What is the Probability of Measuring 2a in State |\varphi (t)\rangle\right?

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In summary, the conversation discusses finding the probability, P_{2a}(t), of measuring the quantity A in the state |\varphi (t)\rangle\right and obtaining the value 2a. The solution involves diagonalizing the given matrix A and finding the eigenvectors and eigenvalues. The correct eigenvectors are [\frac{-i}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0], [0,0,1], and [i,1,0]. The probability is given by P_{2a} = cos^{2}(wt) divided by the dot product of \varphi(t) with its complex conjugate.
  • #1
Denver Dang
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Homework Statement


Find the probability, [itex]P_{2a}(t)[/itex], that a measurement of the quantity [itex]A[/itex] in
the state [itex]|\varphi (t)\rangle\right[/itex] will yield the value [itex]2a[/itex].


Homework Equations


[tex]\hat{A}|1\rangle\right = a(|1\rangle\right - i|2\rangle\right[/tex]
[tex]\hat{A}|2\rangle\right = a(i|1\rangle\right + |2\rangle\right[/tex]
[tex]\hat{A}|3\rangle\right = -2a(|3\rangle\right[/tex]

[tex]A = \[ \left( \begin{array}{ccc}
a & ia & 0 \\
-ia & a & 0 \\
0 & 0 & -2a\end{array} \right)\][/tex]

[tex]|\varphi (t)\rangle\right = \[ \left( \begin{array}{ccc}
cos(wt) \\
0 \\
-isin(wt) \end{array} \right)\][/tex]


The Attempt at a Solution



Well, I kinda suck at finding these probabilities. So I'm not sure what to do, since it asks for [itex]2a[/itex]. Is it just:
[tex]P(2a) = \left|\langle\psi_j|\Psi\rangle\right|^2,[/tex]
where [itex]\psi_j = \varphi[/itex] and [itex]\Psi = A|3\rangle\right[/itex], or am I just not getting it ?


Regards.
 
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  • #2
You need to find the eigenvectors and eigenvalues of the matrix, A, in terms of the basis vectors you are given. So basically you need to diagonalize the matrix you are given. This will give you 3 eigenvectors that are superpositions of the states: [itex]|1\rangle,|2\rangle,|3\rangle[/itex].

Once you have the eigenvectors of 'A': [itex]|1'\rangle, |2'\rangle, |3'\rangle[/itex] (all of which you need to properly normalize) and their corresponding eigenvectors. Then you will choose the eigenvector [itex]|u'\rangle[/itex] with the eigenvalue, +2a. Since this will be the superposition state the wavefunction will collapse to once you measure the property A and find its value +2a.

Finally, as you said before, the probability of finding this superposition state is:

[tex]P(2a) = \frac{\left|\langle u'|\varphi(t)\rangle\right|^2}{\left|\langle \varphi(t)|\varphi(t) \rangle \right|^2}[/tex]
 
Last edited:
  • #3
So I get the eigenvectors to be:

[itex][-i,1,0][/itex], [itex][0,0,1][/itex] and [itex][i,1,0][/itex].

Normalized they will become:

[itex][\frac{-i}{\sqrt{2}}, 0, 1][/itex], [itex][\frac{1}{\sqrt{2},0][/itex], and the last one can't be normalized, or am I wrong ?

The eigenvalues is:

[itex][0, -2a, 2a],[/itex]

So I need to use the 3rd eigenvector ?

And using the formula, I get that the probability must be:

[tex]P_{2a} = \frac{cos^{2}(wt)}{4},[/tex]
or am I way off ?Regards, and sorry for the late reply.
 
  • #4
You have the correct eigenvectors, but the normalized versions of them are wrong. You are correct about the eigenvalues and choosing the third eigenvector. But your probability is wrong. Double check the denominator in that probability.
 
  • #5
Think I screwed up the normalized eigenvectors. It should be:

[itex][\frac{-i}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0][/itex], [itex][0,0,1][/itex], and then again, the 3rd and last, I don't think I can normalize, since [itex]\sqrt{i^{2} + 1^{1} + 0} = \sqrt{0}[/itex]

And I think I used the 3rd eigenvector instead of [itex]\varphi(t)[/itex] in the denominator, so that's why I got it wrong. It should of course be:

[tex]P_{2a} = cos^{2}(wt)[/tex]

Or maybe divided by 2 if I can normalize the eigenvector as the 1st one ?
 
Last edited:
  • #6
The third vector should normalized the same as the first.

You want to multiply the vector by some normalization constant and solve for that constant. Remember that the magnitude of a vector is the dot product of itself with its complex conjugate. I believe you are leaving off the complex conjugate in the multiplication.
 
  • #7
Yup... I see now :)

Once again, thank you very much.
 

What is the difference between probability and statistics?

Probability deals with predicting the likelihood of future events based on previous occurrences, while statistics involves analyzing and interpreting data to make conclusions about a population or sample.

What is the formula for calculating probability?

The formula for calculating probability is: P(A) = n(A) / n(S), where P(A) is the probability of an event occurring, n(A) is the number of favorable outcomes, and n(S) is the total number of possible outcomes.

How is probability used in everyday life?

Probability can be used in everyday life to make decisions about risk, such as choosing an insurance plan or investing in the stock market. It is also used in fields like sports, weather forecasting, and gambling.

What is the difference between theoretical and experimental probability?

Theoretical probability is based on mathematical calculations and assumes that all outcomes are equally likely, while experimental probability is based on actual data collected through experiments or observations.

What is the difference between independent and dependent events?

Independent events are events whose outcomes do not affect each other, while dependent events are events whose outcomes do affect each other. In other words, the occurrence of one event does not change the probability of the other event in independent events, but it does in dependent events.

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