A. In class we found the eigenstate for spin along an arbitrary direction n-sinBcospa + s1ηθsinqy...
The spin operator along an arbitrary direction = (sin 0 cos 0) is defined as sin 0 sin o cos (1.18) h where S 2 _ (a) Find the eigenstates of the above spin operator and show that their eigenvalues are + 2 h h would be measured to have S (b) Find the probability that the state S =+ 2 where n' 2 would be measured to have S 2 (c) Find the probability that the state S =...
2- If the z-component of an electron spin is +h/2, what is the probability that its component along a direction z', that forms an angle θ with the z-axis, equals +h/2 or-h/2? What is the average value of the spin along z'? (Hint. Sz.-S. n where n; sin θ cospi + sin θ sin φ j + cos θ k is a unit vector along z'.) (10 Scores)
2- If the z-component of an electron spin is +h/2, what is...
1. In this problem, we are going to look at a three-level system. A spin-1 particld is placed in a constant magnetic field along the a-direction with strength B,. The spin-1 particle İs initialized in a z-eigenstate with positive eigenvalue h, ie, the i 1,m 1) state. What is the probability to find the negative eigenvalue the spin along the z axis as a function of time? Assume that the spin-1 particle has inagnetic moment 2 × μιι, i.e. that...
Problem 1: Greens Functions In class on Wednesday we found that we could solve the position, r(t), of a mass and spring when an arbitrary force is applied using Green's Functions, r(t)= | G(t,T)f(r)dr and that the particular Green's Function for this case is Gu.T)--sin ) wo where w, h/m and Θ(p) is a step function at p = 0. A: Find the position of the mass as a function of time if a force fo is applied for a...
A spin-3/2 system can have four values of angular momentum (Sz) when measured along the z- 2 2 If we measure the angular momentum of 1000 atoms, each prepared in the following state lp〉, how many times should we expect to measure an angular momentum of a) ? 2 Here 3 h the state of the system with an angular momentum of - etc.
2. [10pts) In considering the EPR experiment, we used the anti-correlation property of the spin-0 state as an essential aspect of the analysis. In this question you are going to show that this property does not depend on the choice of direction of axis of the Stern-Gerlach analyzers. Show that the entangled state: |4= (1 131] 1)2 – 1 1/)2) is physically equivalent to the general state: 4) = - (11)in1)2n - | 1)in 1)2n) where ñ is a unit...
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 ,...
The behavior of a spin-
particle in a uniform magnetic field in the z-direction,
, with the Hamiltonian
You found that the expectation value of the spin vector
undergoes Larmor precession about the z axis. In this sense, we can
view it as an analogue to a rotating coin, choosing the
eigenstate with eigenvalue
to represent heads and the eigenstate with eigenvalue
to represent tails. Under time-evolution in the magnetic field,
these eigenstates will “rotate” between each other.
(a) Suppose...
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...