In 1885 Johann Balmer, a Swiss physicist, discovered, by trial and error, that the energies in the emission spectrum of hydrogen were given by the formula:

Explanation of spectra

Johann Balmer

(1825-1898)

where n may take integer values 3, 4, 5, … and R is a constant number

Since the emitted light from a gas carries energy, it is reasonable to assume that the emitted energy is equal to the difference between the total energy of the atom before and after the emission.

Since the emitted light consists of photons of a specific wavelength, it follows that the emitted energy is also of a specific amount since the energy of a photon is given by:

Explanation of spectra

hc

hE f

This means that the energy of the atom is discrete, that is, not continuous.

If the energy of the atom were continuous the emission of light wouldn't always be a set of specific amounts.

The first attempt to explain these observations came with the “electron in a box” model.

Imagine that an electron is confined in a box of linear size L.

If the electron is treated as a wave, it will have a wavelength given by:

The “electron in a box” model

p

h

x=0 x=L

the electron can only be found somewhere along this line

If the electron behaves as a wave, then: The wave is zero at the edges of the box The wave is a standing wave as the electron does not

lose energy

This means that the wave will have nodes at x=0 and x=L. This implies that the wavelength must be related to the size

of the box through:

The “electron in a box” model

n

2L

Where n is an integer

Therefore, the momentum of the electron is:

The “electron in a box” model

2L

mh

n2Lhh

p

The kinetic energy is then:

2

22

2

2

k 8mL

hn

2m2Lmh

2m

pE

This result shows that, because the electron was treated as a standing wave in a “box”, it was deduced that the electron’s energy is quantized or discrete:

The “electron in a box” model

3n 8mL

h9

2n 8mL

h4

1n 8mL

h1

2

2

2

2

2

2

kE

However, this model is not correct but because it shows that energy can be discrete it points the way to the correct answer.

In 1926, the Austrian physicist Erwin Schrödinger provided a realistic quantum model for the behaviour of electrons in atoms.

The Schrödinger theory assumes that there is a wave associated to the electron (just like de Bröglie had assumed)

This wave is called wavefunction and represented by:

The Schrödinger theory

),( tx

Erwin Schrödinger

(1887-1961)

This wave is a function of position x and time t. Through

differentiation, it can be solved to find the Schrödinger function:

),r(ψ )(),r(ψ2

),r( 22

trVtm

tt

i

The Schrödinger theory

The German physicist Max Born interpreted Schrödinger's equation and suggested that:

2),( tx

can be used to find the probability of finding an electron near position x at time t.

This means that the equation cannot tell exactly where to find the electron.

This notion represented a radical change from classical physics, where objects had well-defined positions.

The Schrödinger theory

Solving for Hydrogen, it is found that:

In other words, this theory predicts that the electron in the hydrogen atom has quantized energy.

The model also predicts that if the electron is at a high energy level, it can make a transition to a lower level.

In that process it emits a photon of energy equal to the difference in energy between the levels of the transition.

eV2n

13.6E

The Schrödinger theory

Because the energy of the photon is given by E = hf, knowing the energy level difference, we can calculate the frequency and wavelength of the emitted photon.Furthermore, the theory also predicts the probability that a particular transition will occur.

energyThis is essential to understand why some spectral lines are brighter than others.Thus, the Schrödinger theory explains atomic spectra.

0 eV

-13.6 eV

n=1

n=2

n=3

n=4n=5

high n energy levels very close to each other