Homework Assignment 4 Canonical Transformations 1. Consider 2019. 5. 15.آ  Homework Assignment 4...

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Transcript of Homework Assignment 4 Canonical Transformations 1. Consider 2019. 5. 15.آ  Homework Assignment 4...

  • Homework Assignment 4 Canonical Transformations Due Tuesday, Feb. 23, 2010

    1. Consider the canonical transformation from {p1, p2, q1, q2} → {P1, P2, Q1, Q2} such that

    Q1 = q 2

    1 , Q2 =

    q2

    cos(p2) ,

    P1 = p1cos(p2) − 2q2

    2q1cos(p2) , P2 = sin(p2) − 2q1.

    a) Show the following is true for the Poisson Brackets: [Qi, Pj ]pq = δij , [Qi, Qj ]pq = 0, and [Pi, Pj ]pq = 0. b) Find a generating function for this transformation.

    2. Show that the following transformation is canonical by showing that the symplectic condition is satisfied.

    Q = ln

    (

    1

    q sin(p)

    )

    , P = q cot(p).

    3. Find a generating function for the following transformation.

    Q1 = q1, P1 = p1 − 2p2

    Q2 = p2, P2 = −2q1 − q2

    Hint: A combination of two types of generating functions will work. Note that the transformation is a combi- nation of the identity transformation and the exchange transformation (which we went over in class).

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