Quantum Theory of the Atom
Particles and waves
What is a particle?
A particle is a discrete unit of matter having the attributes of mass,
momentum (and thus kinetic energy) and optionally of electric charge.
What is a wave?
A wave is a periodic variation of some quantity as a function
of location or time. For example, the wave motion of a vibrating
guitar string is defined by the displacement of the string from
its center as a function of distance along the string.
A sound wave consists of variations in the pressure with location.
A wave is characterized by its wavelength
λ (lambda) and frequency ν (nu), which are connected by
the relation
in which u is the velocity of propagation of the disturbance
in the medium.
Problem example: The velocity of sound in the air is 330 m s–1.
What is the wavelength of A440 on the piano keyboard?
Two other attributes of waves are the amplitude (the height
of the wave crests with respect to the base line) and the
phase, which measures the position of a crest with respect
to some fixed point. The square of the amplitude gives the
intensity of the wave.
Absorption lines of Sodium
Emission lines of Sodium
J J Thompson and the Cathode Ray Tube
Ernest Rutherford
De Broglie
Wave Behaviour of Particles
The Belgian physicist de Broglie (pronounced ‘de Broy’) reasoned
that if waves have a particulate properties, it was reasonable to
suppose that particles had wave properties. He devised the relationship,
which states that particles have wave properties.
It is the logical extension of the particulate nature of electromagnetic
wave phenomena.
He combined the following equations:
Energy of photons:
E = hf
Einstein’s mass equivalence:
E = mc2
Therefore hf = mc2.
Now f = c/λ
So mc = h/λ
λ=h/mc =h/p
The term mc is mass ´ velocity, which is momentum.
We give momentum the code p.
We can rewrite the equation as
λ = h/p
λ = h/mv
Therefore every particle with a momentum has an associated
de Broglie wavelength,
What is the de Broglie wavelength of an electron
travelling at 2 × 10 6 m/s?
l = h/p = 6.63 × 10-34 Js ÷ (9.11 × 10-31
kg × 2 × 106 m/s)
= 3.64 ×10-10 m
In 1885 Johann Balmer (a Swiss school teacher )discovered an
equation which describes the emissionabsorption spectrum of atomic
1 / l = 1.097 x 107 (1 / 4 - 1 / n2)
where n = 3, 4, 5, 6, ...
Balmer found this by trial and error, and
had no understanding of the physics
underlying his equation.
In 1885 a Swiss school teacher figuredout that the frequencies of
the light corresponding to these wavelengths fit a relatively simple
mathematical formula:
where C = 3.29 x 1015 s-1
Niels Bohr
He proposed that only orbits of certain radii, corresponding to defined energies
, are "permitted" An electron orbiting in one of these "allowed" orbits:
1-Has a defined energy state
2-Will not radiate energy
3-Will not spiral into the nucleus
Bohr's Model of the Hydrogen Atom
The negatively charged electron of the hydrogen atom is forced to a
circular motion by the attractive electrostatic force of the positively
charged atomic nucleus. Thus, the electrostatic force is the
centripetal force.
m v2 / r = e2 / (4 Π ε0 r2)
m ... mass of the electron
v ... velocity of the electron
r ... orbital radius
e ... elementary charge
ε0 ... permittivity of vacuum
However, only those orbital radii are allowed, for which
the angular momentum is an integer multiple of h/(2Π).
Bohr's quantum condition:
r m v = n h / (2p)
r ... orbital radius
m ... mass of the electron
v ... velocity of the electron
n ... principal quantum number (n = 1, 2, 3, ...)
h ... Planck's constant
r = (h2 ε0 / (m e2 Π)) · n2
Orbital radius for the state of principal quantum number n:
h ... Planck's constant
ε0 ... permittivity of vacuum
m ... mass of the electron
e ... elementary charge
n ... principal quantum number (n = 1, 2, 3, ...)
Using the formulation
E = Epot + Ekin = - e2 / (4 Π ε0 r) + (m / 2) v2, we get:
Energy of the hydrogen atom for the state of principal
quantum number n:
E = - (m e4 / (8 e0 2 h2)) · 1 / n2
m ... mass of the electron
e ... elementary charge
ε0 ... permittivity of vacuum
h ... Planck's constant
n ... principal quantum number (n = 1, 2, 3, ...)
What is the uncertainty principle?

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