- #1
squigglywolf
- 16
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Is it possible to express ANY observable A(X,P) in terms of the ladder operators?
I know how to evaluate expectation values in the |n> basis given the operators in terms of a & a+, but was trying to figure out <1/X^2>. How do you express 1/X^2 in terms of ladder operators? <ψ|(1/X^2)|ψ> can be done in the X basis fine, by evaluating ∫ψ*(1/x^2)ψ dx , but what about in |n> basis?
i.e
Given X^2 = (h/2mw)[a^2 + aa' + a'a +a'^2] how do I "invert" this into 1/X^2 , and be able to evaluate it in a state <1|(1/X^2)|1> for example?
(a' = a-dagger)
I know how to evaluate expectation values in the |n> basis given the operators in terms of a & a+, but was trying to figure out <1/X^2>. How do you express 1/X^2 in terms of ladder operators? <ψ|(1/X^2)|ψ> can be done in the X basis fine, by evaluating ∫ψ*(1/x^2)ψ dx , but what about in |n> basis?
i.e
Given X^2 = (h/2mw)[a^2 + aa' + a'a +a'^2] how do I "invert" this into 1/X^2 , and be able to evaluate it in a state <1|(1/X^2)|1> for example?
(a' = a-dagger)
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