy - For He, n1, l0, ml0, ms or - 1/2
- For Li, the next electron must have n2
- Build the whole table this way
- Explains why we have shells and subshells

Periodic Table

X-Ray Spectra

- Visible, UV and IR spectra deal with transitions

between outer electrons - If we take a complex atom and kick out an inner

shell electron, lots of energy involved - If an electron goes from far away and falls into

a vacant inner shell, high energy photon is

released - This is an x-ray, so x-ray spectra ought to give

information about inner energy levels

X-Ray Spectra

The lines depend on the element. The continuous

part is due to bremsstrahlung or braking

radiation due to decelerating electrons striking

the target. The left side cutoff is where all

the energy of the electron goes into a photon.

X-Ray Spectra

The lines depend on the element. The continuous

part is due to bremsstrahlung or braking

radiation due to decelerating electrons striking

the targe. The left side cutoff is where all the

energy of the electron goes into a photon.

Fluorescence

- Excite an electron in an atom or molecule to a

higher energy state - If there is an intermediate state between where

the electron is and where it came from, it can

lose energy by emitting a photon and going to

that intermediate state - From there it can decay again into the starting

state

Fluorescence

Phosphorescence

- The intermediate state is metastable
- This means it has a much longer lifetime which

may extend to minutes or even hours - Long delay in emitting the second photon
- Both fluorescence and phosphorescence can be used

to identify atoms or molecules

Lasers

- Give off coherent light
- Means all the light rays have the same phase
- Again, metastable states of atoms are involved

Lasers

Absorption of Light

Stimulated Emission of Light

Lasers

- Need a population inversion

Lasers

Ruby Laser

Using Chromium Atoms in Ruby

He-Ne Laser

CD-ROM

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