# Can the expectation of an operator be imaginary?

• I
• Tspirit
In summary, the expectation $$\int\varPsi^{*}\hat{O}\varPsi$$ can be imaginary if the quantum operator is not Hermitian. However, if the operator is Hermitian, the expectation will always be real. This can be seen from the definition of a Hermitian operator, which states that the expectation value is equal to its conjugate, similar to the equation for a real number.
Tspirit
Assume ##\varPsi## is an arbitrary quantum state, and ##\hat{O}## is an arbitrary quantum operator, can the expectation $$\int\varPsi^{*}\hat{O}\varPsi$$ be imaginary?

Not if the operator is hermitian. Otherwise it can be. For instance, think about the situation where ##\Psi## is normalized to unity and the operator is just a multiplication with ##i## (imaginary unit).

hilbert2 said:
Not if the operator is hermitian. Otherwise it can be. For instance, think about the situation where ##\Psi## is normalized to unity and the operator is just a multiplication with ##i## (imaginary unit).
Thank you very much! I have known the deduction.
According to the definition of Hermitian operator denoted by ##\hat{O}## as follows, $$<f|\hat{O}g>=<\hat{O}f|g>,$$ and making ##f=g##, we have $$<f|\hat{O}f>=<\hat{O}f|f>=(<f|\hat{O}f>)^{*},$$ meaning ##<f|\hat{O}f>## is real, which resembles the equation $$a=a^{*},$$ meaning ##a## is real.

vanhees71

## 1. Can an operator's expectation be imaginary?

Yes, an operator's expectation can be imaginary. This means that the expected value of the operator is a complex number with an imaginary component.

## 2. What does it mean for an operator's expectation to be imaginary?

When an operator's expectation is imaginary, it means that the expected value of the operator involves an imaginary component, which is represented by the letter "i". This indicates that the operator has a non-zero imaginary part.

## 3. How is the expectation of an operator calculated?

The expectation of an operator is calculated by taking the sum of the product of each possible outcome with its corresponding probability. Mathematically, it is represented as E[O] = ∑ P(x)O(x), where E[O] is the expectation of the operator O, P(x) is the probability of each outcome x, and O(x) is the value of the operator for that outcome.

## 4. Can an operator's expectation be both real and imaginary?

No, an operator's expectation cannot be both real and imaginary at the same time. It is either completely real or completely imaginary, depending on the composition of the operator and the probability distribution of its outcomes.

## 5. Why is it important to consider the imaginary part of an operator's expectation?

Considering the imaginary part of an operator's expectation is important in certain applications, such as quantum mechanics, where operators are used to represent physical observables. The imaginary part of an operator's expectation can provide important information about the system being studied and can help in making accurate predictions and measurements.

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