# Question if expectation value is considered a measurement?

• JordanGo
In summary, the expectation value in quantum mechanics is a weighted average of all the possible eigenvalues of an operator. It may or may not be a valid eigenvalue itself, depending on the specific state being measured.
JordanGo
Hello,

I was just curious about expectation values. One of the postulates of quantum mechanics state:
The only possible results of a measurement is an eigenvalue of the operator.

Now, is the expectation value considered a measurement, thus considered an eigenvalue?

Thanks!

An expectation value is a weighted average of all of the eigenvalues. So it may or may not actually be a valid eigenvalue itself.

As an example, the X component of the spin operator has two eigenvalues: -1/2 and 1/2. However, if you take the X expectation value of a Z state (which is a linear combination of both of those eigenstates), the expectation value will be 0, since both -1/2 and 1/2 states appear in equal combination. 0 is not actually a valid eigenvalue of the X operator, but it's the average of the two actual eigenvalues.

## 1. What is an expectation value in measurement?

An expectation value is a calculated average of all possible outcomes of a measurement. It represents the most probable value that will be obtained when a measurement is repeated multiple times.

## 2. How is an expectation value calculated?

To calculate an expectation value, you must first determine all possible outcomes of a measurement and their corresponding probabilities. Then, you multiply each outcome by its probability and add them all together. The resulting sum is the expectation value.

## 3. Is an expectation value always a measurable quantity?

No, an expectation value is not always a measurable quantity. It is a theoretical concept used to predict the most probable value of a measurement, but it may not always be physically measurable.

## 4. Can the expectation value be used to determine the exact value of a measurement?

No, the expectation value cannot be used to determine the exact value of a measurement. It only represents the most probable value and does not take into account any fluctuations or uncertainties in the measurement process.

## 5. How does the expectation value relate to other statistical measures such as standard deviation?

The expectation value is related to other statistical measures, such as standard deviation, through the uncertainty principle. The uncertainty principle states that the more precisely one quantity is known, the less precisely the other can be known. Therefore, a smaller uncertainty in the expectation value corresponds to a larger uncertainty in other statistical measures, and vice versa.

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