(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.

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# Homework Help: Quantum Mechanics pure/unpure states

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