# An Introduction to Quantm Mechanics

## The Blackbody Effect and the Ultraviolet Catastrophe

A blackbody is something that would absorb all of the radiation incident to it and then re-emit it. We see examples of this everyday - objects that are heated will glow with red then white light as the temperature increases. Cool objects have long wavelengths (infrared and below) and hotter objects glow white then blue with increasingly shorter wavelengths (ultraviolet and up). In fact, we can approximate the temperature at the peak intensity by using a formula known as Wein's Law

lp T = 2.9 x 10-3 m K

where lp is the peak intensity wavelength and T is the temperature in Kelvin. For example, we can estimate the surface temperature of the sun. The sun gives off light where the strongest intensity occurs at about 500 nm. From Wein's Law, T = 2.9 x 10-3 m K / lp which is approximately 6000 K.

So we know that objects give off light when heated. Even cold objects will give off radiation (below the visible band) and very hot objects can give off radiation way above the visible band of light. Classical physics (using the wave model of light) attempted to explain this phenomenon. The problem is that physicists expected the intensity to change as objects heated up, not the wavelength. Furthermore, the wave model predicted radiation would be very high frequency for blackbodies - the UV range and above and that this energy would continue to build up and "run away". This was clearly not happening. This was the "ultraviolet catastrophe". Okay - it is probably only catastrophic to a classical physicist. But it did show that these blackbodies were not behaving the way they should have.

Enter Max Planck (generally referred to as the Father of Quantum Mechanics). Planck tried to take experimental data and fit it into a curve or formula that would explain what was happening with blackbody radiation. He came up with a startling revelation - that the energy that was being released by the oscillation of the molecules in the object as it was heated was not being released in a continuous spectrum, but was being released in a series of "quantized" packets (that is, as discrete amounts) based on the frequency of the emitter. Thus, he related the energy released to the frequency by the formula:

E = nhf

where E is the energy released, n is an integer value, f is the frequency of the wave being emitted, and h is Planck's Constant = 6.626 x 10-34 J s. The significance of the integer is to show that energy can only be released in quantized amounts - This is kind of like the difference between sliding a box up a ramp and moving it up stairs. On a ramp, it can be at any position. But on the stairs, there are a finite number of levels that the box can be placed at.

Here comes Einstein. In 1905, while working as a patent clerk, he produced some incredible work. This was the year that he published his theory of relativity. But the work that won him the Noble Prize in physics was on quantum mechanics and the photoelectric effect. Einstein said, basically, that light is emitted by vibrating particles and is itself a particle called a photon. He also stated that when light is emitted by an oscillator, it changes the energy of the oscillator by one integer value. Each photon, therefore, should have an energy given by:

E = hf

Einstein postulated his theory using the photoelectric effect, or the fact that certain metals released electrons when a light source struck them. For example, zinc will produce a current when UV light shines on it. This is what gives rise to photocells, and is one method of generating solar power. (A second method actually focuses solar energy to heat and boil water for steam to drive turbines.)

It would be possible to set up an experiment using a photocell and a voltage source with an ammeter to measure current. As light strikes the photocell, it would kick electrons free and this would cause a current flow in the circuit. If the voltage source is set up to be in opposition to that current flow, then electrons below a certain energy level would not be able to cause current to flow. We call this energy level the stopping voltage ( Vo)and we can use it to determine the KE of the electrons being emitted (KEmax = eVo.) Classical physics (the wave model) predicted:

• Increasing the intensity of the light would increase the kinetic energy of the electrons being produced.
• The frequency of the light should have no effect on the KE.

Photon theory says all photons have the same energy (hf) and that increasing the intensity just increases the number of photons striking the plate in a given time. Consider that electrons are held bonded in the metal by forces and that the minimum energy to break that force is called the work function, Wo. If the frequency of light striking the plate is low, the energy induced by the photon will be low (less than Wo ) and no electrons will be emitted. If E is greater than Wo, then energy in excess of that required to break the electron free will be transformed into kinetic energy in that electron. In other words,

E = KE + Wo,

or

hf = KE + Wo.

## Compton Scattering and Pair Production

In 1923, A. H. Compton performed a series of experiments to show the particle nature of light. In one of these, he scattered x-rays from different materials.

In this experiment, he found that the scattered x-rays (which are really just low frequency light) had a lower frequency than the incident light. (in our diagram, f < fo) The rest mass (mo) of a photon is zero, so we need to use other methods to determine the momentum of a photon. From a derivation based on relativistic mass and velocity, the momentum of a photon, p can be given by the formula

p = E/c

where E is the energy of the photon (related to frequency from Planck's hypothesis) and c is the speed of light. (Also the speed of the photon!) This may also be written as

p = hf/c = h/l.

Compton combined this with the laws of conservation of energy and momentum and determined that the change in wavelength of the photon after a collision can be given by the equation:

where f is the angle between the incident photon and the scattered photon.

In pair production, the photon collides with a nucleus and converts its energy into matter in the form of a positron and an electron. Both need to be created since a photon has no charge, and therefore the sum of the resulting particles (+ and -) is also zero. Energy is conserved in the collision, and a nucleus must be present so that momentum in the collision is conserved. We can actually compute the minimum energy and wavelength of the photon required for pair production to take place using the equation E = mc2. Since the mass comes from a positron and an electron (m = 2 electrons),

E = (2)(9.11 x10-31 kg)(3.0 x 108m/s)2 = 1.64 x 10-13 J = 1.02 MeV.

This is the minimum amount of energy a photon must have to cause pair production. Since E = hf =hc/l, substituting in for Energy, Planck's constant h, and c, we find the wavelength of this photon is 1.2 x 10-3 nm. This is a very short x-ray on the EM spectrum.

In summary, there are four possible outcomes from the collision of a photon with matter (generally a nucleus or an electron):

• The photon loses some of its energy from being scattered off the matter (The Compton effect) and the scattered photon has a lower frequency although it still travels at the speed of light, c.
• The photon may knock an electron out of an atom. This is the photoelectric effect. In this case, we lose the photon.
• The photon may knock an electron into a higher energy state and disappear. When the electron returns to the lower energy state, it emits a photon with energy equal to the difference in the two energy states. We'll talk about this more in our lesson on atomic spectra.
• A photon will create a positron and an electron. This is called pair production.

For more on Compton Scattering, try: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/comptint.html

For more on the Photoelectric Effect, try: http://hyperphysics.phy-astr.gsu.edu/hbase/mod1.html

or for an applet you can play with: http://lectureonline.cl.msu.edu/~mmp/kap28/PhotoEffect/photo.htm

And for more on the ultraviolet catastrophe, try http://spiff.rit.edu/classes/phys314/lectures/bb/bb.html