Lecture One

Obligatory Introduction...


Lecture Two

Principle of Newtonian relativity, Galilean transformation, Galilean addition law for velocities, the Michelson-Morley experiment

Special Theory of Relativity

1. All laws of Physics are the same in all inertial reference frames.

2. The speed of light is constant in all inertial frames regardless of the velocity of the observer or the velocity of the source emitting the light.


Lecture Three

The Einstein postulates, consequences of special relativity, simultaneity and the relativity of time, time dilation

Simultaneity is not an absolute concept but one that depends upon the state of the motion of the observer.

Lecture Four

Length contraction, the twins paradox

Consider a spaceship travelling with speed v to another planet a distance L_p away as measured from Earth.
To the observer on Earth, the time taken is Δt = L_p/v.
To the observer on the spaceship, the time taken due to time dilation is Δt_p = Δt/γ due to time dilation (Δt_p ≤ Δt)
N.B. we cannot reverse the frame of reference and assume the spaceship is stationery and the Earth (and universe) is moving relative to it, since the situation is not symmetrical. The spaceship has to undergo acceleration to get to a speed of v.

Lecture Five

The Lorentz transformation

To transform coordinates in S' to S frames, just replace v by -v and interchange primed and unprimed coordinates.

The Lorentz transformation approximates to the Galilean transformation when v << c (relatively low v).

N.B. Lack of simultaneity, time dilation and length contraction can be derived from the Lorentz Transformation.

Lecture Six

Relativistic momentum, relativistic energy, mass-energy equivalence, energy momentum relationships

The laws of physics must remain unchanged under the Lorentz transformation.

Einstein's mass-energy equivalence - mass is a property of energy.

If a body gives off energy in the form of radiation, then its mass must diminish by E/c².
The mass of a body is a measure of its energy content.


Lecture Seven

Relativity and electromagnetism, general relativity

Charge thought experiment - what was viewed as a pure magnetic field in one frame is transformed into electric + magnetic field in another frame.

Two postulates of Einstein's general relativity:

1. All laws of nature have the same form for observers in any frame of reference, whether accelerated or not

2. (The principle of equivalence) In the vicinity of any given point, a gravitational field is equivalent to an accelerated frame of reference in the absence of gravitational effects

Some consequences:

Time scales are altered by gravity - a clock in the presence of gravity runs more slowly.

Light bends in the presence of gravitational fields.

In extreme cases, light paths may be so curved around a very great concentration of mass, such that it cannot escape (a black hole).

The horizon surrounding that dense mass such that light emitted within it can never reach an external observer is the event horizon.


Lecture Eight

The blackbody radiation puzzle and Planck's hypothesis

Black body - an object that absorbs all electromagnetic radiation falling onto it.

Black body radiation - the light emitted by a black body.

Atoms and moleclues at the surface of the object increasingly vibrate as T increases, emitting radiation.

Unfortunately there is an ultraviolet catastrophy - the classical model predicts I → ∞ as λ → 0. This is at odds with experimental results.

Planck made two assumptions in formulating this theory:

1. The molecules emitting the radiation can only have discrete units of energy E_n, where E_n = nhf, where n (the quantum number) is a positive integer and f is the frequency of vibration of the molecules

2. The molecules emit (or absorb) energy in discrete packets called photons, which each have energy hf

This proved to be the beginning of quantum physics.


Lecture Nine

The photoelectric effect puzzle and Einstein's solution, applications of the photoelectric effect

Photoelectric effect - the emission of electrons from certain metal surfaces due to incident light.

Classical physics could not explain many features of the photoelectric effect:

Stopping potential V_s (the maximum kinetic energy of the emitted electrons) is independent of light intensity but proportional to light frequency.

The induced current (the number of emitted electrons) is proportional to the light intensity.

No electrons are emitted if the frequency of the light is below some cut-off frequency f_c.

Electrons are emitted almost instantaneously (< 10^-9 secs) even at low intensities.

Older applications of the photoelectric effect: Light activated burglar alarm, digital soundtracks on side of film.

Modern application of the photoelectric effect: Image enhancement (night vision goggles).



Lecture Ten

The Compton scattering effect, the Compton scattering formula

Compton and his co-workers accumulated evidence that the scattering of X-rays (λ₀ = 0.071nm) by electrons could not be explained by classical theory.

