## Review of Quantum mechanics approximation results.

Posted by peeterjoot on November 10, 2011

# Motivation.

Here I’ll summarize what I’d put on a cheat sheet for the tests or exam, if one would be allowed. While I can derive these results, memorization unfortunately appears required for good test performance in this class, and this will give me a good reference of what to memorize.

This set of review notes covers all the approximation methods we covered except for Fermi’s golden rule.

# Variational method

We can find an estimate of our ground state energy using

# Time independent perturbation

Given a perturbed Hamiltonian and an associated solution for the unperturbed state

we assume a power series solution for the energy

For a non-degenerate state , with an unperturbed value of , we seek a power series expansion of this ket in the perturbed system

Any states are allowed to have degeneracy. For this case, we found to second order in energy and first order in the kets

# Degeneracy.

When the initial energy eigenvalue has a degeneracy we use a different approach to compute the perturbed energy eigenkets and perturbed energy eigenvalues. Writing the kets as , then we assume that the perturbed ket is a superposition of the kets in the degenerate energy level

We find that we must have

Diagonalizing this matrix (a subset of the complete matrix element)

we find, by taking the determinant, that the perturbed energy eigenvalues are in the set

To compute the perturbed kets we must work in a basis for which the block diagonal matrix elements are diagonal for all , as in

If that is not the case, then the unitary matrices of 4.8 can be computed, and the matrix

can be formed. The kets

will still be energy eigenkets of the unperturbed Hamiltonian

but also ensure that the partial diagonalization condition of 4.8 is satisfied. In this basis, dropping overbars, the first order perturbation results found previously for perturbation about a non-degenerate state also hold, allowing us to write

# Interaction picture.

We split of the Hamiltonian into time independent and time dependent parts, and also factorize the time evolution operator

Plugging into Schr\”{o}dinger’s equation we find

# Time dependent perturbation.

We moved on to time dependent perturbations of the form

where are the energy eigenvalues, and the energy eigenstates of the unperturbed Hamiltonian.

Use of the interaction picture led quickly to the problem of seeking the coefficients describing the perturbed state

and plugging in we found

## Perturbation expansion in series.

Introducing a parametrized dependence in the perturbation above, and assuming a power series expansion of our coefficients

we found, after equating powers of a set of coupled differential equations

Of particular value was the expansion, assuming that we started with an initial state in energy level before the perturbation was “turned on” (ie: ).

So that . We then found a first order approximation for the transition probability coefficient of

# Sudden perturbations.

The idea here is that we integrate Schr\”{o}dinger’s equation over the small interval containing the changing Hamiltonian

and find

An implication is that, say, we start with a system measured in a given energy, that same system after the change to the Hamiltonian will then be in a state that is now a superposition of eigenkets from the new Hamiltonian.

# Adiabatic perturbations.

Given a Hamiltonian that turns on slowly at , a set of instantaneous eigenkets for the duration of the time dependent interval, and a representation in terms of the instantaneous eigenkets

plugging into Schr\”{o}dinger’s equation we find

## Evolution of a given state.

Given a system initially measured with energy before the time dependence is “turned on”

we find that the first order Taylor series expansion for the transition probability coefficients are

If we introduce a perturbation, separating all the (slowly changing) time dependent part of the Hamiltonian from the non time dependent parts as in

then we find our perturbed coefficients are

# WKB.

We write Schr\”{o}dinger’s equation as

and seek solutions of the form . Schr\”{o}dinger’s equation takes the form

Initially setting we refine our approximation to find

To first order, this gives us

What we didn’t cover in class, but required in the problems was the Bohr-Sommerfeld condition described in section 24.1.2 of the text [1].

This was found from the WKB connection formulas, themselves found my some Bessel function arguments that I have to admit that I didn’t understand.

# References

[1] BR Desai. *Quantum mechanics with basic field theory*. Cambridge University Press, 2009.

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