How Are Angular Momentum Operators Calculated in Spherical Polar Coordinates?

In summary, the formulae for the angular momentum operators in spherical polar coordinates are obtained by using the del-operator and substituting in for r and del in spherical polars. This will result in the computation of \hat{L_r}, \hat{L_\theta}, and \hat{L_\phi} rather than \hat{L_x}, \hat{L_y}, and \hat{L_z}. The del-operator is used because it represents the gradient operator in spherical coordinates.
  • #1
latentcorpse
1,444
0
How does one obtain the formulae for the angular momentum operators in spherical polar coordinates i.e.

[itex]\hat{L_x}=i \hbar (\sin{\phi} \frac{\partial}{\partial{\theta}} + \cot{\theta} \cos{\phi} \frac{\partial}{\partial{\phi}}[/itex]
[itex]\hat{L_y}=i \hbar (-\cos{\phi}{\phi} \frac{\partial}{\partial{\theta}} + \cot{\theta} \sin{\phi} \frac{\partial}{\partial{\phi}}[/itex]
[itex]\hat{L_z}=-i \hbar \frac{\partial}{\partial{\phi}}[/itex]
[itex]\hat{L}^2=\hbar^2 \left[\frac{1}{\sin{\theta}} \frac{\partial}{\partial{\theta}} \left(\sin{\theta} \frac{\partial}{\partial{\theta}} \right) +\frac{1}{\sin^2{\theta}} \frac{\partial^2}{\partial{\phi}} \right][/itex]

?
 
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  • #2
Use the del-operator in spherical coordinates.
 
  • #3
so [itex]\hat{L}=\hat{r} \times \hat{p}[/itex]

where [itex]\hat{p}=-i \hbar \nabla[/itex]

do i use [itex]\nabla[/itex] as the gradient operator in spherical polars?

also do i write r in terms of x,y,z or in terms of r,theta,phi?
 
  • #4
have you even tried?
 
  • #5
yes. my notes run through it for the Cartesian case and subsitute r=(x,y,z) and [itex]\nabla=(\partial_x,\partial_y,\partial_z)[/itex].
regardelss of which of the combinations i of r and del iuse above i can't get the right answer.

surely if i sub in for del in spherical polars I'm actually computing [itex]\hat{L_r},\hat{L_\theta},\hat{L_\phi}[/itex] rather than [itex]\hat{L_x},\hat{L_y},\hat{L_z}[/itex], no?
 

1. What is an angular momentum operator?

An angular momentum operator is a mathematical operator used in quantum mechanics to describe the angular momentum of a particle. It is represented by the symbol L and its components are Lx, Ly, and Lz.

2. How is angular momentum operator related to angular momentum?

The angular momentum operator is related to angular momentum through the Heisenberg uncertainty principle. The eigenvalues of the angular momentum operator represent the possible values of the angular momentum of a particle.

3. What is the physical significance of angular momentum operators?

The physical significance of angular momentum operators is that they describe the rotational motion of a particle in quantum mechanics. They play a crucial role in understanding the behavior and properties of atoms and molecules.

4. How do angular momentum operators act on wavefunctions?

Angular momentum operators act on wavefunctions by generating a new wavefunction with the same energy and momentum, but with a different angular momentum state. This process is known as a rotational transformation.

5. What are the commutation relations of angular momentum operators?

The commutation relations of angular momentum operators are [Lx, Ly] = iħLz, [Ly, Lz] = iħLx, and [Lz, Lx] = iħLy. These relations show that the components of angular momentum do not commute, and therefore cannot be simultaneously measured with arbitrary precision.

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