Question

A spin-particle is fixed in space with the Hamiltonian H = as, + b($+$3), where a and b are constants and as usual, Sx, Sy, S, are the operators which gives the x-, y-, z-components of the total spin. a) Write the matrix representation of Hamiltonian, H. [6 marks] b) Determine the energy levels of this system. [2 marks] c) List all possible energy levels of this system [2 marks] (Show proper construction of the matrix representation and detail out all calculation involved)

pls solve asap .. thanks

Answer 1

O i H= la to 26 o 2b O atht 26t² 4 2 O -at + 26 4 at + bth n= - att bt? 2 It is required hamiltonian. (6) Now, energy level can be found by eigenvalues of the Hamiltonian, ludI lzo

So, the energy levels are 5,- (64-a) Ez = h (bhra All the possible energy (0) levels are Ei = h (bh-a Ez = h (6hta All possible levels can be found by substituting different different values of a and b.
Here we first wrote the Hamiltonian in the matrix form by putting the value of the spin matrix for spin half particle. Then we found the eigenvalue of the Hamiltonian to find the different energy levels.

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