Deriving the Schrodinger Equation

The Schrodinger Equation

The Schrodinger equation was formulated in 1925 and is used to describe how the state of a quantum mechanical system changes with time. The equation can be formulated with knowledge of differential calculus and of the quantum action principle.

The quantum action principle gives the following identity to a wavefunction of a free particle (Eq. 1):

Each point in spacetime has a wavefunction corresponding to a certain particle, and initially we will confine it to two dimensions: x and t.

Differentiating the wavefunction with respect to time yields a term for total energy (for a free particle, or when there is no potential energy on the particle), (Eq. 2):

Taking the second differential of the wavefunction with respect to space (in this case, just the x-dimension) yields a term for kinetic energy (Eq. 3):

It is assumed that energy is conserved, and a Hamiltonian from classical mechanics is used (the Hamiltonian is the sum of the kinetic and potential energy). It is also assumed that the potential does not vary with time, but it varies with space (which is why Eq. 2 resulted in the total energy, and Eq. 3 resulted in the kinetic energy).

The Hamiltonian is represented by Eq. 4:

Substituting the values of E and K from Eq. 2 and Eq. 3 respectively into Eq. 4 yields Eq. 5:

This can then be placed into the four dimensions of spacetime by using the del operator,  using a distance 'r' to show the spatial distance from a fixed point in the system, and allowing the potential energy of to change with time yields Eq. 6:

 

This shows that the equation for the wavefunction as shown in Eq.1 is valid for a non-relativistic, free, spinless particle as it has been shown to be equivalent to the Schrodinger equation (which deals with non-relativistic, spinless wavefunctions) when applied to the principle of conservation of energy.

QED

(Equation 1 adapted from 'The Quantum Universe: Anything That Can Happen Does Happen', Cox & Forshaw) - Recommended Read!