Can Schrodinger's Equation be Transformed into Spherical Polar Coordinates?

In summary, to change the Schrodinger's equation into spherical polar coordinates, one must use the chain rule for partial derivatives and the equations for converting x, y, z into r, θ, φ. This requires a lot of algebra. The final result, found in textbooks, includes both first and second-order derivatives. The laplacian operator can also be found in various coordinate systems in textbooks. There is not much physics to be learned from deriving it.
  • #1
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how do you change the schrodinger's equation into the spherical polar coordinates?
 
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  • #2
Look up the chain rule for partial derivatives, and the equations that give you [itex]x, y, z[/itex] in terms of [itex]r, \theta, \phi[/itex] for spherical coordinates. Use these to re-write the derivatives [itex]\partial^2 \psi / \partial x^2[/itex] etc. into the derivatives [itex]\partial^2 \psi / \partial r^2[/itex] etc. There's a lot of algebra. The final result (which you should be able to see in your textbook) contains both first- and second-order derivatives.
 
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  • #3
The coordinate-free form of the S.E. is:

[tex]i\hbar\frac{\partial \Psi(\vec r,t)}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2 \Psi(\vec r,t)+V\Psi(\vec r,t)[/tex]

You can (should) look up the laplacian operator [itex]\nabla^2[/itex] in various coordinate systems in your textbook. There's not much physics to be learned by deriving it.
 
  • #4
thank you very much! :)
 

What are polar coordinates and how do they differ from Cartesian coordinates?

Polar coordinates are a type of coordinate system used in mathematics and science to locate points on a plane. They differ from Cartesian coordinates in that they use a distance from the origin and an angle from a reference direction to describe the location of a point, instead of using x and y coordinates.

How do you convert from polar coordinates to Cartesian coordinates?

To convert from polar coordinates to Cartesian coordinates, you can use the following equations:

x = r * cos(theta) and y = r * sin(theta), where r is the distance from the origin and theta is the angle from the reference direction.

Why are polar coordinates useful in certain applications?

Polar coordinates are useful in applications where the distance and angle from a reference point are more relevant than the x and y coordinates. This includes applications in physics, engineering, and navigation.

Can polar coordinates be used in three-dimensional space?

Yes, polar coordinates can be extended to three-dimensional space by adding a third coordinate, typically denoted by z, to describe the height or depth of a point.

What are some common examples of polar coordinates in real life?

Polar coordinates are commonly used in applications such as radar and sonar systems, where the distance and angle from a target are important. They are also used in navigation systems, astronomy, and in describing the movement of objects in circular or elliptical orbits.

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