Kinetic Energy operator is hermitian

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SUMMARY

The kinetic energy operator, represented as T = -h²/2mΔ, is confirmed to be Hermitian. In one dimension, it can be expressed as T = (-h²/2m)(d²/dx²). The discussion raises the question of whether this proof generalizes to three dimensions, indicating a need for clarity on the Hermitian nature of the operator in higher dimensions. The provided notes on Hermitian operators serve as a foundational resource for understanding this topic.

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The problem asks to show that the kinetic energy operator is Hermitian.
The operator is given as T= -h^2/2mΔ but I know I can also write it as p^2/2m which would be

(- ih∇)(-ih∇). My main question is if I can prove this in 1-D so that T=(-h^2/2m)d^2/dx^2
does that generalize to 3-d? Is it proven in 3 dimensions then? If not I'm not sure where to even begin in 3D
 
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See this set of notes on Hermitian operators: http://faraday.uwyo.edu/~yurid/QM/Lecture%208.pdf
 
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