The orthogonality of the wavefunctions of the quantum harmonic oscillator can be proven using the Hermite differential equation.

Substituting eq13 in eq22, we have

where are the Hermite polynomials.

###### Question

Show that eq23 can be written as

###### Answer

Carrying out the derivative, the LHS of eq24 becomes

Multiplying eq24 (with a change of index from to ) by and subtracting the result from the product of and eq24, we have

Substituting and in eq25 gives

Integrating both sides of the above equation with respect to ,

If , then . Since ,