# Quantum Theory

Black Body Radiation

When a body is heated, it will begin to transfer heat to its surroundings through three mechanisms:

1. Convection
2. Conduction
3. Radiation

Black body radiation is concerned with the last of these mechanisms.

The rate of transfer of heat from or to a body through radiation is given by:

…..(1)

where e = emmissivity of the body, which depends on the material being used.

s = Stefan-Boltzmann constant = 5.67 x 10-8 (W m-2 K-4)

A = surface area (m2)

T = Absolute temperature

Observation shows that as the temperature increases so the object "glows" with different coloured light. For example:

3 K glows in the microwave band

500 K glows in the infrared band

6000 K glows in the yellow part of the spectrum

The peak spectral output is given by:

…..(2)

where l p = wavelength of maximum spectral output

T = Absolute temperature

The full spectrum of radiation was developed by Max Plank. His imperical formula is given as:

……(3)

where I = Intensity as a function of wavelength

h = Plank's constant = 6.626 x 10-34 (J s)

c = speed of light = 3.0 x 108 (m s-1)

l = Wavelength (m)

k = Boltzmann's constant = 1.38 x 10-23 (J K-1)

T = Absolute temperature

Figure 1: Black body spectral distribution for three temperatures

Plank then tried to develop a theoretical basis for his formula. After two months found that he could only obtain it by assuming that the molecules did not behave according to classical wave theory, but shed their energy as a finite number of discrete amounts related to the frequency of the radiated light such that:

…..(4)

where E = Energy (J)

n = an integer 1, 2, 3, ….

h = Plank's Constant

f = Frequency of radiated light

Einstein then suggested that the light emitted from the body would be made up of a series of discrete bundles of energy. This is given by:

….(5)

These bundles of energy were called "photons". A series of experiments then confirmed that light did exhibit a "particle" nature as well as a wave nature.

If light was a particle, then it should possess some momentum. To calculate the momentum of a photon, we must explore the relativistic properties of this particle. From relativity we know that:

Therefore

Using a useful mathematical trick:

Therefore

Now where p = momentum of the particle.

Therefore:

At relativistic speeds where mo is the rest mass of the particle.

Therefore

…….(6)

Now a photon is never "at rest", therefore mo = 0. Thus

or ……..(7)

So we have an equation for the momentum of a photon. A. H. Compton showed in a series of experiments that collisions between photons and other small particles resulted in a transfer of momentum, which matched these predictions.

Wave – Particle Duality

It had already been shown that light behaved as a wave and was able to be defracted and refracted. Now it had been shown that light behaved as a particle. The only possible conclusion was that light was both a wave and a particle at the same time. Although this is almost impossible to visualise, the experimental evidence supported this.

In 1923 Louis de Broglie extended this concept to all particles, proposing that a moving particle would also exhibit wave properties. Starting with equation (7) above we see that:

Now therefore .

From wave theory, , therefore

Rearranging gives …..(8)

Thus the wavelength of the "particle wave" is related to the momentum of the particle. For massive objects, the wavelengths are so small that they are undetectable, but for objects the size of electrons and protons, these wave should be detectable and useful. Many experiments have shown that electrons (or any other subatomic particles) does exhibit wave behaviour including defraction and refraction. This has given rise to a series of new technologies, including electron microscopes, which use the wave properties of electrons to magnify very small objects.

The Heisenberg Uncertainty Principle

When measuring the position and movement of an object we need to use photons (or some other particle) to observe them. In order to "see" an object clearly the object must be much greater in dimension that the wavelength used to observe it. But the smaller the wavelength, the more energy and momentum the photon will impart to the object in the process of observation. So, if we use light of wavelength , then the most accurate measurement will have an error of approximately one wavelength.

Similarly, if only one photon is needed to make the observation, it will impart its momentum of to the object. Therefore the error in measurement will be approximately .

Thus …..(9)

It could be argued that the position of an object could be related to the "phase" change experienced by the wave as it passed the object, in which case .

Thus ….(10)

This is Heisenberg’s uncertainty. Therefore, the more accurately we know one parameter of the particle, the less accurately we know the other.

