- #1
renec112
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I am practicing old exams. I tried my best but looking at an old and a bit unreliable answer list, and i am not getting the same result.
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 ##
##a_+ \psi_n = \sqrt{n+1} \psi_{n+1}##
##a_- \psi_n = \sqrt{n} \psi_{n-1}##
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.
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.