Recurrence relations of the Hermite polynomials

The recurrence relations of the Hermite polynomials describe how each polynomial in the sequence can be obtained from its predecessors.

Expanding eq18 and assuming  is an even number,

Taking the derivative of eq26, we have

Substituting eq27 in eq28,

If we repeat the steps from eq26 through eq29 on the assumption that  is an odd number, we too end up with eq29. Therefore, in eq29 represents any number. Taking the derivative of eq29 again,

Substitute 29 and 30 in eq23, we have

Replacing the dummy index in eq31 with gives

Eq29 and eq32 are the recurrence relations of the Hermite polynomials.



Given , show that eq32 can be used to generate the Hermite polynomials.


Substituting in eq32, we have . Substituting  in eq32, we have . Repeating the logic, we can generate the rest of the polynomials.



Using eq32,  and eq46, show that .


Substituting eq32 in eq46, we have

Multiplying the above equation by and integrating over all space,



Next article: Taylor series of single-variable and multi-variable functions
Previous article: Orthogonality of the wavefunctions of The quantum harmonic oscillator
Content page of vibrational spectroscopy
Content page of advanced chemistry
Main content page

Leave a Reply

Your email address will not be published. Required fields are marked *