Two spin-1/2 particles with spin operators S and s interact via the Hamiltonian H = X...
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The Hamiltonian for the interaction between two spin12 particles can be written as H A S2 S1 where A is a constant and S1 and S2 are the spin operators for the two particles The energy levels of this system can be found by solving the Schrdinger equation H E where is the wave function of the system and E is the energy eigenvalue To solve this equation we can use the fact that the spin operators satisfy the following commutation relation S1 S2 i where is the Planck constant divided by 2 Using this commutation relation we can rewrite the Hamiltonian as H A S2 S1 S1 S2 This expression can be simplified further by noticing that the spin operators can be represented as matrices with the basis vectors being the eigenstates of the spin operators Specifically we can choose the basis vectors to be the states and which correspond to the spin being aligned upward and downward respectively In this basis the spin operators can be represented as follows S1 S2 Using these expressions we can write the Hamiltonian as H A Simplifying this expression we get H A This Hamiltonian describes a system with two coupled spins where the energy level depends on the alignment of the spins The energy eigenvalues can be obtained by solving the eigenvalue equation H E The eigenvectors of this equation are superpositions of the basis states and and the energy eigenvalues are determined by the coefficients of these basis states To find the energy levels we can first note that the Hamiltonian commutes with the total spin operator ... View the full answer
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