Friday, 28 December 2012

Parallel Universes and Surface Hopping

In a sense Surface Hopping is like the Kopenhagen interpretation of quantum mechanics. Instead of following all branches of the universal wavefunction (as introduced by Hugh Everett), the wavefunction is collapsed in a stochastic way and just one branch is considered. To show you what I mean I will consider non-adiabatic dynamics simulations of Schrödinger's cat.

To simplify the math, we will consider the following situation. There is a cat locked in a box. Additionally there is a flask of poison and an electron of unknown spin in the box. At a time t0 the spin of the electron is measured. If the spin is up (u) nothing happens. If it is down (d) the poison is released.

The wavefunction of the electron, a superposition between spin up and down, can be written as

It is time-independent (containing only a phase factor, which we can cancel out by appropriately setting the reference energy).

The wavefunction of the cat for t < t0 is given by

where Hl denotes the Hamiltonian for the living cat and Φcat,0 is the state of the cat at t = 0. Alternatively we could of course solve the classical equations of motion for the cat (possibly coarse grained), it does not affect the discussion and this can be viewed as a rather formal expression...

For  t < t0 the electron and the cat are assumed not to interact with each other and the total wavefunction is simply given as

i.e. there is an electron in its superposed state and an independent living cat.

At t0 the spin of the electron is measured. After that we either have spin up and a living cat or spin down and a dead cat (the latter represented by a Hamiltonian Hd). The fate of the cat becomes entangled with the spin of the electron.

One way to read this equation is that we are now considering two parallel universes. In one of them the cat is living and moving according to Hl, in the other one the cat is dead described by Hd. This is just the result of the laws of quantum mechanics.

From the point of view of the poor cat or any observer the wavefunction "collapsed". However, considering the maths this only happens because any observer is now also entangled with the electron. And in fact both alternate histories would be part of the truth. The Kopenhagen interpretation of quantum mechanics tells us that the branch of the wavefunction we are experiencing is somehow the "real" branch, Everett's "universal wavefunction" or "many world" interpretation says that in fact all branches are equal.

Getting back to my teaser line: Surface Hopping is somewhat similar to the Kopenhagen interpretation of quantum mechanics. Instead of considering all branches of the wavefunction, only one branch is continued after every potential quantum transition. Of course in practical implementations there are also numerical differences related to more complex interactions between the different states and decoherence effects. But aside from that it just follows the laws in which we perceive nature.

Finally, it is important to remember that the cat is either alive or dead. It is not half-dead, i.e. it is not possible to use the average Hamiltonian for the wavefunction propagation

In computational science this is called Ehrenfest dynamics. It apparently does work in other cases though, if the different states are of similar nature.

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