Calculate the probability that a measure on S_y yields h/2

In summary, the probability of finding an electron with spin h/2 when measuring sy in the spin state (a,B) can be calculated by taking the modulus squared of (a-iB)/Sqrt[2], which is equivalent to (|a-i*B|^2)/2. This can also be calculated by finding the amplitude of the state (a,b) with the eigenvector of the pauli matrix for y, resulting in the same answer.
  • #1
QFT25
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Homework Statement


. Suppose an electron is in the spin state (a,B) If sy is measured, what is the probability of the result h/2?

Homework Equations


Eigenvectors of the pauli matrix for y are (1,i)/Sqrt[2] (1,-i)/Sqrt[2] and if you are given a wave function of the sort a | +> +b |-> then the probability of getting state | +> is a^2/(a^2+b^2)

The Attempt at a Solution



I wrote out (a,B) as a linear combination of the of the two eigenvectors for the pauli matrix and got that the probability of finding the electron with spin h bar/2 to be (|a-ib|^2)/2. I just want to check with all of you if that is right. [/B]
 
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  • #2
Doesn't look right to me. I assume you're just being sloppy and are using b and B to be the same variable.

Please show the calculations you used to arrive at your answer.
 
  • #3
Certainly (a,B)=(x/Sqrt[2])(1,i)+(y/Sqrt[2])(1,-i). I solved for x and y on Mathematica and got x=(a/Sqrt[2] - iB/Sqrt[2]) and for y= a/Sqrt[2]+iB/Sqrt[2]. I then assuming a^2+B^2=1 I just took the mod square of x and got (|a-i*B|^2)/2 to be my answer. Did I do something wrong?
 
  • #4
Nope, my mistake. Your answer is correct.

Because you already worked out ##\lvert +_y \rangle = \frac{1}{\sqrt{2}}(\lvert + \rangle + i\lvert - \rangle)##, an easier way to arrive at the same result is to calculate the amplitude ##\lvert \langle +_y \vert (a,b) \rangle \rvert^2##.
 

1. What is the formula for calculating the probability of a measure on S_y yielding h/2?

The formula for calculating the probability of a measure on S_y yielding h/2 is P(h/2) = |S_y(h/2)|^2 / ||S_y||^2, where S_y(h/2) is the projection of the state vector onto the subspace of eigenstates with eigenvalue h/2 and ||S_y||^2 is the total probability of all possible outcomes.

2. How do you find the projection of the state vector onto the subspace of eigenstates?

To find the projection of the state vector onto the subspace of eigenstates, you can use the formula S_y(h/2) = |e_i>e_i|Ψ>, where e_i represents the eigenstates and Ψ is the state vector.

3. What is the significance of the eigenvalue in calculating the probability?

The eigenvalue represents the possible outcomes of the measurement on the system. In this case, h/2 is the specific eigenvalue we are interested in and the projection of the state vector onto the subspace of eigenstates with eigenvalue h/2 gives us the probability of measuring that specific outcome.

4. How does the probability change if the state vector is in a superposition of eigenstates?

If the state vector is in a superposition of eigenstates, then the probability of measuring a specific outcome would be the sum of the individual probabilities for each eigenstate that makes up the superposition. This is because the state vector can collapse into any one of the eigenstates upon measurement, and the probability of each eigenstate is additive.

5. Can you calculate the probability for any measure on S_y or only for h/2?

You can calculate the probability for any measure on S_y as long as you have the projection of the state vector onto the subspace of eigenstates for that specific measure. The formula for calculating the probability remains the same, but the specific eigenvalue and projection will vary depending on the measure you are interested in.

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