How Do You Find the Energies and Wavefunctions for a Particle on a Sphere?

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In summary, the problem at hand involves finding the energies and wavefunctions of stationary states for a particle of mass m moving on the surface of a sphere of radius R. The Hamiltonian for this system is given by H = L^2/(2mR^2). The approach to solving this problem involves using the eigenfunctions of Lz, which are also eigenfunctions of L^2, and operating on them with L^2 to obtain the energy values. The rest of the stationary states can then be obtained by using the raising and lowering operators on the initial state. Despite the straightforward nature of this problem, the individual is struggling to find a starting point and is seeking assistance.
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land
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OK, I have that a particle of mass m is moving on the surface of a sphere of radius R but is otherwise free. The Hamiltonian is H = L^2/(2mR^2). All I have to do is find the energies and wavefunctions of the stationary states...

this seems like it should be really easy, but I am struggling mightily for some reason. To be honest I don't even know how to get started.. I know eigenfunctions of Lz are also eigenfunctions of L^2. I know L^2 operating on [tex]Y^m_l[/tex] is [tex]\hbar^2l(l+1)Y^m_l[/tex]. I know once I get one stationary state I should be able to get the rest by operating the raising and lowering operators on it. But I just don't know how to get there. :(

Thanks as always for the help.
 
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Anybody have a clue how to go about this? I feel like I should be able to do this but I just can't. I've looked at it for hours and don't know how to begin.. nothing I've tried works. sigh. thanks for the help :)
 
  • #3
Anybody? This should be a straightforward problem, which is what makes it so frustrating :(
 

FAQ: How Do You Find the Energies and Wavefunctions for a Particle on a Sphere?

What are spherical eigenvalues/functions?

Spherical eigenvalues/functions are a set of mathematical concepts used to describe the behavior of waves on a spherical surface or in a spherical region. They are solutions to a specific type of differential equation known as the spherical wave equation.

How are spherical eigenvalues/functions different from regular eigenvalues/functions?

Spherical eigenvalues/functions are unique because they take into account the curvature of the spherical surface, whereas regular eigenvalues/functions do not. This makes them particularly useful for studying phenomena such as light waves on the surface of a sphere or sound waves in a spherical room.

What are some applications of spherical eigenvalues/functions?

Spherical eigenvalues/functions have numerous applications in physics and engineering, including in the study of acoustics, electromagnetism, and quantum mechanics. They are also used in geophysics and seismology to model seismic waves traveling through the Earth.

How are spherical eigenvalues/functions calculated?

Spherical eigenvalues/functions are typically calculated using specialized mathematical techniques, such as separation of variables or series solutions. These methods involve breaking down the spherical wave equation into simpler equations that can be solved using known mathematical techniques.

What is the significance of spherical eigenvalues/functions in science?

Spherical eigenvalues/functions play a crucial role in understanding and predicting the behavior of waves on spherical surfaces or in spherical regions. They also have practical applications in various fields of science and engineering, making them an important area of study in mathematics and physics.

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