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## Homework Statement

At time ##t=0## the nomralized harmonic oscialtor wavefunction is given by:

## \Psi(x,0) = \frac{1}{\sqrt{3}}(\psi_0(x) + \psi_1(x) + i \psi_2(x))##

## a_+ ## and ##a_-## is respectively the raising and lower operator in the harmonic oscilatordefine ## Y = i(a_+ - a_-)##

find the expectation value ## \langle Y \rangle ##

## Homework Equations

##a_+ \psi_n = \sqrt{n+1} \psi_{n+1}##

##a_- \psi_n = \sqrt{n} \psi_{n-1}##

## The Attempt at a Solution

Do a sandwich:

## \langle \Psi | Y | \Psi \rangle ##

insert operator

## = \langle \Psi | i(a_+ - a_-) | \Psi \rangle ##

Split inner product

## = \langle \Psi | i a_+ | \Psi \rangle - \langle \Psi | i a_- | \Psi \rangle ##

take ##i## constant out

## =i \langle \Psi | a_+ | \Psi \rangle - i\langle \Psi | i a_-| \Psi \rangle ##

Insert wavefunction the a's er operatring on

## = i\langle \Psi | a_+ | \frac{1}{\sqrt{3}}(\psi_0(x) + \psi_1(x) + i \psi_2(x)) \rangle - i \langle \Psi | a_- | \frac{1}{\sqrt{3}}(\psi_0(x) + \psi_1(x) + i \psi2(x))\rangle ##

Operate with a's

## = i \langle \Psi | \frac{1}{\sqrt{3}}(\psi_1(x) + \sqrt{2} \psi_2(x) + \sqrt{3} i \psi_3(x)) \rangle - i \langle \Psi | \frac{1}{\sqrt{3}}( \psi_0(x) + i \sqrt{2} \psi_1(x))\rangle ##

insert ##\Psi## on bra as well

## = i \langle \frac{1}{\sqrt{3}}(\psi_0(x) + \psi_1(x) + i \psi_2(x)) | \frac{1}{\sqrt{3}}(\psi_1(x) + \sqrt{2} \psi_2(x) + \sqrt{3} i \psi_3(x)) \rangle - i \langle \frac{1}{\sqrt{3}}(\psi_0(x) + \psi_1(x) + i \psi_2(x)) | \frac{1}{\sqrt{3}}( \psi_0(x) + i \sqrt{2} \psi_1(x))\rangle ##

Take ##1/\sqrt{3} ## out of inner product

## = i \frac{1}{3} \langle (\psi_0(x) + \psi_1(x) + i \psi_2(x)) | (\psi_1(x) + \sqrt{2} \psi_2(x) + \sqrt{3} i \psi_3(x)) \rangle - i \frac{1}{3} \langle (\psi_0(x) + \psi_1(x) + i \psi_2(x)) | ( \psi_0(x) + i \sqrt{2} \psi_1(x))\rangle ##

Do inner product

## = i \frac{1}{3} (1 + i \sqrt{2}) - i \frac{1}{3} (1 + i \sqrt{2})##

Take minus inside parenthesis in the last term

## = i \frac{1}{3} (1 + i \sqrt{2}) + i \frac{1}{3} (-1 - i \sqrt{2})##

factorize

## = i \frac{1}{3} (1 + i \sqrt{2} -1 - i \sqrt{2}) ##

## = 0 ##

I got the same result last time. Maybe I'm doing the same mistakes? Would love you input.