# Expectation value of an anti-Hermitian operator

• meteorologist1
In summary, the conversation discusses how to prove that the expectation value of an anti-Hermitian operator is a pure imaginary number. Daniel provides a solution by considering an antiself-adjoint linear operator and its corresponding eigenvalues and eigenvectors. Seratend clarifies the difference between "expectation" and "expected", and provides further explanation on the use of "expectation value". Overall, the conversation highlights the importance of understanding and using correct terminology in mathematical and scientific discussions.
meteorologist1
Hi, could anyone tell me how one would show that the expectation value of a anti-Hermitian operator is a pure imaginary number? Thanks.

I solved this problem once right on this site...(Dunno if in the QM forum,or college homework).Use the site's search engine,or google to find it...

Daniel.

Yes,post #2 of that thread.Consider an antiself-adjoint linear operator $\hat{A}$ for which u wish to prove that it has a spectrum made up of 0 & purely imaginary #-s...

$$\hat{A}|\psi\rangle=\lambda|\psi\rangle$$

for an arbitrary eigenvector $|\psi\rangle$ corresponding to an eigenvalue $\lambda$

Then,the expectation value for this eigenstate is the eigenvalue,because

$$\langle\hat{A}\rangle_{|\psi\rangle}=\langle\psi|\hat{A}|\psi\rangle=\lambda$$(1)

The matrix element involved in (1) has the property

$$\langle\psi|\hat{A}|\psi\rangle=\left(\langle\psi|\hat{A}^{\dagger}|\psi\rangle\right)^{*}=\left(-\lambda\right)^{*}$$ (2)

Equating (1) & (2),you get that

$$\lambda=-\lambda^{*}$$ (3)

which means $\mbox{Re}(\lambda) =0$,Q.e.d.

Daniel.

Last edited:
In another form:

A anti hermitian => i.A is hermitian
=> eigenvalues of A= (eigenvalues of i.A)/i= -i.(real number)= imaginary number.
QED.

Seratend.

Ok thanks Daniel and Seratend.

By the way, "expectation" is a noun. "Expected" is an adjective.
The "expected value" is the "expectation".

HallsofIvy said:
By the way, "expectation" is a noun. "Expected" is an adjective.
The "expected value" is the "expectation".

I believe that used in this context, "expectation" becomes genitive, i.e. "value of an expectation". Other examples: economics textbook, price theory, etc.

Also - "expectation value" seems to be normal usage:

http://mathworld.wolfram.com/ExpectationValue.html

## 1. What is the expectation value of an anti-Hermitian operator?

The expectation value of an anti-Hermitian operator is a mathematical concept in quantum mechanics that represents the average measurement of a physical quantity in a quantum system. It is calculated by taking the inner product of the state vector with the operator and the state vector again.

## 2. How is the expectation value of an anti-Hermitian operator different from that of a Hermitian operator?

The expectation value of an anti-Hermitian operator is purely imaginary, while the expectation value of a Hermitian operator is a real number. This is because anti-Hermitian operators represent anti-symmetric or imaginary physical quantities, while Hermitian operators represent symmetric or real physical quantities.

## 3. What is the significance of the expectation value of an anti-Hermitian operator in quantum mechanics?

The expectation value of an anti-Hermitian operator is significant because it allows us to predict the average measurement of a physical quantity in a quantum system. It also plays a crucial role in determining the evolution of quantum states over time, as it is related to the Hamiltonian operator which governs the time evolution of quantum systems.

## 4. Can the expectation value of an anti-Hermitian operator be negative?

Yes, the expectation value of an anti-Hermitian operator can be negative. This is because anti-Hermitian operators can represent imaginary or anti-symmetric physical quantities, which can have negative values.

## 5. How is the expectation value of an anti-Hermitian operator related to uncertainty in quantum mechanics?

The expectation value of an anti-Hermitian operator is related to uncertainty in quantum mechanics through the Heisenberg uncertainty principle. This principle states that the product of the uncertainties in two non-commuting operators, such as an anti-Hermitian operator and a Hermitian operator, must be greater than or equal to the absolute value of their commutator. This means that the expectation value of an anti-Hermitian operator can give us information about the uncertainty in a corresponding physical quantity.

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