Principles of Quantum Mechanics · Problem 2.8.3

All Is Not Well with Classical Mechanics — R. Shankar

Physics — Quantum Mechanics · symmetries-conservation-laws

Problem

Consider H=px2+py22m+12mω2(x2+y2)\mathscr{H} = \frac{p_x^2 + p_y^2}{2m} + \frac{1}{2}m\omega^2(x^2 + y^2) whose invariance under the rotation of the coordinates and momenta leads to the conservation of lzl_z. But H\mathscr{H} is also invariant under the rotation of just the coordinates. Verify that this is a noncanonical transformation. Convince yourself that in this case it is not possible to write δH\delta\mathscr{H} as ε{H,g}\varepsilon\{\mathscr{H}, g\} for any gg, i.e., that no conservation law follows.

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