Getting Schrödinger's equation from basic assumptions

The basic laws which are used are:

• Energy is conserved. The total energy, E, of a particle moving in a potential is maintained as a sum of kinetic & potential energy.

  E = p²/2 m + V
  ( p is the momentum, m is the mass of the particle, V is the potential energy )

• Planck's quantum law: The energy of radiation (or any particle) is only allowed discrete values no finer than: E = h f
  (h is Planck's constant, f is the frequency of the radiation)

• deBroglie's momentum law: The momentum of a wave or particle,
  p = h / l
  (l is the wavelength of radiation)

• A particle moving freely in a zero potential (no forces acting) maintains a constant momentum and hence wavelength.
  It is reasonable to guess that the wave associated with such a particle can be described by the following equation:

  y=y0 e ^{2 PI i ( f t - x / l )}

  (x is distance, i is the square root of -1)

These four rules can be combined together using the logical rules of mathematics to derive Schrödinger's equation (S.E.).

• Putting the Planck & deBroglie equations into the energy conservation equation gives:

  h f = h² / (2 m) / l² + V

Differentiating y with respect to t gives: dy/dt = 2 PI i f y so f = dy/dt / (2 PI i y)

Differentiating y with respect to x twice gives: dy/dx = (-2 PI i / l ) y d2y/dx2 = (-4 PI2 / l2) y so 1 / l2 = d2y/dx2 / (-4 PI2 y)

• Putting the equations for f & 1/l² into the modified energy equation gves:

  h dy/dt / (2 PI i y) = h² / (2 m) . d²y/dx² / (-4 PI² y) + V

• Multiplying this equation by y & replacing h/(2 PI) with hbar gives:

  - i hbar dy/dt = hbar² / (2 m) . d²y/dx² + V y

This equation describes how a quantum wave, y, will change in time given it's current state over space (x) and the potential, V, it is in.

Mathematical solutions to Schrödinger's equation can then be found and compared with observations of the physical world. It is found that the complex solutions match closely with the results of measuring the wavelegths of light that are emitted by atoms and molecules as electrons change there positions in the atoms.

infact the behaviour of atoms & molecules is very successfully described by this equation (extended to 3 dimensions & many particles).

Hence a few simple laws are shown to be sufficient to describe how the much of the physical world behaves.