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
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