- #1
K.J.Healey
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Please read the following posts. I know what is contained in this original is wrong, and I have since approached it differently.
Consider a measurement of spin in the x-y plane, with the operator [tex]\hat{Q} = \frac{3\hat{S_x}+4\hat{S_y}}{5} [/tex]
a) What are the eigenvalues and eigenstates (eigenspinors) of this operator?
b) Given a spinor [tex]X = \frac{1}{\sqrt{5}} \binom{2}{1} [/tex], what is the probability that the above measurement yields the result of [tex]\frac{-\hbar}{2}[/tex]?
?
Ok, for part a, I am not sure where to start.
Can I "say" that +-h/2 (from now on h=hbar) are the eigenvalues for each of the Sx,Sy operators in the main operator, and then just sum them up over all possibilities?
3/5 * h/2 + 4/5 * h/2 = 7h/10
3/5 * h/2 - 4/5 * h/2 = -h/10
-3/5 * h/2 + 4/5 * h/2 = h/10
-3/5 * h/2 - 4/5 * h/2 = -7h/10
Or do I have to say that,
[tex]QX^+ = (3/5)S_x X^+_x + (4/5)S_ yX^+_y [/tex]
And thus when [tex]Q X^+ [/tex] is considered I use separate vectors for each of the operators?
Where:
[tex]X^+_x = \frac{1}{\sqrt{2}} \binom{1}{1}[/tex]
[tex]X^-_x = \frac{1}{\sqrt{2}} \binom{1}{-1}[/tex]
[tex]X^+_y = \frac{1}{\sqrt{2}} \binom{1}{i}[/tex]
[tex]X^-_y = \frac{1}{\sqrt{2}} \binom{1}{-i}[/tex]
Doing it that way, using the Pauli spin matrices for Sx,Sy, I get the possible eigenvalues to be:
[tex]\frac{7\hbar}{10\sqrt{2}}[/tex] for [tex]X^+[/tex]
[tex]\frac{-7\hbar}{10\sqrt{2}}[/tex] for [tex]X^-[/tex]
Which of these methods are correct? Or am I doing this all wrong?
The reason I ask, is part (b):
To measure a probability of the [tex]\frac{-\hbar}{2}[/tex], I would assume that means that [tex]\hat{Q}X^? = \frac{-\hbar}{2}[/tex]
and I believe from there you would do something like
[tex]C_? = X^? X = X^? \frac{1}{\sqrt{5}} \binom{2}{1}[/tex]
and then [tex]|C_?|^2 [/tex] would be the probability.
Any hints or suggestions please?
I feel like I understand what they're asking, but my concept of the eigenspinor may not be that clear.
Wouldn't the eigenspinors of some operator Q be whatever eigenvectors you can use with it? But what are the constraints? Do I have to normalize the results that I got? Or are they correct?
If so how would I ever measure -h/2 for part b, if my eigenspinors don't contain that value? So I assume they should. But how?
Thank you!
-Kristopher Healey
Homework Statement
Consider a measurement of spin in the x-y plane, with the operator [tex]\hat{Q} = \frac{3\hat{S_x}+4\hat{S_y}}{5} [/tex]
a) What are the eigenvalues and eigenstates (eigenspinors) of this operator?
b) Given a spinor [tex]X = \frac{1}{\sqrt{5}} \binom{2}{1} [/tex], what is the probability that the above measurement yields the result of [tex]\frac{-\hbar}{2}[/tex]?
Homework Equations
?
The Attempt at a Solution
Ok, for part a, I am not sure where to start.
Can I "say" that +-h/2 (from now on h=hbar) are the eigenvalues for each of the Sx,Sy operators in the main operator, and then just sum them up over all possibilities?
3/5 * h/2 + 4/5 * h/2 = 7h/10
3/5 * h/2 - 4/5 * h/2 = -h/10
-3/5 * h/2 + 4/5 * h/2 = h/10
-3/5 * h/2 - 4/5 * h/2 = -7h/10
Or do I have to say that,
[tex]QX^+ = (3/5)S_x X^+_x + (4/5)S_ yX^+_y [/tex]
And thus when [tex]Q X^+ [/tex] is considered I use separate vectors for each of the operators?
Where:
[tex]X^+_x = \frac{1}{\sqrt{2}} \binom{1}{1}[/tex]
[tex]X^-_x = \frac{1}{\sqrt{2}} \binom{1}{-1}[/tex]
[tex]X^+_y = \frac{1}{\sqrt{2}} \binom{1}{i}[/tex]
[tex]X^-_y = \frac{1}{\sqrt{2}} \binom{1}{-i}[/tex]
Doing it that way, using the Pauli spin matrices for Sx,Sy, I get the possible eigenvalues to be:
[tex]\frac{7\hbar}{10\sqrt{2}}[/tex] for [tex]X^+[/tex]
[tex]\frac{-7\hbar}{10\sqrt{2}}[/tex] for [tex]X^-[/tex]
Which of these methods are correct? Or am I doing this all wrong?
The reason I ask, is part (b):
To measure a probability of the [tex]\frac{-\hbar}{2}[/tex], I would assume that means that [tex]\hat{Q}X^? = \frac{-\hbar}{2}[/tex]
and I believe from there you would do something like
[tex]C_? = X^? X = X^? \frac{1}{\sqrt{5}} \binom{2}{1}[/tex]
and then [tex]|C_?|^2 [/tex] would be the probability.
Any hints or suggestions please?
I feel like I understand what they're asking, but my concept of the eigenspinor may not be that clear.
Wouldn't the eigenspinors of some operator Q be whatever eigenvectors you can use with it? But what are the constraints? Do I have to normalize the results that I got? Or are they correct?
If so how would I ever measure -h/2 for part b, if my eigenspinors don't contain that value? So I assume they should. But how?
Thank you!
-Kristopher Healey
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