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Determine whether the ground state is an eigenfunction of [p] and of [p^2] [closed]

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Consider a particle confined in an infinite square well potential of width L,

$$V(x)=left{ begin{array}{ll}infty, &{rm for} (x le 0)vee (x ge L) , &{rm for} 0 < x < L end{array}right. $$

The ground state solution of the time independent Schrödinger equation is given by $psi$(x)=Asin(kx), where k = $frac{pi}{L}$ and A = $sqrt{frac2L}$.

Determine whether the ground state is an eigenfunction of $[p]$ and of $[p^2]$. Discuss the implication of the results.

So far I have guessed the expectation value to be $frac{L}{2}$ and confirmed it by evaluating the integral (from an earlier part of the question). I’m stuck on how to do the last part though. From a previous second year quantum mechanics exam, any help would be greatly appreciated.


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