# Measurement with respect to the observable Y

• I
• Peter_Newman
In summary, the conversation discusses the calculation of the average value or expectation of a measurement with respect to observable Y for a given system state. The eigenvalues and projection operators for +1 and -1 are derived, and it is shown that the probabilities for each outcome are 0.5. The conversation then addresses the question of how the average value can be 0, and it is explained that this is a result of the 50% chance for each outcome. An alternative method for calculating the expectation value is also provided.
Peter_Newman
Hello,

I would like to start with an assumption. Suppose a system is in the state:
$$|\psi\rangle=\frac{1}{\sqrt{6}}|0\rangle+\sqrt{\frac{5}{6}}|1\rangle$$

The question is now: A measurement is made with respect to the observable Y. The expectation or average value is to calculate.
My first ideas are this:

$$Y=\begin{pmatrix}0&i\\-i&0\end{pmatrix}$$
the eigenvalues are:
$$|u_1\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\i\end{pmatrix}, |u_2\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\-i\end{pmatrix}$$
The projection operator corresponding to a measurement of +1 is:
$$P_{u_1}=|u_1\rangle\langle u_1|=\left(\frac{|0\rangle+i|1\rangle}{\sqrt{2}}\right)\left(\frac{\langle0|-i\langle 1|}{\sqrt{2}}\right)$$
$$=\frac{1}{2}(|0\rangle\langle 0|-i|0\rangle\langle 1| +i|1\rangle\langle 0| +|1\rangle\langle 1| )$$
for ##P_{u_2}##
$$P_{u_2}=|u_2\rangle\langle u_2|=\frac{1}{2}(|0\rangle\langle 0|+i|0\rangle\langle 1| -i|1\rangle\langle 0| +|1\rangle\langle 1| )$$
accordingly
$$Pr(+1)=\langle \psi|P_{u_1}|\psi\rangle=\left(\frac{1}{\sqrt{6}}\langle 0|+\sqrt{\frac{5}{6}}\langle 1|\right)\left(\frac{1}{2}(|0\rangle\langle 0|-i|0\rangle\langle 1| +i|1\rangle\langle 0| +|1\rangle\langle 1| \right)\left(\frac{1}{\sqrt{6}}|0\rangle +\sqrt{\frac{5}{6}}|1\rangle\right)$$
I've calculated that and get to the following result:
$$Pr(+1)=0.5, Pr(-1)=0.5$$
The probabilities sum up to 1:
$$\langle \psi | P_{u_1}|\psi\rangle+\langle \psi | P_{u_2}|\psi\rangle=1$$
$$0.5+0.5=1$$
I would be interested to know if this is okay so far? My problem arises here:
The average value is: ##\langle X \rangle = (+1)Pr(+1)+(-1)Pr(-1)=0##

My question is, how can the average value be 0?

If something does not fit with the notation here, I would be very happy to be corrected.

So I would be very happy about answers and criticism. Thank you!

You may calculate the expectation value more directly without resorting to eigen-value decomposition:
$$\langle Y\rangle =\psi^\dagger Y \psi = \left(\begin{array}{cc}\sqrt{1/6} & \sqrt{5/6}\end{array}\right) \left(\begin{array}{rr} 0 & i \\ -i & 0\end{array}\right)\left(\begin{array}{r}\sqrt{1/6}\\ \sqrt{5/6}\end{array}\right)=$$
$$= \left(\begin{array}{cc}\sqrt{1/6} & \sqrt{5/6}\end{array}\right) \left(\begin{array}{r}i\sqrt{5/6}\\ -i\sqrt{1/6}\end{array}\right)= i\frac{\sqrt{5}}{6} - i\frac{\sqrt{5}}{6} =0$$

So you see I am getting 0 as well. That's not impossible. You gave the two eigen-vectors for Y and their respective eigen-values were +1 and -1. You also noted that their separate probabilities, given your initial system "state" were each 1/2 as you verified the total probability was 1. It is then quite clear that if you have outcome +1 50% of the time and outcome -1 50% of the time you'll have an average outcome of 0. Quite literally "you win some, you lose some".

vanhees71, Peter_Newman and jim mcnamara

## 1. What is measurement with respect to the observable Y?

Measurement with respect to the observable Y refers to the process of quantitatively determining the value of a specific observable Y. This can involve using various tools and techniques to collect data and make numerical observations.

## 2. Why is measurement with respect to the observable Y important?

Measurement with respect to the observable Y is important because it allows us to better understand and analyze the behavior of the observable Y. It also helps us make predictions and draw conclusions about the relationship between different variables.

## 3. What are some examples of observable Y?

Observable Y can refer to any measurable quantity or characteristic, such as temperature, weight, height, or time. It can also include more abstract concepts like happiness, intelligence, or success.

## 4. How do you ensure accuracy in measurement with respect to the observable Y?

To ensure accuracy in measurement with respect to the observable Y, it is important to use reliable and calibrated instruments, follow proper measurement techniques, and repeat measurements multiple times to reduce errors. It is also important to consider any potential sources of error and try to minimize their impact.

## 5. How does measurement with respect to the observable Y contribute to scientific research?

Measurement with respect to the observable Y is a crucial aspect of scientific research as it allows for the collection of quantitative data, which can be analyzed and used to support or refute hypotheses. It also helps to establish a standard for comparing and replicating results, leading to further advancements in scientific understanding.

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