Possible Results and Probabilities of a Measurement of Operator Q

[koil_]

Homework Statement

A quantum system is prepared in a superposition state of the normalized eigenfunctions
$$\phi^{Q}$$ associated with the dynamical variable Q. The superposition may be written as,
$$ \Psi = 2\phi_1^Q+3\phi_2^Q+6\phi_3^Q $$

If the eigenvalues are $$Q_1 =1, Q_2 =0, Q_3 =−1$$ , what are the possible results of a
measurement of Q on this quantum system and what is the probability of obtaining each
possible result?

Find the expected value of Q

Relevant Equations

$$ \Psi = 2\phi_1^Q+3\phi_2^Q+6\phi_3^Q $$ $$Q_1 =1, Q_2 =0, Q_3 =−1$$

I first Normalise the wavefunction:
$$ \Psi_N = A*\Psi, \textrm{ where } A = (\frac{1}{\sum {|a_n^{'}|^{2}}})^{1/2} $$
$$ \Psi_N = \frac{2}{7}\phi_1^Q+\frac{3}{7}\phi_2^Q+\frac{6}{7}\phi_3^Q $$

The Eigenstate Equation is:
$$\hat{Q}\phi_n=q_n\phi_n$$

The eigenvalues are the set of possible outcomes from the dynamic variable Q so is that not the answer to the first part of the question?

For the second part, the fourth postulate states that after the dynamic variable is carried out the probability the result is equal to a particular eigenvalue is:
$$P(q_n) = |a_n|^2 \textrm{ Where } \Psi_N = \sum{a_n}{\phi_n}$$

which would simply give:

$$P(\phi_1)=\frac{4}{49}, P(\phi_2)=\frac{9}{49}, P(\phi_3)=\frac{36}{49} $$

But nowhere in my answers have I used the eigenvalues provided and the question is of the form "if the eigenvalues are X, find this" so I'm sure I've missed something, I'm just not sure what. The last part of the question talks about the expectation value of $Q$ which I would think to be:
$$<x>=q_1P(\phi_1)+q_2P(\phi_2)+q_3P(\phi_3)$$
$$<x>=1*\frac{4}{49}+0+(-)1*\frac{36}{49}$$
$$<x>=0.653$$

This part does (at least to my understanding) utilise the eigenvalues but for the second part discussed above I'm not sure how it comes into it.

Many thanks

 

[TSny]

Your work looks good, but check the sign of your answer for $<x>$.

The eigenvalues come in for the first part where you are asked to list the possible results for a measurement of Q.

Also, the eigenvalues are relevant when considering the probabilities. For example, $\frac{36}{49}$ is the probability that a measurement of Q yields the result -1.

 

[vela]

The eigenvalues are the set of possible outcomes from the dynamic variable Q so is that not the answer to the first part of the question?

Your answer is generally true, but the question is asking you to list the possible outcomes for this particular state. If the probability of a particular outcome turned out to be 0 for the given state, you shouldn't list the outcome as one of the possible results of the measurement. For example, if you had instead been given the state $\Psi = \frac{1}{\sqrt 2}\phi_1^Q + \frac{1}{\sqrt 2}\phi_2^Q$, you would not list $-1$ as one of the the possible outcomes because $a_3 = 0$ for this particular state.

 

[koil_]

Your work looks good, but check the sign of your answer for $<x>$.

The eigenvalues come in for the first part where you are asked to list the possible results for a measurement of Q.

Also, the eigenvalues are relevant when considering the probabilities. For example, $\frac{36}{49}$ is the probability that a measurement of Q yields the result -1.

Thanks for pointing that mistake out and for the clarification :)

 

[PeroK]

$$P(\phi_1)=\frac{4}{49}, P(\phi_2)=\frac{9}{49}, P(\phi_3)=\frac{36}{49} $$

This isn't wrong, as such, but instead I would have written:
$$P(Q_1)=\frac{4}{49}, P(Q_2)=\frac{9}{49}, P(Q_3)=\frac{36}{49} $$
Can you see why?

 

[koil_]

Your answer is generally true, but the question is asking you to list the possible outcomes for this particular state. If the probability of a particular outcome turned out to be 0 for the given state, you shouldn't list the outcome as one of the possible results of the measurement. For example, if you had instead been given the state $\Psi = \frac{1}{\sqrt 2}\phi_1^Q + \frac{1}{\sqrt 2}\phi_2^Q$, you would not list $-1$ as one of the the possible outcomes because $a_3 = 0$ for this particular state.

I see thank you - I think what threw me off was just the wording of the question (as this was the only question in a series of superposition question that involved the eigenvalues) and I expected to utilise more information than I needed.

On from that could you possibly recommend any resources that are good at getting the introductrory level understanding in QM particularly with the terminology. I'm a wee Y2 Astrophys student and I'm not feeling terribly confident in my work using university resources just due to the "choppyness" of content covered instead of a thorough coverage.

We use the Quantum Mechanics by Alastair I. M. Rae and Jim Napolitano but the lack of examples restrict my practical abilities when it comes to questions. Many thanks

 

[koil_]

This isn't wrong, as such, but instead I would have written:
$$P(Q_1)=\frac{4}{49}, P(Q_2)=\frac{9}{49}, P(Q_3)=\frac{36}{49} $$
Can you see why?

I would like to say because, when a measurement is made, the wavefunction will collapse into one of the superposition eigenfunctions $\phi_n$ where the operator $\hat{Q}$ on $\phi_n$ will 'output' the eigenvalue $Q_n$?

 

[PeroK]

I would like to say because, when a measurement is made, the wavefunction will collapse into one of the superposition eigenfunctions $\phi_n$ where the operator $\hat{Q}$ on $\phi_n$ will 'output' the eigenvalue $Q_n$?

I would say because the probabilities are fundamentally related to the outcome of the experiment: and the outcome of the experiment is the measurement $Q_n$. The probability of each measurement value describes the experimental outcomes. The wave-function and its collapse are the underlying QM machinery, as it were. So, it seems more pointed to me to relate probabilities to what you measure, rather than what the wave function may or may not be doing in the background.