Section IV, Quantum Mechanics II
Symmetries and conservations laws in quantum mechanics
1) Show that the invariancy of Hamiltonian H with respect to infinitesimal shifts in space leads to the momentum conservation in the quantum system governed by H, i.e. the mean value hP iψ(t)= const, where ψ(t) is an arbitrary state.
Use the Heisenberg equation dSdt = i
~[H, S].
2) Show that the invariancy of Hamiltonian with respect to infinitesimal rotations leads to the angular momentum conservation.
3) Consider the parity inversion operator P ψ(x) = ψ(−x). Formulate the conservation law being a consequence of invariancy of the Hamiltonian with respect to parity inversion.
4) Consider the Hamiltonian
Hf (x) = −d2f (x)
dx2 , f ∈ L2(0, 2π) ,
acting in L2(0, 2π); where f (0) = f (2π) and f0(0) = f0(2π). Define the momentum operator and formulate the momentum conservation in the con- sidered model.
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