Question

Derive Schrodinger time dependent wave equation

Answer 1

Considering a complex plane wave:

Ψ (x, t) = Ae^{i(kx - ωt)}.

Now the Hamiltonian of a system is

H = T + V

Where ‘V’ is the potential energy and ‘T’ is the kinetic energy. As we already know that ‘H’ is the total energy, we can rewrite the equation as:

E = p²/2m + V(x)

Now taking the derivatives,

We know that,

p = 2πh/λ and k = 2π/λ

where ‘λ’ is the wavelength and ‘k’ is the wave number.

We have

k = p/h

Therefore,

Now multiplying Ψ (x, t) to the Hamiltonian we get,

The above expression can be written as:

We already know that the energy wave of a matter wave is written as

E = hω,

So we can say that

Now combining the right parts, we can get the Schrodinger Wave Equation.