**Hellman-Feynman** Theorem investigates how the energy of a system varies as the Hamiltonian varies.

If a system is characterised by a Hamiltonian that depends on a parameter , then the** Hellmann-Feynman** theorem states that,

where, is the energy of the system.

### Proof:

Since the wavefunctions must be normalised,

Differentiating both sides w.r.t some parameter

Since, ,

The 2nd and 3rd terms on the L.H.S. can be replaced by

Since (Normalisation condition)

### Applications:

**Expectation value of 1/r :**for Hydrogen, using Hellmann-Feynam Theorem:

**Hamiltonian:**

and **Energy:**

Taking, , in the Hellmann-Feynman theorem,

Since, Bohr radius,

Therefore,

**Expectation value of 1/r^2 :**for Hydrogen, using Hellmann-Feynam Theorem:

**Hamiltonian:**

and **Energy:**

Taking, , in the Hellmann-Feynman theorem,

- Harmonic Oscillator:

Relation between and :

Take

Relation between and :

Take

Relation between and :

Take