1. More on Spin-1/2 system: (10 points) The rising and lowering operators for a spin-1/2 system...
2. Spin-1/2 system: (20 points) The Pauli matrices are, 0 -1 from which we can define the spin matrices, s.-슬&z, Šv = , S.-출.. We'll use the eigenkets of S that, for the spin half system, they can be represented by the spinors, a) Show, by matrix multiplication that |+) and |-) are eigenstates of the S operator and determine the eigenvalues. Show that they are not eigenstates of S and Sy b) Show that the matrix squares s ,...
A spin-1 particle interacts with an external magnetic field B = B. The interaction Hamiltonian for the system is H = gB-S, where S-Si + Sỳ + SE is the spin operator. (Ignore all degrees of freedom other than spin.) (a) Find the spin matrices in the basis of the S. S eigenstates, |s, m)) . (Hint: Use the ladder operators, S -S, iS, and S_-S-iS,, and show first that s_ | 1,0-ћ /2 | 1.-1)) . Then use these...
4. 10 points The Spin operators for a spin-1/2 particle can be described by the Pauli matrices: 0 1 0 0 ,02= 0 -1 1 ¿ a) Write the normalized eigenvectors of Oz, I+) and 1-) which are defined such that 0z|+) = 1+) and 0z1-) = -1-), as column vectors in the same basis as the Pauli matrices given above. (You can assume without loss of generality that these eigenvectors are real.) (3 pts) b) Consider an eigenvector (V)...
1. (30 points). Coupled spins. Spin-1/2 particles A and B evolve under the influence of the following Hamiltonian (for simplicity takeh-1 so that energies are expressed in frequency units): We work in the uncoupled basis aba) Ib), where a,b E 0,1 and where states 0) (1)) correspond to single spins aligned (antialigned) with the z-direction. As we discussed in lecture, the eigenstates of the Hamiltonian are 100), 111), and 2-1/2 (101) 110)). a. We prepare the initial state t01). Since...
2. (25 points). Rabi oscillations. Consider a spin-1/2 particle in a magnetic field B - Bo2 such that the spin eigenstates are split in energy by hwo (let's label the ground state |0) and the excited state |1)). The Hamiltonian for the system is written as hwo Zeeman - _ here and below. ơng,z are the usual Pauli matrices. A second, oscillating field is applied in the transverse plane, giving rise to a time-dependent term in the Hamiltoniain hw Rabi-...