Compton's results: Two intensity peaks: one at λ₀, and a (higher) one at a longer wavelength λ'.

This disagreed with the classical prediction of a distribution of Doppler shifted values.

Had to be explained by assuming that light consists of photons with definite energy and momentum - light as a beam of particles.

The Compton scattering equation can then be derived by using the conservation of energy and momentum.

Lecture Eleven

Atomic spectra, emission spectra, absorption spectra, Bohr's quantum model of the atom

All substances emit radiation depending on their temperature. This radiation is characterised by a continuous distribution of wavelengths. The hotter the substance, the more likely it is to emit visible radiation.

Emission spectrum - the few discrete wavelengths emitted when passing an electric current through a low pressure gas (electric discharge).

Absorption spectrum - the few discrete wavelengths removed from a continuous spectrum when passing light from a continuous source through a substance.

Bohr's quantum model of the atom

The four Bohr postulates:

1. The electron moves in circular orbits around the proton under the influence of the Coulomb attraction

2. Only certain electron orbits are stable. These stable orbits do not emit any radiation. The total energy of the electron in the orbit remains constant, and classical mechanics can be used to describe its motion

3. Radiation is emitted by the atom when the electrom "jumps" from a more energetic initial orbit to a lower orbit. This jump cannot be treated classically. The frequency of emitted radiation is E_i - E_f = hf, E_i > E_f

4. The size of the allowed electron orbits is constrained by the electron's orbital angular momentum. The allowed orbits are those for which the electron angular momentum about the nucleus is an integral multiple of h/2π, i.e. mvr = nh/2π where n = 1,2,3...



Lecture Twelve

Energy levels of the electron in the hydrogen atom, radii of Bohr orbits in hydrogen, energy quantisation of electron orbits, Lyman, Balmer, Paschen series

Using the four Bohr postulates, we can calculate the allowed energy levels and emission wavelengths of the hydrogen atom.

The electrical potential energy of the hydrogen atom is U = -k_ee²/r.

The total energy E of the atom is E = K + U = mv²/2 - k_ee²/r.

Since the Columb attractive force on the electron equals the centripedal force, k_ee²/r² = mv²/r,

therefore K = mv²/2 = k_ee²/2r.

Substituting into the expression of E above, we get:

We can square and rearrange mvr = nh/2π to get

v² = n²h²/(2π)²m²r².

We can also use K = mv²/2 = k_ee²/2r to get

v² = k_ee²/mr.

By equating these two equations (thus eliminating v²) and solving for r, we have:

It follows that the quantisation of orbit radii leads to the quantisation of the electron energy.

By substituting the expression for the radius r_n = n²a₀ into E = -k_ee²/2r, we have

The lowest energy level (ground state) corresponds to n = 1 (and the smallest orbit). E₁ = -13.6 eV

The highest energy level (n = ∞) represents the state where the electron is removed from the atom. E_∞ = 0 eV.

Thus the ionisation energy for the hydrogen atom is 13.6 eV.

Using the third Bohr postulate E_i - E_f = hf, f = (E_i - E_f)/h.

Substituting E_n = -[1/n²]k_ee²/2a₀, we get

f = (k_ee²/2a₀h)[(1/n²_f) - (1/n²_i)]

and thus

1/λ = f/c = (k_ee²/2ca₀h)[(1/n²_f) - (1/n²_i)]

which is exactly the empirical general formula for spectral lines in hydrogen when the Rydberg constant R_H = (k_ee²/2ca₀h)!



Lecture Thirteen

The wave properties of particles, the de Broglie hypothesis, quantisation of angular momentum in the Bohr model of hydrogen, the Davisson-Germer experiment, Young's double slit experiment using light, Young's double slit experiment using electrons, probability distributions (whew!)

Principle of complementarity - the wave and particle descriptions of light are complementary. We need both descriptions to explain nature, but only one at a time.

Like light, electrons also produce an interference pattern.

If the electrons go through the slit one at a time, we still get the interference pattern!

The electron, in going through one slit, interferes with itself going through the other one!

We have to turn to a mathematical interpretation: Quantum mechanics.

