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?Solution:                                          

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

millikan

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/λSo 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 or 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?  

= 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 emission-absorption spectrum of atomic hydrogen:

1 / l = 1.097 x 107 (1 / 4 - 1 / n2)

        where n = 3, 4, 5, 6, ...

Balmer found this by trial and error, andhad no understanding of the physicsunderlying 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 / (2)

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 vacuumm ... 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 0 2 h2)) · 1 / n2

m ... mass of the electrone ... elementary charge ε0 ... permittivity of vacuum h ... Planck's constant n ... principal quantum number (n = 1, 2, 3, ...)

What is the uncertainty principle?