# Understanding interference and decoherence

1. Aug 8, 2007

### Talisman

I'm trying to make sense of various explanations of why decoherence causes interference to disappear, but I'm afraid I don't quite grok it.

There's a bit of explanation in this thread.

And some more on wikipedia.

The first link starts with $$|\psi{\rangle} = |a{\rangle} + |b{\rangle}$$ and considers $${\langle}\psi | \psi{\rangle} = {\langle}a|a{\rangle} + {\langle}b|b{\rangle} + 2 Re({\langle}a|b{\rangle})$$. Clearly a and b are not meant to be basis vectors here; in that case, why did we write $$\psi$$ as a sum of them to begin with?

Wikipedia shows the following:
$$prob(\psi \Rightarrow \phi) = |{\langle} \psi |\phi {\rangle}|^2 = |\sum_i\psi^*_i \phi_i |^2 = \sum_{ij} \psi^*_i \psi_j \phi^*_j\phi_i= \sum_{i} |\psi_i|^2|\phi_i|^2 + \sum_{ij;i \ne j} \psi^*_i \psi_j \phi^*_j\phi_i$$
In both cases, when we work out the algebra in the inner product, we find that the expansion contains terms involving the product of components of each state, which are called "cross terms."

What I don't understand is the significance of taking the inner product of a state with itself in the first case, or what the "transition probability" refers to in the second. Also, how does this relate to the double slit experiment? Aren't the position states represented by the electron going through each slit orthogonal?

Last edited: Aug 8, 2007