Schrodinger Time Independent Wave Equation (Derivation)

SCHRODINGER TIME INDEPENDENT WAVE EQUATION (Derivation)

Consider a wave associated with a moving particle.

Let x, y, z be the coordinates of the particle and Ψ wave function for de - Broglie's waves at any given instant of time t. (Fig 6.6)

The classical differential equation for wave motion is given by

Here, v is wave velocity. 

The eqn (1) is written as

The solution of eqn (2) gives Ψ as a periodic variations in terms of time t,

Here, Ψ0 (x, y, z) is a function of x, y, z only which is the amplitude at the point considered. ω is angular velocity of the wave.

Differentiating eqn (3) with respect to t, we get

Again differentiating with respect to t, we have

Substituting eqn (4) in eqn (2), we have

We know that angular frequency

Squaring eqn (6) on both sides, we get

Substituting eqn (7) in eqn (5), we have

If E is total energy of the particle, V is potential energy and 1/2 mv2 is kinetic energy, then

Total energy = Potential energy + Kinetic energy

Multiplying by m on both sides, we have

Substituting eqn (10) in eqn (9), we get

The eqn (11) is known as Schrodinger time independent wave equation for three dimensions.

on substituting eqn (12) in eqn (13), Schrodinger time-independent wave equation is written as

Note: In eqn (14), there is no term representing time. That is why it is called as time independent equation.

Special case

If we consider one-dimensional motion ie., particle moving along only X - direction, then Schrodinger time independent equation (14) reduces to