Y: Calculating Expectation Value of O with Orthonormal State

[qtp]

I have a question... if anyone can maybe help

you have psi = a(psi1) + b(psi2) + c(psi3) and the state is orthonormal...
what is the expectation value of O if O (O is an operator) yields known eigenvalues for psi1 psi2 and psi3 i tried to say that psi1*Opsi1 over all space is (a)(eigenvalue of psi1) times integral of psi1*psi1 which is 1 since the state is orthonormal but that didn't give me the right answer. The correct answer is 1 with a= 1/(root(6)) b= 1/(root(2)) c= 1/(root(3)) and Opsi1 = 1psi1, Opsi2 = -1psi2, Opsi3 = 2psi3 any help would be greatly appreciated

qtP

 

[Jimmy Snyder]

The correct answer is 1

I get 1/3 as follows:

$<O> = <\psi|O|\psi>$
$= <a\psi_1 + b\psi_2 + c\psi_3|O|a\psi_1 + b\psi_2 + c\psi_3>$
$= <a\psi_1 + b\psi_2 + c\psi_3|a\psi_1 - b\psi_2 + 2c\psi_3>$
$= a^2 - b^2 + 2c^2$
$= \frac{1}{6} - \frac{1}{2} + \frac{2}{3}$
$= \frac{1}{3}$

 

[qtp]

hey thank you very much :) you are correct but the answer is 1 because i switched Opsi1 and Opsi2 Opsi1= -1psi1 and Opsi2= 1psi2 so it is -1/6+ 1/2+ 2/3 = 1 thank you again :)