I've been trying to form a proof using , using majorly dirac notation.There has been claims that its much better to use in QM.(adsbygoogle = window.adsbygoogle || []).push({});

The question i wanted to generally show that the expected value is Zero for all odd energy levels.I believe i have solved the question but im a bit Iffy about a step i took:

<x>_{n}= <Ψ_{n}|x|Ψ_{n}> = L

for a given Ψ_{n}= (A^{+})^{n}(n!)^{-2}

Energy eigen functions have definite parity, assume for all odd n's if one is zero the rest should follow.

Take n = 1

=> L = <(A^{+})(n!)^{-2}|x|(A^{+})(n!)^{-2}>

= (n!)^{-1}<(A^{+})|x|(A^{+})>

B = <(A^{+})|x|(A^{+})>

Define A^{+}= Lx + iC : B,C are Real

=> <Lx+iC|x|Lx+iC>

(Bit iffy after these steps)

= <Lx|x|Lx|> + <iC|x|iC>

= <L|x^{3}|L>+<C|x|C>

as ∫x^{2n+1}dx for limits [-∞,∞] and n =0,1,2,3....

=> 0

we find B=0

therefore

<x>_{n}= 0

For odd ns.

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# I Expectation for the Harmonic Oscillator ( using dirac)

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