I Strange Hamiltonian of two particles on the surface of a sphere

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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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.
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
If we release an electron around a positively charged sphere, the initial state of electron is a linear combination of Hydrogen-like states. According to quantum mechanics, evolution of time would not change this initial state because the potential is time independent. However, classically we expect the electron to collide with the sphere. So, it seems that the quantum and classics predict different behaviours!
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