About momentum operator in other coordinate system

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SUMMARY

The discussion focuses on deriving the momentum operator in non-linear coordinate systems, specifically spherical coordinates. Participants emphasize the importance of expressing other coordinates in terms of Cartesian coordinates (x, y, z) and applying the chain rule for differentiation. The formula provided illustrates the transformation of the partial derivative with respect to x into spherical coordinates using the variables r, φ, and θ. This method is essential for understanding quantum mechanics in various coordinate systems.

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  • Understanding of quantum mechanics and operators
  • Familiarity with spherical coordinate systems
  • Knowledge of partial derivatives and the chain rule
  • Basic proficiency in mathematical notation and transformations
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  • Study the application of the chain rule in multi-variable calculus
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Students and professionals in physics, particularly those studying quantum mechanics, as well as mathematicians interested in coordinate transformations and their applications in physics.

KFC
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Hi there, does anyone know where can I found a material or note about how to deduce momentum operator in coordinate system other than linear coordinate (especially in spherical coordinate system)?

Thanks in advanced.
 
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KFC said:
Hi there, does anyone know where can I found a material or note about how to deduce momentum operator in coordinate system other than linear coordinate (especially in spherical coordinate system)?

Thanks in advanced.

Express the other coordinates in terms of [itex]x[/itex], and [itex]y[/itex], and [itex]z[/itex], and then use the chain rule. For example,

[tex] \frac{\partial}{\partial x} = \frac{\partial r}{\partial x}\frac{\partial}{\partial r} + \frac{\partial \phi}{\partial x}\frac{\partial}{\partial \phi} + \frac{\partial \theta}{\partial x}\frac{\partial}{\partial \theta}.[/tex]
 

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