Question

Explain briefly the time independent Schrodinger wave equation?

Answer 1

The time independent Schrodinger equation can be expressed as

`hat"H" ψ = "E"ψ` ……………(1)

Where `hat"H"` is called Hamiltonian operator.

Ψ is the wave function.

E is the energy of the system.

`hat"H" = [(-"h"^2)/(8π^2"m")((∂^2)/(∂"x"^2) + (∂^2)/(∂"y"^2) + (∂^2)/(∂"z"^2)) + "V"]` ..........(2)

Since Ψ is a function of position coordinates of the particle and is denoted by Ψ (x, y, z)

∴ Equation (1) can be written as,

`[(-"h"^2)/(8π^2"m")((∂^2ψ)/(∂"x"^2) + (∂^2ψ)/(∂"y"^2) + (∂^2ψ)/(∂"z"^2)) + "V"ψ] = "E"ψ` ..........(3)

Multiply the equation (3) by `hat"H"` and rearranging

`(∂^2ψ)/(∂"x"^2) + (∂^2ψ)/(∂"y"^2) + (∂^2ψ)/(∂"z"^2) + (8π^2"m")/"h"^2 ("E" - "V")ψ = 0` .........(4)

The above equation (4) Schrodinger wave equation does not contain time as a variable and is referred to as time independent Schrodinger wave equation.