When I was learning translational symmetry I saw that for translation invariance, i.e(adsbygoogle = window.adsbygoogle || []).push({});

the momentum P needs to be conserved[T,H]=0

.[P,H]=0

This momentum is actually the generator of small translations defined as

.T:x→x+ε

Now, I was solving some problems and I met one which is interesting. I am given a potential

That potential is actually invariant under the translation defined above. In that problem they say that consequently the "momentum is not conserved". Can anybody tell me why?V(x)=Asin(2πx/ε)(where A is a constant)

I tried to understand it by saying that, if I can show that the Hamiltonian is not conserved hence the momentum is not. But it looks that I don't have enough information about H.

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# Potential Invariant under translation

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