Scroedinger time dependent wave equation

SCHROEDINGER TIME DEPENDENT WAVE EQUATION

Schroedinger describes the wave nature of a particle in mathematical form and is known as Schroedinger wave equation. There are two types of wave equations, viz.

(i) Time dependent wave equation.

(ii) Time independent wave equation.

Time dependent wave equation

A particle can behave as a wave only under motion. So, it should be accelerated by a potential field. Therefore, the total energy (E) of the particle is equal to the sum of its potential energy (V) and kinetic energy.

 (or) E = V + ½  mv²

(or) E = V +  ½  (m²v²/m)

(or) E = V + p²/2m        [p = mv]

(or) Eᴪ = Vᴪ + (p²/2m)ᴪ          ………(1)

According to classical mechanics, if 'x' is the position of the particle moving with the velocity 'v', then the displacement of the particle at any time 't' is given by

 y=A e^-iω(t-x/v)

where ω is the Angular frequency of the particle.

Similarly, in Quantum Mechanics the wave function ᴪ(x, y, z, t) represents the position (x, y, z) of a moving particle at any time 't' and is given by

 ᴪ (x, y, z, t) = A e^-iω(t-x/v)       ………..(2)

We know angular frequency ω = 2πv

Equation (2) becomes

ᴪ (x, y, z, t) = A e^{-2πi( vt - vx/v )}    ………..(3)

We know E = hv (or) v=E/h    ………..(4)

Also, if 'v' is the velocity of the particle behaving as a wave, then the frequency v = v/ λ (or) v/v = 1/λ         ………..(5)

Substituting equations (4) and (5) in equation (3), we get,

ᴪ(x, y, z, t) = A e^{-2πi( Et/h - x/λ )}    ………..(6)

If 'p' is the momentum of the particle, then the de‒Broglie wavelength is given by

 λ = h / mv = h/p             ………..(7)

Substituting equation (7) in (6) we get

Differentiating equation (8) partially with respect to 'x' we get

Differentiating once again partially with respect to 'x' we get

Differentiating equation (8) partially with respect to 't', we get

Substituting equations (9) and (10) in equation (1), we get,

Equation (11) represents the one dimensional (along 'x' direction) Schroedinger time dependent equation. It is called time dependent wave equation, because here the wave function ᴪ(x, y, z, t) depends both on position (x, y, z) and time (t).

Similarly, the 3‒dimensional Schroedinger time dependent wave equation can be written as

Equation (12) can also be written as

Eᴪ = Hᴪ                ...(13)

where E is the energy operator given by  and

H is called Hamiltonian Operator, given by .