Quantum Mechanics pure/unpure states

[kreil]

Homework Statement

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.

Homework Equations

$$\rho_a = a |u><u| + (1-a)|v><v|$$..(1)

If we now define the two vectors,
$$|x> = \sqrt{a} |u> + \sqrt{1-a}|v>$$
$$|y> = \sqrt{a} |u> - \sqrt{1-a}|v>$$

Then rho can also be written

$$\rho_a = \frac{1}{2} |x><x| + \frac{1}{2} |y><y|$$..(2)

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 $\hat w |w> = w |w>$, or are they looking for something like $\hat w |w> = e^{i \theta} |w>$? 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.

 

[arkajad]

Let $$\bf{n}$$ be a unit vector in 3d. In 2-dimensional space let $$|\bf{n}>$$ be the eigenvector of $$\sigma ({\bf n})=n_x\sigma_x+n_y\sigma_y+n_z\sigma_z$$ to the eigenvalue +1. Now $$|{\bf n}>$$ depends on two continuous parameters defining a direction in $${\bf R}^3$$.
That is more than you need. But, in fact, once you master this fact, you will be able to use it for solving your problem.

 

[kreil]

Let $$\bf{n}$$ be a unit vector in 3d. In 2-dimensional space let $$|\bf{n}>$$ be the eigenvector of $$\sigma ({\bf n})=n_x\sigma_x+n_y\sigma_y+n_z\sigma_z$$ to the eigenvalue +1. Now $$|{\bf n}>$$ depends on two continuous parameters defining a direction in $${\bf R}^3$$.

This makes sense to me; I'm just not sure how to implement it in relation to the problem. Am I looking to define an infinitely large set of vectors?

 

[arkajad]

After reading again the formulation of your problem, I think I misunderstood it. Perhaps the point is to construct not any example but to construct an example using the original density matrix of the problem. Then my previous comment was irrelevant.

What will be the result if you manipulate the phase of one of the terms:

$$|x>_\theta = \sqrt{a}|u>+e^{i\theta}\sqrt{1-a}|v>$$

$$|y>_\theta = ...$$

 

[kreil]

Clearly that leaves the outer product unchanged..do you think it is sufficient to say that since theta can vary that constitutes an infinite set of pure states?

 

[arkajad]

It should be sufficient provided you understand clearly why

$$|x>_\theta = e^{i\theta}\sqrt{a}|u>+e^{i\theta}\sqrt{1-a}|v>$$
...

though also good, would not be a solution of your problem.
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[kreil]

I believe it is because "global" phase factors represent the same state (i.e. Multiplying a state by a constant doesn't change the state, but only doing so to one term does)

 

[arkajad]

Indeed, and that was the point.