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# 06_580_hw_2 - Physics 580 Quantum Mechanics I Prof P M...

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Physics 580 Handout 6 7 September 2010 Quantum Mechanics I webusers.physics.illinois.edu/ goldbart/ Prof. P. M. Goldbart 3135 (& 2115) ESB University of Illinois 1) Conservation of angular momentum in Hamiltonian mechanics : A particle moves in three dimensions and its state is specified by the position q and the momentum p . Under an infinitesimal rotation of the state, q and p transform according to: q q + δ q = q + a × q p p + δ p = p + a × p where a is a vector of infinitesimal magnitude describing a rotation through angle | a | about the axis specified by a / | a | . a) Show that G a · ( q × p ) is the generator of this infinitesimal transformation, and that the transformation is canonical. b) Consider the hamiltonian H = 1 2 m | p | 2 + U ( | q | ) . Show that the Poisson bracket {H , G } vanishes for arbitrary infinitesimal a , and hence that angular momentum is conserved. c) The angular momentum vector L has cartesian components L μ (with µ = 1 , 2 , 3). Evaluate the Poisson brackets { L μ , L ν } between them. Explain why the transformation to a pair of components of angular momentum cannot be a canonical transformation. 2) Hamiltonian mechanics and Poisson brackets – only parts (g) to (l) are manda- tory : As Hamilton discovered, there are significant virtues associated with making yet an- other reformulation of mechanics, beyond the Newton Lagrange Stationary Action reformulations that we have already encountered. Amongst these virtues are: a set of dy- namical equations – the so-called canonical equations – that is even more simple and sym- metrical in its structure than Lagrange’s equations; the freedom to make transformations not only between convenient sets of generalised coordinates, but also to make so-called canonical transformations, i.e. , transformations that mix the generalised momenta and coordinates under which the canonical equations retain their form (thus breaking the link between coor- dinates and velocities that is inherent in the Lagrangian framework); and a formulation for classical mechanics that provides the most straightforward platform from which to embark upon quantum mechanics. In Hamilton’s formulation the coordinates and momenta appear on what is an essentially equal footing. Recall that in the Lagrangian formulation of mechanics we focus on the generalised coordinates { q j } n j =1 and the associated generalised velocities { ˙ q j } n j =1 . In particular, the 1

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dynamics is governed by the Lagrangian function L , which is a function of the generalised coordinates, the generalised velocities and, possibly, the time.
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