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Physics
Quantum Physics
Expectation for the Harmonic Oscillator ( using dirac)
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[QUOTE="Somali_Physicist, post: 6012075, member: 639437"] 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. 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 I am a bit Iffy about a step i took: <x>[SUB]n[/SUB] = <Ψ[SUB]n[/SUB]|x|Ψ[SUB]n[/SUB]> = L for a given Ψ[SUB]n[/SUB] = (A[SUP]+[/SUP])[SUP]n[/SUP](n!)[SUP]-2[/SUP] 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[SUP]+[/SUP])(n!)[SUP]-2[/SUP]|x|(A[SUP]+[/SUP])(n!)[SUP]-2[/SUP]> = (n!)[SUP]-1[/SUP] <(A[SUP]+[/SUP])|x|(A[SUP]+[/SUP])> B = <(A[SUP]+[/SUP])|x|(A[SUP]+[/SUP])> Define A[SUP]+[/SUP] = Lx + iC : B,C are Real => <Lx+iC|x|Lx+iC> (Bit iffy after these steps) = <Lx|x|Lx|> + <iC|x|iC> = <L|x[SUP]3[/SUP]|L>+<C|x|C> as ∫x[SUP]2n+1[/SUP]dx for limits [-∞,∞] and n =0,1,2,3... => 0 we find B=0 therefore <x>[SUB]n[/SUB] = 0 For odd ns. [/QUOTE]
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Physics
Quantum Physics
Expectation for the Harmonic Oscillator ( using dirac)
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