Consider the state of a spin-1/2 particle 14) = v1o (31+z) + i] – z)) where...
qm 2019.3 3. The Hamiltonian corresponding to the magnetic interaction of a spin 1/2 particle with charge e and mass m in a magnetic field B is À eB B. Ŝ, m where Ŝ are the spin angular momentum operators. You should make use of expres- sions for the spin operators that are given at the end of the question. (i) Write down the energy eigenvalue equation for this particle in a field directed along the y axis, i.e. B...
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-...
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...
[3] A spin-1/2 particle is in the state IW) 1/311) +i2/3|). (a) A measurement is made of the x component of the spin. What is the probability that the spin will be in the +z direction? (b) Suppose a measurement is made of the spin in the z direction and it is found that the particle has m,#1/2. what is the state after the measurement? (c) Now a second measurement is made immediately after to determine the spin in the...
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...
(10 points) A spin-1/2 particle is originally in the ground state of the Hamiltonian Ho woS At time t - 0 the system is perturbed by Here and above s, are the spin matrices. Consider H, as a small perturbation of Ho i.e., ao > wi, Find the probability for the particle to flip its spin under the perturbation at t n oo.
2. Consider a spin-1/2 particle. The physical quantity K is represented by the operator (written in the S-basis): k=ko –2i) * +2i 0 where k is a real number with the appropriate dimensions. (a) What are the eigenvalues and normalized eigenstates of K? (b) What value(s) of K could you measure? (c) What state(s) could the particle be in immediately after you measured K? (d) For a single particle, could you simultaneously know both the z-component of spin and the...
1 2. Consider the normalized spin state To (31t) +i\L)) (2) 10 (a) Is this state lx) an eigenstate of $2 ? Is it an eigenstate of Ŝe ? (Justify your answers.) In each case, if it is an eigenstate, give the eigenvalue. (b) If the spin state is as given above, and a measurement is made of the 2-component of the angular momentum, what are the possible results of that measurement and what are probabilities of each possible result?...
Consider a two-dimensional state vector space and a basis in this space lay), laz), eigenvectors of an observable A: Ala) = aja) Alaz) = azlaz) A representation of Hamiltonian operator in this basis is: H = (8 5) Find: -Eigenstates and eigenvalues of H. -If the system is in state |az) at time t=0, What is the state vector of the system at time t? -What is the probability of finding the system in the state |az) at time t?...
(7) Consider a spin-1/2 particle that is in the eigenstate of S,. The sys- tem is rotated by an angle 0 = T/4 about z axis. What is. (a) State of the system after rotation?? (b) What are the expectation values of S2, S, and S̟ after rotation ??