Question

Determine the Time-Dependent Schrodinger Equation (TDSE) from the classical non-relativistic expression for the energy of a particle and de Broglie’s claim that all particles can be represented as waves. (Important: Describe as much as possible and must use your own words to explain it.)

Answer 1

According to De-Broglie's claim, all particles with masses have a wave associated with it (matter waves) whose wavelength is related to its momentum as:

or,

For a particle, the wavefunction describing this matter wave is given by:

=>

taking differentials in time and space gives the respective operators as:

and

now total energy = potential energy + kinetic energy

so,

=>

substituting the corresponding operators in the above equation gives:

this is the time dependent Schrodinger's equation.