Recall:

Max Planck assumed that light energy came in little "bundles" called photons, with the energy of each bundle depending on the frequency of the light

E = hn,

h = 6.626 ´ 10-³⁴ J s.

Note that the energy of one photon is likely to be pretty small.

With this assumption Planck was able to derive the formula for the frequency distribution in black body radiation (thermal radiation).

Einstein used the Planck assumption that the energy of a photon was hnto solve the problem of the photoelectric effect.

That is, light impinging on a metal surface ejects electrons. The electrons have a kinetic energy given by

,

Thus the kinetic energy of the electron depends on frequency and not the intensity of the light. (The intensity determines how many electrons are ejected per second.) Note that if hn< WF there would be no electrons ejected.

(While we are here, Einstein made another interesting application of the Planck law. There was a well known formula for the heat capacity of metals and monatomic solids. It is called the law of DuLong and Petite. It says that the heat capacity of a metal or monatomic solid is 3R, where R - 8.3145 J/K mol is the universal gas constant - you probably know it as 0.08206 L atm /K mol. The law of DuLong and Petite works pretty well for most metals at room temperature and above, but it fails for some monatomic solids - like diamond - and it fails for all metals at low temperatures.

Einstein applied Planck's quantization rule to the mechanical vibration of the atoms in a solid [rather than the vibrations of the electric field in a light wave] and obtained an expression which gave the correct qualitative behavior at low temperatures. [For diamond, room temperature is a low temperature.] )
 

Line Spectra

At the same time people were looking at black body radiation the were also looking at the light emitted from hot gases, like H, and He, etc. Instead of finding a smooth distribution of emitted frequencies they found that gases, H for example, emitted only at a small number of frequencies.

The emission from hot gasses were diffracted through a prism to give a series of lines of different colors instead of a smooth distribution of frequencies. The came to be known as line spectra.

Although there was no explanation for why this should be, people were able to find a formula which fit the wavelengths of light emitted by atomic hydrogen. The wavelengths fit into groups called "series."

One such group called the Balmer series fit an equation, called the Rydberg equation, of the form

,

The constant, R, is called the "Rydberg constant." It has the value, 1.0974 ´ 10⁷ m^-1 (or 0.010974 nm^-1).

People suspected that some of the other series might be obtained by changing the 2² to, maybe, 1², or 3², and so on. That is, maybe there were series whose wavelengths fit the equation

,

,

When people looked for spectral lines with wavelengths given by these formulas the found them!

The series from the first formula is called the Lyman series after the physicist who found it, and the series from the second formula is called the Paschen series after its discoverer. There were even series found with the 2² replaced with 4² and 5².
 
 

The Bohr Model of the Hydrogen Atom

In 1913 Niels Bohr formulated a successful theory of the hydrogen atom. By his time it was known it atoms consisted of a central heavy and relatively small core - called the nucleus - surrounded by the relatively light electrons.

The question is, how are the electrons arranged?

Bohr assumed that the electrons move around the nucleus like the planets move around the sun.

But the physics of Newton and Maxwell predicted that such an atom would radiate away its energy and collapse.

Bohr assumed that for unknown reasons the atom wouldn't radiate away its energy and collapse, but that there were only certain allowed orbits.

He assumed that Planck's formula for the energy of a photon was correct and then converted the Rydberg equation for wavelength into an equation in terms of energy rather than wavelength.

Remember that n = c/l and that Planck's equations gives the energy of a photon as E = hn, so E = hc/l .

Multiply the Rydberg formula by hc and we get an equation that has units of energy.

Based on his modified Rydberg formula, he deduced that the energy of the allowed orbits had to fit a formula of the form

,

(In actual fact, Bohr made another logical leap in his application of quantum principles. He made the intellectual jump from the quantization of light to the quantization of angular momentum. It turns out that h/2p has the same units as angular momentum. Bohr assumed that the angular momentum of the motion of the electron around the nucleus must be an integer multiple of h/2p . With this assumption and the usual coulomb's law and Newton's mechanics he derived the above expression for the energy of an electron in a hydrogen atom.)
 
 

Energy Level Diagrams

Formulas like Bohr's formula for the allowed energies of the electron in a hydrogen atom were a new thing in physics.

However, these formulas have a relatively simple graphical representation. We make a graph with energy plotted on the vertical axis and then we draw a horizontal line to represent each allowed energy of the system. Each different line is said to represent a "state" of the system. This graph is called an energy level diagram.

For example, the energy level diagram for the hydrogen atom would look something like this:

__________ m = ¥

__________ m = 3
 
 

__________ m = 2
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

__________ m = 1
 
 
 
 

The de Broglie equation and the Bohr Atom.

In 1924 Louis de Broglie looked at the results of Planck and Einstein in which it seemed like electromagnetic vibrations (waves) and mechanical vibration waves (in crystals) have particle properties.

He reasoned that perhaps particles might display wave properties. So what is the wavelength of a particle?

He put together Planck's result for photons,

Putting these two together we find that,

cp = hn,

.

de Broglie used this equation to rederive Bohr's equation for the energy of the electron in a hydrogen atom.

The argument is as follows: The electron moves in a circular orbit around the nucleus. This orbit must contain an integral number of wavelengths of the particle, otherwise the particle would cancel itself out due to wave interference.

so

Combining this expression for momentum with Coulomb's law for the attraction between the electron and proton and using standard Newtonian mechanics, de Broglie arrived at Bohr's expression for the energy,

.

Furthermore, he could calculate the value of the Rydberg constant, R, from other constants like the mass of the electron, the charge on the electron, Planck's constant and so on. (You don't have to know this, but for the record, in SI units, the Rydberg constant is given by

.

The New Quantum Theory

The above description of matter is sometimes now referred to as the "old quantum mechanics." The old quantum mechanics did very well in come cases but was not easily extendable to more complicated systems. It worked well for some aspects of the hydrogen atom, like line spectra, but failed in some areas. It couldn't be extended to more complicated atoms, like helium. Also, it did not provide a consistent explanation for Planck's and Einstein's applications of the theory.

In 1926 the Austrian physicist, Erwin Schrödinger, was pursuing de Broglie's idea that matter shared some of the properties of waves. In classical physics wave motion was well known and well studied. All wave motion could be described by an appropriate "wave equation."

The details of a classical wave equation need not concern us here. Suffice it to say that one could write the amplitude of the wave as a function,

Classical physicists knew about wave equations and they knew how to solve them.

Schrödinger set out to find the wave equation which described the wave nature of particles. He found it! The wave equation that describes the wave nature of particles is called Schrödinger's equation and it looks different from the classical wave equation.

Schrödinger's equation replaces the amplitude of the classical wave equation with a "wave function," usually symbolized by the Greek letter psi, y , or Y .

The wave function provides a description of the position of a particle in terms of probabilities.

The square of the magnitude of y tells us where the particle is. At a point in space where y is large the particle is likely to be there. At a point in space where y is small the particle is not likely to be there. And where ever in space y is zero the particle isn't there.

The Schrödinger equation for a system will, in general, have many solutions (usually in infinite number of solutions). Each solution has its own unique wave function, y , and we distinguish them from each other by adding a subscript to give y_i.

Schrödinger's equation does more than just tell us where the particle is likely to be. It also gives us the allowed energies of the system. There is an energy associated with each y_i which we indicate by E_i.