Lecture 15: The Hydrogen Atom

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Lecture 15 The Hydrogen Atom
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J.J. Thomsons Plum Pudding Model of the Atom
(1897)
He proposed that the electrons are embedded in a
positively charged pudding
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Rutherfords a Scattering Experiment (1911)
He found that, once in a while, the a-particles
were scattered backwards by the target
video clip
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Discovery of the Atomic Nucleus
To explain the backscattering, the positive
charge must be concentrated in a small region
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Rutherfords Solar System Model of the Atom
The atom consists of electrons orbiting around a
small but dense central nucleus
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Hydrogen Atom is Unstable?

• It is known that accelerating charges emit
  radiation
• Thus, electron should emit radiation, lose energy
  and eventually fall into the nucleus!
• Why doesnt this happen? Shows that something was
  wrong with this model of the hydrogen atom

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Absorption Spectrum of a Gas
Dark lines will appear in the light spectrum
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Absorption spectrum of Sun
Emission spectra of various elements
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Balmers Formula for Hydrogen

• Notice there are four bright lines in the
  hydrogen emission spectrum
• Balmer guessed the following formula for the
  wavelength of these four lineswhere n 3,
  4, 5 and 6

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Bohrs Model of the Hydrogen Atom(1913)
He proposed that only certain orbits for the
electron are allowed
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Bohrs Empirical Explanation

• Electrons can only take discrete energies
  (energy is related to radius of the orbit)
• Electrons can jump between different orbitsdue
  to the absorption or emission of photons
• Dark lines in the absorption spectra are due to
  photons being absorbed
• Bright lines in the emission spectra are due to
  photons being emitted

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Absorption / Emission of Photonsand Conservation
of Energy
Ef - Ei hf
Ei - Ef hf
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Energy Levels of Hydrogen
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Electron jumping to a higher energy level
E 12.08 eV
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Spectrum of Hydrogen
Bohrs formula
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Hydrogen is therefore a fussy absorber / emitter
of light
It only absorbs or emits photons with precisely
the right energies dictated by energy conservation
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This explains why some nebulae are red or pink
in colour
One of the transitions in the Balmer
series corresponds to the emission of red light
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Schrödingers Improvement to Bohrs Model

• Showed how to obtain Bohrs formula using the
  Schrödinger equation
• Electron is described by a wave function y
• Solved for y in the electric potential due to the
  nucleus of the hydrogen atom

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Square Well

• Approximate electric (roller coaster) potential
  by a square well
• System is then identical to the wave equation for
  a string that is fixed at both ends

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Vibrational Modes of a String
fundamental
2nd harmonic
3rd harmonic
4th harmonic
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Energy Levels in a Box
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Quantum Numbers

• Energy levels can only take discrete values
• Labelled by a quantum number n, which takes
  values 1, 2, 3, ...
• Each level has energy that increases with n

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Ground State (n1)

• Lowest or ground-state energy is non-zero
• Electron cannot sit still but must be forever
  jiggling around
• Expected from the Heisenberg uncertainty principle

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Vibrational Modes of a Rectangular Membrane
(1,1) mode
(1,2) mode
(2,1) mode
(2,2) mode
Vibrational modes of a circular membrane (drum)
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Electron in a Hydrogen Atom

• Wave function is like a vibrating string or
  membrane, but the vibration is in three
  dimensions
• Labelled by three quantum numbers
• n 1, 2, 3,
• l 0, 1, , n-1
• m -l, -l1, , l-1, l
• For historical reasons, l 0, 1, 2, 3 is also
  known as s, p, d, f

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1s Orbital
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Density of the cloud gives probability of where
the electron is located
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2s and 2p Orbitals
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Another diagram of 2p orbitals
Note that there are three different
configurations corresponding to m -1, 0, 1
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3d Orbitals
Now there are five different configurations
corresponding to m -2, -1, 0, 1, 2
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4f Orbitals
There are seven different configurations
corresponding to m -3, -2, -1, 0, 1, 2, 3
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Summary

• Electron does not fly round the nucleus like the
  Earth around the Sun (Rutherford, Bohr)
• Depending on which energy level it is in, the
  electron can take one of a number of stationary
  probability cloud configurations (Schrödinger)