Quantum mechanics

To extend Bohr theory to He, Li, we need Quantum Mechanics which requires

Wave Particle Duality of De Broglie

Total energy , where m is the mass of a photon.

Therefore, photon mass = and

Photon momentum .

Therefore an argument could be made that an electron with a mass and momentum also has a wavelength , and can then

be represented by wave, where for this particle momentum p can be associated with a wave of wavelength , where particle momentum

Schrödinger and Heisenberg quantum mechanics to describe small particles like electrons, etc.

Postulates of Quantum Mechanics

.  There exist a wave function of a particle where x, y and z are the space variables and t is the time. From this one can find the dynamic behaviour of a particle in system, is complex.

.  The for a given system is determined by Schrödinger equation:

, where m is the mass of the particle and   is the potential energy operator for the system.

.  and must be finite, continuous and single-valued for all x, y, z and t.

.  probability that the particle is in a spatial volume element dV. So,  ,  integration over all        space.

.  A mathematical operator can be associated with each dynamic system variable i.e. position, momentum, velocity, etc.

The expectation value, when this operator operates on    is given as

As examples, the space variables x, y, z are given as

The momentum are given as

where

Similarly

Energy E is derived with an operator

Therefore the expected value of energy from postulate 5 is

If energy = E 0 (constant)

or

or