Sunday, 15 November 2020

homework and exercises - How to prove dp/dt = -dV/dx? Quantum mechanics



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I got this problem from a book called Introduction to quantum mechanics, griffin 2nd edition. and I did not get why the solution says


first term integrates to zero, integration by parts twice?!



Please see the solution below! Thanks everyone for helping!



Answer



He uses


(xf(x)) g(x) dx= f(x)(xg(x)) dx,


to get


(2x2Ψ)xΨ dx=(xΨ)(2x2Ψ) dx=Ψ3x3Ψ dx.


The fact that you can shift the sqare of the derivative 2x2 around like this is already suggested by Δ=3n=12x2n being an observable in several contexts. If it's an observable, then it's hermitean and ΔΦ|Φ=Φ|ΔΦ.


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