## PHY456H1F: Quantum Mechanics II. Lecture 8 (Taught by Prof J.E. Sipe). Time dependent pertubation (cont.)

Posted by peeterjoot on October 8, 2011

# Disclaimer.

Peeter’s lecture notes from class. May not be entirely coherent.

# Time dependent pertubation.

We’d gotten as far as calculating

where

and

Graphically, these frequencies are illustrated in figure (\ref{fig:qmTwoL8fig0FrequenciesAbsorbtionAndEmission})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{qmTwoL8fig0FrequenciesAbsorbtionAndEmission}

\caption{Positive and negative frequencies.}

\end{figure}

The probability for a transition from to is therefore

Recall that because the electric field is real we had

Suppose that we have a wave pulse, where our field magnitude is perhaps of the form

as illustated with in figure (\ref{fig:gaussianWavePacket}).

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{gaussianWavePacket}

\caption{Gaussian wave packet}

\end{figure}

We expect this to have a two lobe Fourier spectrum, with the lobes centered at , and width proportional to .

For reference, as calculated using Mathematica this Fourier transform is

This is illustrated, again for , in figure (\ref{fig:FTgaussianWavePacket})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{FTgaussianWavePacket}

\caption{FTgaussianWavePacket}

\end{figure}

where we see the expected Gaussian result, since the Fourier transform of a Gaussian is a Gaussian.

FIXME: not sure what the point of this was?

# Sudden pertubations.

Given our wave equation

and a sudden pertubation in the Hamiltonian, as illustrated in figure (\ref{fig:suddenStepHamiltonian})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{suddenStepHamiltonian}

\caption{Sudden step Hamiltonian.}

\end{figure}

Consider and fixed, and decrease . We can formally integrate 3.8

For

While this is an exact solution, it is also not terribly useful since we don’t know . However, we can select the small interval , and write

Note that we could use the integral kernel iteration technique here and substitute and then develop this, to generate a power series with dependence. However, we note that 3.11 is still an exact relation, and if , with the integration limits narrowing (provided is well behaved) we are left with just

Or

provided that we change the Hamiltonian fast enough. On the surface there appears to be no consequences, but there are some very serious ones!

## Example: Harmonic oscillator.

Consider our harmonic oscillator Hamiltonian, with

Here continuously, but *very quickly*. In effect, we have tightened the spring constant. Note that there are cases in linear optics when you can actually do exactly that.

Imagine that is in the ground state of the harmonic oscillator as in figure (\ref{fig:suddenHamiltonianPertubationHO})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{suddenHamiltonianPertubationHO}

\caption{Harmonic oscillator sudden Hamiltonian pertubation.}

\end{figure}

and we suddenly change the Hamilontian with potential (weakening the “spring”). Professor Sipe gives us a graphical demo of this, by impersonating a constrained wavefunction with his arms, doing weak chicken-flapping of them. Now with the potential weakended, he wiggles and flaps his arms with more freedom and somewhat chaotically. His “wave function” arms are now bouncing around in the new limiting potential (initally over doing it and then bouncing back).

We had in this case the exact relation

but we also have

and

So

and at later times

whereas

So, while the wave functions may be exactly the same after such a sudden change in Hamiltonian, the dynamics of the situation change for all future times, since we now have a wavefunction that has a different set of components in the basis for the new Hamiltonian. In particular, the evolution of the wave function is now significantly more complex.

FIXME: plot an example of this.

# Adiabatic pertubations.

FIXME: what does Adiabatic mean in this context. The usage in class sounds like it was just “really slow and gradual”, yet this has a definition “Of, relating to, or being a reversible thermodynamic process that occurs without gain or loss of heat and without a change in entropy”.

This is treated in section 17.5.2 of the text [1].

This is the reverse case, and we now vary the Hamiltonian *very slowly*.

We first consider only non-degenerate states, and at write

and

Imagine that at each time we can find the “instantaneous” energy eigenstates

These states do not satisfy Schr\”{o}dinger’s equation, but are simply solutions to the eigen problem. Our standard strategy in pertubation is based on analysis of

Here instead

we will expand, not using our initial basis, but instead using the instananeous kets. Plugging into Schr\”{o}dinger’s equation we have

This was complicated before with matrix elements all over the place. Now it is easy, however, the time derivative becomes harder. Doing that we find

We bra into this

and find

If the Hamiltonian is changed *very very* slowly in time, we can imagine that is also changing very very slowly, but we are not quite there yet. Let’s first split our sum of bra and ket products

into and terms. Looking at just the term

we note

Something plus its complex conjugate equals 0

so must be purely imaginary. We write

where is real.

# References

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

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