Consider a free particle in a plane of dimensions L x and L y in a plane of impenetrable walls. The time-independent Schrödinger equation for this system with wave function | ψ ψ ⟩ ⟩ can be written as − − ℏ ℏ 2 2 m ( ∂ ∂ 2 ψ ψ ∂ ∂ x 2 + ∂ ∂ 2 ψ ψ ∂ ∂ y 2 ) = E ψ ψ The permitted energy values are E n x , n y = π π 2 ℏ ℏ 2 2 m ( n x 2 L x 2 + n y 2 L y 2 ) The normalized wave function is ψ ψ n x , n y ( x , y ) = 2 L x L y sin ⁡ ⁡ ( n x π π x L x ) sin ⁡ ⁡ ( n y π π y L y ) where n x , n y = 1 , 2 , 3 , … …