In case of time-independent perturbation theory in Quantum mechanics, we find that, the first order correction to the energy is the expectation value of the perturbation in the unperturbed state.
What is the physical implication of this?
Dear Basabendu,
The result can be interpreted in the following way: suppose the perturbation is applied, but we keep the system in the original quantum state, which is a valid quantum state though no longer a stationary energy eigenstate. The perturbation causes the average energy of this state to increase by its expectation value in this state. However, the true energy shift is slightly different, because the perturbed eigenstate will in reality also change. These further shifts are then given by the second and higher order corrections to the energy.
Dynamically, you might also think of the situation in terms of an instantaneous change in the Hamiltonian, or a quantum quench, whereby the perturbation is suddenly applied. Before the original state vector has time to evolve under the Schrödinger equation with the Hamiltonian including the perturbation term, the "expected" shift to the energy immediately after the quench is given by the expectation value of the perturbation in the original state.
Regards,
Pekko
Dear Pekko,
Thank you so much for such a nice explanation. I have one doubt though: does the first-order correction in energy depend on how "fast" or how "slow" the perturbation is applied? (what I mean to say, is it valid only under adiabetic approximation or independent of that)
Regards,
Basabendu
Dear Basabendu,
Glad to be of assistance. If you would include corrections to all orders in the time-independent perturbation, you would obtain (exactly) the new stationary energy eigenstates of the full, perturbed Hamiltonian. In that sense, there is no reference to time dependence in the theory itself - after all, it's a time-independent theory. But for a fully adiabatic perturbation, you could also apply it to track the evolution of a given state vector, since knowing the instantaneous eigenpairs would be enough. (Having said that, my earlier remark on the quench may seem a bit contradictory, but it was just a suggestion where the lowest-order energy correction could have a concrete sense, albeit only very momentarily.)
Basabendu; Eigenstates of the unperturbed Hamiltonian spans a complete basis, in which any eigenfunction of the full Hamiltonian can be expanded. This part is exact. Then we can calculate the expectation value of the perturbing part with respect to the ground state wave function of the unperturbed Hamiltonian. This part is also exact. Then we interpret the expectation value of perturbation potential (w.r.t the ground state wave function of unperturbed case) as the first order correction to ground state energy. Physical interpretation is that this is the most dominant piece of correction to the ground state energy. Higher order corrections are sub-leading to this dominant piece.
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