(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

It is shown in the following two equations that any nonpure state operator can be decomposed into a mixture of pure states in at least two ways. Show, by constructing an example depending on a continuous parameter, that this can be done in infinitely many ways.

2. Relevant equations

[tex] \rho_a = a |u><u| + (1-a)|v><v| [/tex]..(1)

If we now define the two vectors,

[tex] |x> = \sqrt{a} |u> + \sqrt{1-a}|v> [/tex]

[tex] |y> = \sqrt{a} |u> - \sqrt{1-a}|v> [/tex]

Then rho can also be written

[tex]\rho_a = \frac{1}{2} |x><x| + \frac{1}{2} |y><y| [/tex]..(2)

3. The attempt at a solution

Can someone give me an example of a state operator that depends on a continuous parameter? Is it as simple as [itex] \hat w |w> = w |w> [/itex], or are they looking for something like [itex] \hat w |w> = e^{i \theta} |w>[/itex]? Also any hints would be appreciated. I'm sure the problem is simple I'm just having a hard time getting started.

Thank you for your time.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Quantum Mechanics pure/unpure states

**Physics Forums | Science Articles, Homework Help, Discussion**