1. Setting the scene
and deriving the Uncertainty Principle
The Photoelectric Effect
The photoelectric effect was the first example of a quantum phenomenon to be seen at the end of the 19th Century. Light or other forms of electromagnetic radiation shone onto metals release electrons; the energy supplied by the radiation frees the electrons from the metal.
The way in which the numbers and energies of electrons released changes when the frequency and intensity of the radiation changes cannot be explained using the classical wave model of light.
- Increasing the intensity of the radiation does not increase the energy of the electrons but releases more of them per second.
- Increasing the frequency of the light increases the energy of the electrons.
- Below a certain frequency of radiation, f0, no electrons are emitted no matter how intense the radiation.
These facts are explained using the photon model of light. Light (and all EM radiation) is emitted and absorbed in little packets or quanta called photons. The energy of a photon is equal to its frequency multiplied by Planck's constant, h = 6.63 x 10-34 Js.
E = h f
The photoelectric effect is summed up by Einstein's photoelectric equation for which he won the Nobel Prize.
KEmax of electrons = hf - W where W is the energy needed to escape from the metal.
This phenomenon introduces the
wave-particle duality of nature: light behaves as a wave at times
(e.g. Young's slits) and as a particle at times. This duality is
central to the way quantum mechanics explains nature as it applies
to everything.
Young's slits
Thomas Young used this experiment to 'prove' that light was a wave at a time when light was thought to be a particle. The light going through two slits interferes and produces a pattern that is easy to explain using a wave model but which cannot be explained if light acts like particles.
In 1924 Louis de Broglie suggested that if light behaves as a wave and a particle then perhaps electrons, thought to be particles, might behave like waves. He put forward the following equation:
for the wavelength of particles. In 1928 Thompson showed that electrons diffracted through crystals. He won the Nobel Prize for showing that electrons were waves whereas his father had won the Nobel Prize for showing that electrons were particles!
In 1974 the Young slit experiment was carried out with electrons. [ref: Jonsson, Electron Diffraction at Multiple Slits, Am. Journal of Physics, 42,4-11,1974] We have electron microscopes which exploit the wavelike properties of electrons to produce a picture.
Making an assumption
These 'probability-waves' obey exactly
the same rules as 'normal' waves such as water or sound
waves.
Thinking about pulses of sound
Listen to these three pairs of pulses. In each pair one of the pulses is higher in pitch.
- Which one is higher?
- With which of the three pairs is it easiest to tell this?
- Why is it easier to tell?
In the first pair both pulses lasted 0.01s, in the second they lasted 0.05s and 0.5s in the third. You should have heard that the second pulse was slightly higher (805Hz rather than 800Hz) and you should have found it a lot easier to detect this with the longer pulses.
The longer the sound wave lasts for, the easier it is to measure its frequency.
To measure the frequency of a pulse to a certain accuracy Df therefore we need it to last at least a certain time:
Some maths that is more difficult than we need to worry about gives
so that the formula becomes
We got this result thinking about sound waves but it should apply to all waves including our quantum probability ones. Remembering that E=hf we can substitute this in and get
which can be rewritten as
This equation is very important in
quantum mechanics. It is one form of Heisenberg's Uncertainty
Principle In the form we are going to use in order to measure the
energy of a system to within DE
needs a time interval of at least Dt.