Expectation value of an anti-Hermitian operator

[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.

 

[dextercioby]

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.

 

[meteorologist1]

Are you referring to this post: https://www.physicsforums.com/showthread.php?t=68937

I have trouble understanding what you wrote. Could you please explain it in detail? Thanks.

 

[dextercioby]

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.

 

[seratend]

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.

 

[meteorologist1]

Ok thanks Daniel and Seratend.

 

[HallsofIvy]

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

 

[juvenal]

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