1) The document explains Johann Balmer's empirical formula for the emission spectrum of hydrogen and how it relates the energies of emitted photons to integer values. 2) It then discusses early quantum models like the "electron in a box" model which showed energy must be quantized. 3) Finally, it describes Erwin Schrödinger's wave equation theory of quantum mechanics which successfully explained the quantization of energy levels in hydrogen and allowed prediction of atomic emission spectra.

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