The particle in one dimensional potential well
∴ V = 0 for 0 < x < L
V = ∞ for x ≤ 0 and x ≥ L
The particle present in a box so outside wave function Ψ is zero.
Ψ = 0 for x ≤ 0 x ≥ L
Schrodinger's equation time independent
\(\frac{∂^2\psi}{∂x^2}+\frac{2m}{h^2}\) (E - V) Ψ = 0
one dimensional V = 0
\(\frac{∂^2\psi}{∂x^2}+\frac{2m}{h^2}\) (E) Ψ = 0
Let the factor 2m/h2 E = k2 then
\(\frac{∂^2\psi}{∂x^2}\) + k2 Ψ = 0 ...(1)
then general solution of equation (1), we get
Ψ = A sin kx + B cos kx