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Lecture 15: The Hydrogen Atom

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Title: Lecture 15: The Hydrogen Atom


1
Lecture 15 The Hydrogen Atom
2
J.J. Thomsons Plum Pudding Model of the Atom
(1897)
He proposed that the electrons are embedded in a
positively charged pudding
3
Rutherfords a Scattering Experiment (1911)
He found that, once in a while, the a-particles
were scattered backwards by the target
video clip
4
Discovery of the Atomic Nucleus
To explain the backscattering, the positive
charge must be concentrated in a small region
5
Rutherfords Solar System Model of the Atom
The atom consists of electrons orbiting around a
small but dense central nucleus
6
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

7
Absorption Spectrum of a Gas
Dark lines will appear in the light spectrum
8
Absorption spectrum of Sun
Emission spectra of various elements
9
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

10
Bohrs Model of the Hydrogen Atom(1913)
He proposed that only certain orbits for the
electron are allowed
11
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

12
Absorption / Emission of Photonsand Conservation
of Energy
Ef - Ei hf
Ei - Ef hf
13
Energy Levels of Hydrogen
14
Electron jumping to a higher energy level
E 12.08 eV
15
Spectrum of Hydrogen
Bohrs formula
16
Hydrogen is therefore a fussy absorber / emitter
of light
It only absorbs or emits photons with precisely
the right energies dictated by energy conservation
17
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
18
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

19
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

20
Vibrational Modes of a String
fundamental
2nd harmonic
3rd harmonic
4th harmonic
21
Energy Levels in a Box
22
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

23
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

24
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)
25
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

26
1s Orbital
27
Density of the cloud gives probability of where
the electron is located
28
2s and 2p Orbitals
29
Another diagram of 2p orbitals
Note that there are three different
configurations corresponding to m -1, 0, 1
30
3d Orbitals
Now there are five different configurations
corresponding to m -2, -1, 0, 1, 2
31
4f Orbitals
There are seven different configurations
corresponding to m -3, -2, -1, 0, 1, 2, 3
32
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)
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