A beam of electrons in an eigenstate of Sz with eigenvalue ½ħ is fed into a Stern-Gerlach apparatus, which measures the component of spin along an axis at an angle θ to the z-axis and separates the particles into distinct beams according to the value of this component. Find the ratio of the intensities of the emerging beams.Problem 2:
A quantum system can exist in two states |ψ1>
and |ψ2>, which are eigenstates of the
Hamiltonian with eigenvalues E1 and E2.
An observable A has eigenvalues ±1 and eigenstates |ψ±>= (1/√2)(|ψ1> ± |ψ2>).
This observable is measured at times t = 0, T, 2T, ... . The normalized state of the system at t = 0, just before the first measurement, is c1|ψ1>+ c2|ψ2>. If pn denotes the probability that the measurement at t = nT gives the result A = 1, show that pn+1 = ½(1 - cosα) + pn cosα, where
α/T = (E1 - E2)/ħ
and deduce that pn = ½ (1 - cosnα) + ½|c1 + c2|2 cosnα.
What happens in the limit as n --> ∞ with nT = t fixed?
Consider a spin ½ particle with magnetic moment m = γS. Let
|+> and |-> denote the eigenvectors of Sz and let the state of the
system at t = 0 be |ψ(0)> = |+>.
(a) At t = 0 we measure Sy and find +½ħ. What is the state vector |ψ(0)> immediately after the measurement?
(b) Immediately after this measurement we apply a uniform, time-dependent field parallel to the z-axis.
The Hamiltonian operator becomes H(t) = ω0(t)Sz.
Assume ω0(t) = 0 for t < 0 and for t > T, and increases linearly from 0 to ω0 when 0 < t < T. Show that at time t the state vector can be written as
|ψ(t)> = 2-½[exp(iθ(t))|+> + iexp(-iθ(t))|->]
and calculate the real function θ(t).
(c) At time t = τ > T, we measure Sy. What results can we find and with what probability?
Determine the relation that must exist between ω0 and T in order for us to be sure of the result. Give a physical interpretation.