Stated more completely:

…(11)

Schrodinger’s Wave Equation

A wave can be described by the general harmonic function:

…(12)

where a and b are constants and k is the phase of the wave.

I.e. ….(13)

Since for a particle

…(14)

where p = momentum of the particle

h = Plank’s constant

By conservation of energy

….(15)

where U = potential energy

p = momentum

m = mass of the particle

Substituting for p gives

….(16)

So we are looking for a differential equation which satisfies conservation of energy and has as its solution. So

Multiplying both sides by gives

Now , therefore

or

…(18)

This is the differential form of Schrodinger’s equation.

Example

Now let us consider a particular solution to this equation. Consider a particle trapped inside an infinitely deep potential well.

U = 0 for 0 < x < L

U = infinity for x < 0 and x > L.

For x = 0 to x = L

From equation (16)

….(19)

Outside this well, U = infinity, therefore k = 0 and =0.

Now consider the boundary conditions:

At x = 0 =0.

Therefore

At x = L,

Now "a" can not be zero or there will be no particle. Therefore

This is satisfied when where n = 1, 2, 3, ...

Note: n can not be zero, or there will be no particle.

Thus

So from equation (19)

or

...(20) where n = 1, 2, 3, ...

Note: when n = 1 this is the lowest "state" that the particle can possibly have. The energy at this state will be:

This is called the "zero point" energy. One outcome of this result is that even at absolute zero temperature (0 K) particles will not have zero energy as would be expected from classical mechanics.

Wave Amplitude

One last stipulation of the wave equation is that the ...(21)

Now

So

Now let

Note: when x = L and

Thus

By definition

Therefore

...(22)

So the full form of the wave equation is:

...(23)

for 0 < x < L

Some of the outcomes of this analysis are:

1. The particle can only occupy discrete energy states, each varying in energy and particle frequency.
2. The particle can never truly be "at rest". There will be a "zero point" energy associated with the particle, which corresponds to the lowest state which the particle may exist in.
3. When considering the case of an electron orbiting around the nucleus of an atom, this explains why an electron does not loose all its energy and callapse into the nucleus. It also explains why discrete quanta of energy are involved in all electron interactions in the atom.

Atomic Theory

We note in the analysis above that Schrodinger's equation in one dimension gives rise to one quantum number. In four dimensions (Relativistic Space-Time), there are four quantum numbers needed to describe the behaviour of a particle.

Thus, for an atom, the behaviour of the electrons must be described by four unique quantum numbers. These are:

1. n = principle quantum number = 1, 2, 3, 4, ....
2. l = orbital quantum number = 0, 1, 2, ... n – 1. This is related to the angular momentum of the electron.
3. ml = magnetic quantum number = -l -(l-1) ... -1, 0, 1, ...(l-1), l
4. ms = "spin" quantum number = -0.5 or 0.5 only.

The two most important quantum numbers are n and l. In atomic configuration tables, "l" is refered to by a letter of the alpherbet such that:

1. l = 0 is called the "s" subshell
2. l = 1 is called the "p" subshell
3. l = 2 is called the "d" subshell

The Exclusion Principle

It has been shown that no two electrons in an atom can occupy the same quantum state.

Electron Configuration

From this it is possible to predict the configuration of the ground states of all the elements in the periodic table. For example:

The quantum state for the Helium atom is:

 n l ml ms 1 0 0 0.5 1 0 0 -0.5

The electron configuration is described as:

1s2 – which indicates that the primary quantum number n = 1, l = 0 and there are two electrons in this subshell.

For Sodium it will be:

 n l ml ms 1 0 0 0.5 1 0 0 -0.5 2 0 0 0.5 2 0 0 -0.5 2 1 1 0.5 2 1 1 -0.5 2 1 0 0.5 2 1 0 -0.5 2 1 -1 0.5 2 1 -1 -0.5 3 0 0 0.5

The electron configuration is described as:

1s22s22p63s1

The same result can be derived for each of the elements of the periodic table.