Hydrogen Atom

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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

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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

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Principle Quantum Number

• The principle quantum number defines the energy
  of the electron just as in the Bohr theory

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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

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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

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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.
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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

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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!
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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

10
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,

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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)

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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

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The Whole Picture
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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

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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

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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

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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

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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

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Periodic Table
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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

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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.
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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.
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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

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Fluorescence
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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

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Lasers

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

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Lasers
Absorption of Light
Stimulated Emission of Light
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Lasers

• Need a population inversion

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Lasers
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Ruby Laser
Using Chromium Atoms in Ruby
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He-Ne Laser
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