How to Prove Momentum Operator is Hermite Operator?

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SUMMARY

The momentum operator in quantum mechanics is proven to be a Hermitian operator by applying the definition of a Hermitian operator to the inner product, which involves integration. Partial integration is utilized, considering that wave functions must belong to the L2 space and adhere to specific boundary conditions at infinity. The self-adjoint nature of the momentum operator in one-dimensional space is contingent upon these boundary conditions, as discussed in the referenced material from Caltech.

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Karmerlo
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Hi, I have little trouble in proving a proving problem ---


How to Prove Momentum Operator is Hermite Operator?


Thanks.
 
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Apply the definition of a Hermitian operator on the inner product (which is an integral), do a partial integration and use that wave functions have to be in L2 (and thus have certain boundary conditions at infinity).
 

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