Transcript Lecture 15: The Hydrogen Atom

Lecture 15: The Hydrogen Atom
J.J. Thomson’s Plum Pudding Model
of the Atom (1897)
He proposed that the electrons are embedded
in a positively charged ‘pudding’
Rutherford’s a Scattering Experiment
(1911)
He found that, once in a while, the a-particles
were scattered backwards by the target
video clip
Discovery of the Atomic Nucleus
To explain the backscattering, the positive charge
must be concentrated in a small region
Rutherford’s Solar System Model
of the Atom
The atom consists of electrons orbiting around
a small but dense central nucleus
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 doesn’t this happen? Shows that something was
wrong with this model of the hydrogen atom
Absorption Spectrum of a Gas
Dark lines will appear in the light spectrum
Absorption
spectrum of Sun
Emission spectra of
various elements
Balmer’s 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 lines:
where n = 3, 4, 5 and 6
Bohr’s Model of the Hydrogen Atom
(1913)
He proposed that only certain orbits for the
electron are allowed
Bohr’s Empirical Explanation
 Electrons can only take discrete energies
(energy is related to radius of the orbit)
 Electrons can jump between different orbits
due 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
Absorption / Emission of Photons
and Conservation of Energy
Ef - Ei = hf
Ei - Ef = hf
Energy Levels of Hydrogen
Electron jumping to
a higher energy level
E = 12.08 eV
Spectrum of Hydrogen
Bohr’s formula:
Hydrogen is therefore a fussy absorber /
emitter of light
It only absorbs or emits photons with precisely the
right energies dictated by energy conservation
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
Schrödinger’s Improvement
to Bohr’s Model
 Showed how to obtain Bohr’s 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
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
Vibrational Modes of a String
fundamental
2nd harmonic
3rd harmonic
4th harmonic
Energy Levels in a Box
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
Ground State (n=1)
 Lowest or ground-state energy is non-zero
 Electron cannot sit still but must be forever
‘jiggling around’
 Expected from the Heisenberg uncertainty principle
Vibrational Modes of a Rectangular
Membrane
(1,1) mode
(1,2) mode
(2,1) mode
Vibrational modes of a circular membrane (drum)
(2,2) mode
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, -ℓ+1, …, ℓ-1, ℓ
 For historical reasons, ℓ = 0, 1, 2, 3 is also known
as s, p, d, f
1s Orbital
Density of the cloud gives probability of
where the electron is located
2s and 2p Orbitals
Another diagram of 2p orbitals
Note that there are three different configurations
corresponding to m = -1, 0, 1
3d Orbitals
Now there are five different configurations
corresponding to m = -2, -1, 0, 1, 2
4f Orbitals
There are seven different configurations
corresponding to m = -3, -2, -1, 0, 1, 2, 3
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)