Phase factors with eigenstates

[Hazzattack]

Hi guys, I'm currently writing a document about decoherence and I'm trying to make a point about phase factors and how they are altered by an interaction with a surrounding environment. However, i don't want to only state this, i want to show it through basic QM. Is it possible to say the following;

If we represent a superposition state the following way;

$\varphi_{s}$ = $\Sigma_{n}$$\alpha_{n}^{*}\varphi_{n}$

and since each component basis eigenstate has an associated complex coefficient, this can be written;

$\alpha^{*}$ = x+ iy = re^iphi

I'm basically trying to show how each component eigenstate (that makes up the superposition state) have a relative phase and when this is coherent we observe interference patterns etc.

If anyone could help with any pointers to make this more clear it would be very much appreciated.

Thanks

 

[HomogenousCow]

I thought people usually demonstrated interference when they calculated the probability density and noted the cross term which comes out.

 

[Hazzattack]

Yep, that's true. I'm probably not making myself clear. In order to obtain a superposition all the component eigenstates (that make up that superposition state) but have a coherent phase relationship. I just want to show where that phase comes from (i.e the coefficient).

 

[Jilang]

I always thought it was is the component eigenstates not in the coefficients.

 

[Hazzattack]

This is what is confusing me. So if we were to assume that this particular superposition was made of three component basis eigenstates such that;

$\varphi_{s}$ = $\alpha_{1}$$\varphi_{1}$ + $\alpha_{2}$$\varphi_{2}$ + $\alpha_{3}$$\varphi_{3}$

is the relative phase factor (between each of the components) in the $\varphi_{3}$ part (for example)?

I would have thought that it was in the complex coefficient and since this can be re-written in terms of euler's identity you impose a relative phase factor where phi would be the same for each component (when its coherent). However, this principle kind of works for what you are saying also i guess.

If anyone could give me a mathematical definition (by example) that would be very useful.

 

[Jilang]

If A and B are the components, would the relative phase not just be just be log (A/B)/i. ?

 

[bhobba]

I'm basically trying to show how each component eigenstate (that makes up the superposition state) have a relative phase and when this is coherent we observe interference patterns etc.

How is trivial - its because pure states form a complex vector space.

I just want to show where that phase comes from (i.e the coefficient).

Why is another matter - but still answerable - check out:
http://www.scottaaronson.com/democritus/lec9.html

I suspect what you are really after is a simple way to look at decoherencre. The following may help:
http://www.ipod.org.uk/reality/reality_decoherence.asp

Thanks
Bill

 

[Hazzattack]

Thanks a lot for the response. I believe this is what i was trying to say;

'In other words, if the amplitude for some measurement outcome is α = β + γi, where β and γ are real, then the probability of seeing the outcome is |α|2 = β2 + γ2.'

α being the co-efficient corresponding to a component eigenstate. Thus, it makes sense that this has a relative phase as it can be represented using eulers.

 

[Jilang]

This may help.
http://en.wikipedia.org/wiki/Bloch_sphere