Why does classical physics predicts infinite energy when
applied to the ultra–violet range of black body radiation – the so called
black body catastrophe. Unfortunately
a rigorous mathematical treatment involves integration that is out of the scope
of A-level

I have freely adapted work from an excellent book – “In
search of Schrodinger’s cat” by John Gribbin ISBN 0-552-12555-5– which is
almost like a novel in structure and I cannot recommend it enough in it’s
style as a non mathematical introduction to quantum mechanics

**The catastrophe:**

Classical physics starts with assumption that hot object
emit e/m radiation, and the hotter they are the more they emit in the higher
frequency end. (red hot – white
hot – blue hot – to U.V. hot). This
e/m radiation must be associated with vibrations of electrons within their
atoms.

In the 19^{th} century the physicists tried to
combined e/m theory with statistical mechanics (the kind of maths that Boltzman
used to predict the behaviour of an ideal gas.
He considered the velocity of an atom in three dimension – and then
adding up all possible atoms and their possible motion to get the total pressure
of a hot gas)

It didn’t work -
and predicted infinite electromagnetic energy in emitted in the UV spectrum.

Why?

They considered the movement (the vibration) of the
electrons orbiting the nucleus like the vibration on a guitar string.
If the electron wobbles on it’s orbit around the atom
- then they could try to predict the e/m energies that would create.
So picture (incorrectly I must add) the electron not just orbiting the
atom but bouncing up and down on it’s way round the nucleus.

Then think it might vibrate twice on its way round in one complete circle or three times or four time or any number of infinite wobbles possible in on orbit of the atom. So we add up all possible ways of vibrating – with the possible energies associated with each electron – and get a very big number. It’s not too bad a model for low frequency e/m waves – the theory isn’t that far off – as each extra mode of vibration we have to add up makes less and less difference to the overall energy. But for a certain range of frequencies (that just so happen to be around the UV range) adding up all of the available energies due to the infinite number of ways that an electron could vibrate (all the different waves that could fit into the atom) comes up with an infinite value – because the energy of each vibration does not tail off as we get into smaller and smaller wavelengths of vibrations of the electrons that could possible fit into the atom.

Eg. I’ll make
up some numbers to help illustrate the point:

- for low
frequency e/m waves – the energy of them could be made up by adding together
these numbers … 10J + 1J + 0.01J +
0.0001J + 0.000001J which may not equal infinity as you add them up – even if
you carry on adding up the sequence forever.
Each number represents the energy equivalent of the 1^{st} and 2^{nd}
and 3^{rd} (and so on)
harmonics.

- for U.V. e/m waves – the energy sequence might be like
this: 100J + 99.99999J + 99.99998J
+ 99.99997J and so on – which may well total up to infinite energy if you
carry on adding up forever.

Planck’s response was to **stop **the sequence – and
say that there is a **minimum** energy that anything can vibrate at.
The smallest possible energy that these mini vibrations could ever have
is E= h f.
The f is the smallest mode of vibration possible and at that point the
sequence ends because there can be no difference of energy less than that value.

Or put another way – energy is quantised – like charge is – and comes in little packets.