In quantum mechanics, the wave function ψ corresponds to the variable y of wave motion. We know that the wave function for a particle is given by

Ψ(x,t)=A exp[-i(ωt-kx)]

Put ω=2πv and K=2π/h

Ψ(x,t)=A exp[ -i(2πvt-2π/h x)] (1)

If E= total energy of the particle

P= momentum of the particle then

E=hv=2πЋ/p

Where Ћ = h/2π

Putting in equation (1), we get

Ψ(x,t)= A exp[-i(E/Ћ t –p/ Ћ x)]

Ψ(x,t) = A exp[-i/ Ћ(Et-px)] (2)

This equation is a description of the wave equivalent of a free particle moving in the +ve x- direction. But generally we are interested in situations where particle is not free i.e it is subjected to some external force.

Differentiating equation (2) w.r.t. x

dΨ/d x =A exp[-i/ Ћ(Et-px] / x[-i/h(Et-px)]

= A exp[-i/ Ћ(Et-px)](i/ Ћ p)

dΨ/ dx = Aip/ Ћ exp[-i/ Ћ(Et-px)]

Again differentiating wr.t. x

dΨ/d x^{2} =Aip/ Ћ exp[ -i/ Ћ(Et-px)](i/ Ћ p)

= A(ip/ Ћ)^{2} exp [-i/ Ћ(Et-px)]

dΨ/dx^{2}= -p^{2}/h^{2} Ψ (3)[ using equation(2)]

Or p^{2} Ψ= – Ћ^{2} d^{2}Ψ/ dx^{2}

Also on differentiating (2) wr.t. t ,we get

dΨ(x,t)/dt= A exp [-i/ Ћ(Et-px)] (-i/ Ћ E)

dΨ/dt =-(i/ Ћ) E Ψ

or E Ψ=(i Ћ) Ψ/ t (4)

When the particle is acted upon by a force then its total energy is the sum of Kinetic and potential energies i.e.

Total energy = Kinetic energy + potential energy

E=p^{2}/2m +V

E Ψ= p^{2}/2m Ψ+ V Ψ (5)

Putting equations (3) and (4) in equations (5), we get

-Ћ/t dΨ/dt=-h^{2}/2m d^{2}Ψ/dx^{2}+V Ψ

i Ћ dΨ/dt= – Ћ^{2}/2m(^{ }Ψ/x^{2}) + V Ψ (6)

Which is time dependent form of Schroedinger wave equation

In three –dimensional form

i Ћ Ψ/t =- Ћ ^{2}/2m(d^{2}Ψ/dx^{2}+ d^{2}Ψ/dy^{2}+ dΨ/dz^{2})+ V Ψ

where the particle potential V is a function of x,y,z and t . Any restriction on the particle motion will effect the potential energy V. once V is known, Schroedinger equation may be solved for the wave function Ψ of the particle form where Ψ^{2} may be determined.