Assume the particle can be represented by a mathematical expression - a wavefunction, ψ (which may be complex)

Also assume that the intensity profile of the interference pattern (num of particles/sec) can be expressed by the square of the absolute value of this wavefunction, |ψ|²

If we open slit 1 (with slit 2 closed), the wavefunction of electrons passing through it is ψ₁, and its intensity profile is |ψ₁|²

If we open slit 2 (with slit 1 closed), the wavefunction of electrons passing through it is ψ₂, and its intensity profile is |ψ₂|²

If we open slit 1 for half the time (with slit 2 closed), and then open slit 2 for half the time (with slit 1 closed), the intensity profile is |ψ₁|² + |ψ₂|²

If we open both slits, the electron wavefunctions are superimposed (similar to light), and the combined wavefunction is (ψ₁ + ψ₂). The intensity profile is then |ψ₁ + ψ₂|².

|ψ₁ + ψ₂|² = |ψ₁|² + |ψ₂|² + 2(ψ₁.ψ₂)

The term 2(ψ₁.ψ₂) represents the interference term.

The wavefunction does not have any physical meaning - they are just mathematical interpretations which appear to work.

The square of the wavefunction is however physically meaningful, representing the probability of a particle being detected at a particular position (the probability density).



Lecture Fourteen

Wave packets, the Heisenberg uncertainty principle

We cannot envisage what a particle/wave looks like if it is small - we need a mathematical representation.

The particle wavefunction should have wave properties, and yet be localised in space.

The wave packet representation of a particle - the particle is "fuzzy".

An isolated wave group is the result of superimposing an infinite number of waves with different wavelengths.

For example, at a certain time, the wave group as a function of x is represented by:

ψ = a₀sin(2πx/λ₀) + a₁sin(2πx/λ₁) + a₂sin(2πx/λ₂) + ...

or

ψ = a₀sink₀x + a₁sink₁x + a₂sink₂x + ...

where k = 2π/λ is the wave number, and a₀ ... etc are constants.


Suppose a particle has a very well defined size. This has to be represented by a very localised wave packet. Its mathematical representation requires very many superimposed waves with a wide range of k.

Therefore Δx (the spatial extent of the wave group, or its fuzziness) is small, and Δk (the range of x) is large.

Since p = h/λand k = 2π/λ, then k = 2πp/h, and Δk = 2πΔp/h - Δp is proportional to Δk.

i.e. when Δx is small, then Δp is large.

conversely, when Δx is large, then Δp is small by the same argument in reverse.

Interpretation: If the position of a particle is well defined, we do not know its momentum very well, and vice versa.

Heisenberg stated in 1927 that if a measurement of position is made with precision Δx and a simultaneous measurement of momentum is made with a precision Δp_x, then the product of the two uncertainties can never be smaller than h/2π.

Note that the Bohr model of the atom is unrealistic according the the Heisenberg principle, since the uncertainty in the radial position of the electron is double the Bohr radius assuming that the uncertainty in electron momentum is less than mv.



Lecture Fifteen

Wave functions, probability density, expectation value, particle in a box of infinite sides

The probability of experimentally finding a particle described by the wave function ψ at (x,y,z) is |ψ|².

Normalisation condition - the sum of the probabilities that the particle must be somewhere along the x-axis is 1 in a one-dimensional system.

The probability of finding a particle in the interval b ≥ x ≥ a is:

P_ab = (integrate from a to b) |ψ|² dx.

Expectation value - the average value of the position of the particle represented by ψ within the region a to b

< x > = (integrate from a to b) x|ψ|² dx.

Consider a particle confined to moving along the x-axis, bouncing back and forth between two impenetratable walls.

If its speed is v, then the magnitude of its momentum mv is constant, as is its kinetic energy mv/2.

The quantum mechanical treatment of a particle in a box is similar to that of standing waves, i.e. only those functions that satisfy the conditions of a node at each wall are allowed.

Note that the wavelength of a standing wave on a string is quantised.

In analogy with standing waves, the allowed wave functions for the particle in the box are sinusoidal and are given by:

since the wavelengths of the particle in the box are quantised and restricted to the condition λ = 2L/n, the magnitude of the momentum is also quantised:

And if the momentum is quantised, then so is the energy:

E_n = mv²/2 = p²/2m = (nh/2L)²/2m