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Angular momentum polar coordinates

  1. Sep 24, 2012 #1
    1. The problem statement, all variables and given/known data

    from the cartesian definition of angular momentum, derive the operator for the z component in polar coordinates

    L_z = -ih[x(d/dy) - y(d/dx)]

    to

    L_z = -ih(d/dθ)

    2. Relevant equations
    x = rcosθ
    y = rsinθ

    r^2 = x^2 + y^2

    r = (x^2 + y^2)^1/2

    3. The attempt at a solution

    first of all I'm not sure how this is even possible. Every derivation of the angular momentum operator I've seen requires spherical coordinates, not polar.

    I tried taking the derivative of r with respect to x to get cosθ and with respect to y to get sinθ

    and dx/dθ = -rsinθ dy/dθ = rcosθ but it's not getting me anywhere.

    is there something i should be rewriting d/dx and d/dy as?
     
  2. jcsd
  3. Sep 24, 2012 #2
    Since you are transforming from one basis to another, you can use the identity (written for your particular case):

    [itex]\frac{∂}{∂θ}[/itex] = [itex]\frac{∂x}{∂θ}[/itex][itex]\frac{∂}{∂x}[/itex] + [itex]\frac{∂y}{∂θ}[/itex][itex]\frac{∂}{∂y}[/itex].

    Does that help?
     
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