Strange Hamiltonian of two particles on the surface of a sphere

In summary, there is a problem with the given Hamiltonian for two identical particles on the surface of a sphere, as it does not include any interaction between the particles. To find the eigenvalues of this Hamiltonian, one may separate it into four parts and consider orbital and spin eigenstates, taking into account the Pauli principle. However, there are infinitely many states to consider due to the unbounded values of ##l_1## and ##l_2##.
  • #1
Salmone
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I have a problem with this Hamiltonian: two identical particles of mass ##m## and spin half are constrained to move on the surface of a sphere of radius ##R##. Their Hamiltonian is ##H=\frac{1}{2}mR^2(L_1^2+L_2^2+\frac{1}{2}L_1L_2+\frac{1}{2}S_1S_2)##. By introducing the two operators
##L=L_1+L_2## and ##S=S_1+S_2## I was able to rewrite the Hamiltonian as: ##H=\frac{1}{8}mR^2(3L_1^2+3L_2^2+L^2+S^2-\frac{3}{2}\hbar^2)## this looks to me very strange since the Hamiltonian for two spinless particles on the surface of a sphere is ##H=\frac{L_1^2+L_2^2}{2mR^2}## so how can this be the Hamiltonian of two particles on the surface of a sphere?

And how can I find the eigenvalues of this Hamiltonian? For the resolution I thought I can separate the Hamiltonian into four parts: ##H_1=\frac{3}{8}mR^2L_1^2##, ##H_2=\frac{3}{8}mR^2L_2^2##, ##H_3=\frac{3}{8}mR^2L^2##, ##H_4=\frac{1}{8}mr^2S^2-\frac{3}{16}\hbar^2mR^2## but still I don't know how to go on.
 
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  • #2
Salmone said:
this looks to me very strange since the Hamiltonian for two spinless particles on the surface of a sphere is ##H=\frac{L_1^2+L_2^2}{2mR^2}## so how can this be the Hamiltonian of two particles on the surface of a sphere?
This Hamiltonian doesn't include any interaction between the particles. The other one clearly does.

Salmone said:
And how can I find the eigenvalues of this Hamiltonian? For the resolution I thought I can separate the Hamiltonian into four parts: ##H_1=\frac{3}{8}mR^2L_1^2##, ##H_2=\frac{3}{8}mR^2L_2^2##, ##H_3=\frac{3}{8}mR^2L^2##, ##H_4=\frac{1}{8}mr^2S^2-\frac{3}{16}\hbar^2mR^2## but still I don't know how to go on.
There are an infinite number of states, as ##l_1## and ##l_2## are unbounded. You can separate spin from orbital angular momentum, resulting in the usual singlet and triplet states. Then you can find orbital eigenstates for each value of ##L## starting at 0. Make sure that you only consider orbital + spin combinations that satisfy the Pauli principle.
 
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FAQ: Strange Hamiltonian of two particles on the surface of a sphere

1. What is a Hamiltonian in physics?

A Hamiltonian is a mathematical operator used in classical mechanics to describe the total energy of a physical system. It takes into account the kinetic and potential energies of the system and can be used to predict the future behavior of the system.

2. How is the Hamiltonian of two particles on a sphere different from other systems?

The Hamiltonian of two particles on a sphere is unique because it takes into account the curvature of the surface. This means that the particles are subject to a different gravitational force compared to particles on a flat surface, resulting in different equations of motion.

3. How is the Hamiltonian of two particles on a sphere calculated?

The Hamiltonian of two particles on a sphere is calculated by taking into account the kinetic and potential energies of the particles, as well as the distance between them and the curvature of the sphere. This can be done using mathematical equations and principles from classical mechanics.

4. What are some real-world applications of the Hamiltonian of two particles on a sphere?

The Hamiltonian of two particles on a sphere can be used to model the behavior of particles on curved surfaces, such as the Earth's surface. This can be useful in fields such as geophysics and astronomy, where understanding the motion of particles on curved surfaces is important.

5. Are there any limitations to using the Hamiltonian of two particles on a sphere?

Like any mathematical model, the Hamiltonian of two particles on a sphere has its limitations. It assumes that the particles are point masses and does not take into account other factors such as air resistance or external forces. It is also based on classical mechanics and may not accurately describe the behavior of particles at the quantum level.

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