Cylindrically symmetric potential function

In summary, a cylindrically symmetric potential function is a mathematical function that describes the potential energy in a system with cylindrical symmetry. It only depends on distance from the central axis, and not on the angle around it. It differs from a spherically symmetric potential function, which has symmetry along all three axes. Some real-world examples include a long wire with a point charge, a charged cylinder, or a system of charged particles in a cylindrical shape. It is used in physics to model and analyze systems with cylindrical symmetry, particularly in electrostatics and magnetostatics. It can also be extended to include multiple central axes, resulting in an axially symmetric potential function, commonly seen in systems with multiple cylindrical or rotational symmetries.
  • #1
Amith2006
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Homework Statement



I am trying solve the 3-D Schrodinger equation for a particle in a cylindrically symmetric potential. If it was the case of spherically symmetric potential, then we can approximate it to a central potential. But what will be the form of the potential in the cylindrically symmetric case? Can any suggest a way to derive it?

Homework Equations


central potential function is given by,
V=Ze^2/r



The Attempt at a Solution

 
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  • #2
Have you tried anything yet?

P.S. The obvious first step is to use cylindrical coordinates ;-)
 

FAQ: Cylindrically symmetric potential function

What is a cylindrically symmetric potential function?

A cylindrically symmetric potential function is a mathematical function that describes the potential energy of a point in space in a system that has cylindrical symmetry. This means that the potential energy at a point is only dependent on the distance from the central axis and not on the angle around the axis.

How is a cylindrically symmetric potential function different from a spherically symmetric potential function?

A cylindrically symmetric potential function is different from a spherically symmetric potential function in that it only has symmetry along one axis, while a spherically symmetric potential function has symmetry along all three axes. This means that in a cylindrically symmetric system, the potential energy only changes with distance from the central axis, while in a spherically symmetric system, the potential energy changes with both distance from the center and direction from the center.

What are some real-world examples of systems with cylindrically symmetric potential functions?

Some real-world examples of systems with cylindrically symmetric potential functions include a long wire with a point charge at one end, a charged cylinder, or a system of charged particles arranged in a cylindrical shape.

How is a cylindrically symmetric potential function used in physics?

A cylindrically symmetric potential function is used in physics to model and analyze systems that have cylindrical symmetry, such as those mentioned in the previous question. It is particularly useful in electrostatics and magnetostatics, as it simplifies the calculations and allows for easier visualization of the system.

Can a cylindrically symmetric potential function be extended to include more than one central axis?

Yes, a cylindrically symmetric potential function can be extended to include more than one central axis, resulting in an axially symmetric potential function. This means that the potential energy at a point is only dependent on the distance from multiple central axes and not on the angles around those axes. This is often seen in systems with multiple cylindrical or rotational symmetries, such as a charged ring or a spinning top.

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