Elementary questions about inner product interpretation

[nomadreid]

When one says that <[itex]\varphi[/itex]|[itex]\psi[/itex]> is the probability that [itex]\psi[/itex] collapses to [itex]\varphi[/itex], does this "collapse" necessarily involve a measurement (so that one would have to find the implicit Hamiltonian)? Or does this just exist as part of the evolution of the wave function, perhaps the vacuum energy playing a role in the Schrödinger equation?

 

[kith]

When one says that <[itex]\varphi[/itex]|[itex]\psi[/itex]> is the probability that [itex]\psi[/itex] collapses to [itex]\varphi[/itex], does this "collapse" necessarily involve a measurement

Yes. (Also it's not the probability but the probability amplitude.)

(so that one would have to find the implicit Hamiltonian)?

In order to do what?

Or does this just exist as part of the evolution of the wave function, perhaps the vacuum energy playing a role in the Schrödinger equation?

The vacuum energy is a concept from quantum field theory. It doesn't play a role in ordinary QM. However, there are some speculative extensions to QM which use an explicit physical mechanism to achieve collapse. See http://en.wikipedia.org/wiki/Interpretation_of_quantum_mechanics#Objective_collapse_theories.

 

[nomadreid]

Thanks, kith.

Also it's not the probability but the probability amplitude.

Ah, oops, right. There's many a slip 'twixt cup and lip.

to find the implicit Hamiltonian

In order to do what?

Sorry, I mean the implicit operator. I was considering that the collapse would be brought about by some operator in the measuring process, so that one would be looking at <[itex]\varphi[/itex]|[itex]\psi[/itex]> = <M[itex]\psi[/itex]|[itex]\psi[/itex]> for some operator M.

Thanks for the link.

 

[kith]

Sorry, I mean the implicit operator. I was considering that the collapse would be brought about by some operator in the measuring process, so that one would be looking at <[itex]\varphi[/itex]|[itex]\psi[/itex]> = <M[itex]\psi[/itex]|[itex]\psi[/itex]> for some operator M.

This is a good question. In the textbook description of measurements, there is no such operator M. We have an initial state |ψ> and an observable O and are able to calculate the probabilities for the occurances of the eigenstates of O as final states. The measurement itself has no representation in the mathematical framework.

Now we can try to describe the measurement dynamically. A measuring device and a system interact, so in principle, we should be able to write down an interaction Hamiltonian for them. If we look at the time evolution of the system only, this interaction leads to decoherence in one basis. From this basis, we can construct the observable O. The emergence of the preferred basis is called "environmentally induced superselection". Decoherence means that an initial superposition loses its ability to interfere. This leads to several possible interpretations which get rid of the collapse. But note that your operator M is not a possible description of what happens. The dynamical description doesn't yield a single outcome but contains a probabilistic element.

 

[VantagePoint72]

When one says that <[itex]\varphi[/itex]|[itex]\psi[/itex]> is the probability that [itex]\psi[/itex] collapses to [itex]\varphi[/itex]

Just a clarification, the probability is [itex]|<\varphi[/itex]|[itex]\psi>|^2[/itex], not <[itex]\varphi[/itex]|[itex]\psi[/itex]>. The latter is the probability amplitude.

 

[nomadreid]

Thanks, kith. Very interesting. Now I shall work on the gritty details ...

thanks for the clarification, LastOneStanding.