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Angular momentum polar coordinates

  • Thread starter johnnyies
  • Start date
  • #1
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Homework Statement



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)

Homework Equations


x = rcosθ
y = rsinθ

r^2 = x^2 + y^2

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

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?
 

Answers and Replies

  • #2
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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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