Eigenvalues of operator in dirac not* (measurement outcomes)

[12x4]

Homework Statement

A measurement is described by the operator:

|0⟩⟨1| + |1⟩⟨0|

where, |0⟩ and |1⟩ represent orthonormal states.

What are the possible measurement outcomes?

Homework Equations

[/B]
Eigenvalue Equation: A|Ψ> = a|Ψ>

The Attempt at a Solution

Apologies for the basic question but just very unsure of myself when it comes to this stuff. I have had a go and come up with a solution but I'm not sure if its right so any help would be much appreciated.

We're told that:
A = |0⟩⟨1| + |1⟩⟨0|

Can I then assume something like: Ψ = α|1> + β|0>?

using this I've then solved the eigenvalue equation, AΨ=aΨ, and found:

α|0> + β|1> = aα|1> + aβ|0>

giving:

α=aβ & β = aα

thus,

β=a²β

a = (+-) 1

hence, my eigenvalues are -1 and 1.

and these are the possible outcomes?

 

[jitu16]

You don't assume that $|\psi>=\alpha |1>+ \beta |2>$, there is a reason for that. The completeness relation gives,

$I=|1><1|+|2><2|$ [look up completeness relation if you don't know about it.]

which means, $|\psi>=I|\psi>=|1><1|\psi>+|2><2|\psi>$

or, $|\psi>=\alpha |1>+ \beta |2>$,

where $\alpha=<1|\psi>$

and $\beta=<2|\psi>$ are c-number.

Except that there are no more problem with your work.

 

[Dick]

Homework Statement

A measurement is described by the operator:

|0⟩⟨1| + |1⟩⟨0|

where, |0⟩ and |1⟩ represent orthonormal states.

What are the possible measurement outcomes?

Homework Equations

[/B]
Eigenvalue Equation: A|Ψ> = a|Ψ>

The Attempt at a Solution

Apologies for the basic question but just very unsure of myself when it comes to this stuff. I have had a go and come up with a solution but I'm not sure if its right so any help would be much appreciated.

We're told that:
A = |0⟩⟨1| + |1⟩⟨0|

Can I then assume something like: Ψ = α|1> + β|0>?

using this I've then solved the eigenvalue equation, AΨ=aΨ, and found:

α|0> + β|1> = aα|1> + aβ|0>

giving:

α=aβ & β = aα

thus,

β=a²β

a = (+-) 1

hence, my eigenvalues are -1 and 1.

and these are the possible outcomes?

Yes, that looks just fine to me. The possible measurements of an experiment are the eigenvalues of the operator.

 

[12x4]

thank you. will look up the completeness relation