1. Explanation of spectra
 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:
Johann Balmer
⎛1 1⎞
(1825-1898)
∆E = −Rz gZ ⎜ 2 − 2 ⎟
2
⎝ n f ni ⎠
where n may take integer values 3, 4, 5, … and R is a
constant number

2. Explanation of spectra
 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:
hc
E = hf =
λ
 This means that the energy of the atom is discrete, that is, not
continuous.

3. The “electron in a box” model
 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:
h
λ= the electron can only be found somewhere
p along this line
x=0 x=L

4. The “electron in a box” model
 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:
2L
λ=
n Where n is an integer

5. The “electron in a box” model
 Therefore, the momentum of the electron is:
h h mh
p= = =
λ 2L 2L
n
 The kinetic energy is then:
2
 mh 
2   2 2
p  2L  = n h
Ek = =
2m 2m 8mL2

6. The “electron in a box” model
 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:
h2
1× n =1
8mL2
h2
Ek = 4×
8mL2
n=2
h2
9× n=3
8mL2
 However, this model is not correct but because it shows
that energy can be discrete it points the way to the correct
answer.

7. The Schrödinger theory
 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) Erwin
Schrödinger
 This wave is called wavefunction and
(1887-1961)
represented by:
ψ( x, t )
 This wave is a function of position x and time t. Through
differentiation, it can be solved to find the Schrödinger function:
∂ 2
i ψ( r , t ) = − ∇ ψ( r , t ) +V ( r ) ψ( r , t )
2
∂t 2m

8. The Schrödinger theory
 The German physicist Max Born interpreted Schrödinger's
equation and suggested that:
2
ψ ( x, t )
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.

9. The Schrödinger theory
 Solving for Hydrogen, it is found that:
13.6
E = − 2 eV
n
 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.

10. 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.
This high n
is essential to energy energy
understand why levels very
0 eV close to
some spectral lines each other
are brighter than n=5
others. n=4
n=3
Thus, the
Schrödinger theory n=2
explains atomic
-13.6 eV
spectra.
n=1