A single spin ½ particle

Problem:

Explain why, in a Stern-Gerlach (SG) apparatus, a beam consisting of neutral particles in different spin states is split into different beams.

Solution:

Problem:

An electron is in a state with z-component of spin angular momentum ħ/2.  An observation designed to measure the component of spin angular momentum along an arbitrary direction n is made.  What is the probability of observing a component of spin angular momentum ħ/2 along n?

Solution:

Problem:

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.

Solution:

Problem:

An electron is in the spin state  |χ> = A(3i|+> + 4|->) = Aimage.
(a)  Determine the normalization constant A.
(b)  Find the expectation values <Sx>, <Sy>, and <Sz>.
(c)  Find the root-mean-square deviations ΔSx, ΔSy, and ΔSz.

Solution:

Problem:

Two states of a spin ½ particle are represented in the eigenbasis of Sz by
1> = (1/√2)(|+> + i|->),  |ψ2> = (1/√3)(-i|+> + √2|->).
(a)  Find their representation in the eigenbasis of Sy.
(b)  Find the amplitude <ψ12> in the Sz basis and show that this amplitude remains unchanged when calculated in the Sy basis.  (Show your work.)
(c)  The Hamiltonian for the particle is H = ω0Sz.  Find |ψ1(t)>.  At what times t is |ψ1(t)> an eigenvector of Sx?

Solution:

Problem:

At t = 0 the x-component of the spin of a spin ½ particle is measured and found to be ħ/2.  At t = 0 the particle is therefore in the |+>x eigenstate of the Sx operator.  The particle is confined to a region with a uniform magnetic field B = B0k, its Hamiltonian is H = ω0Sz.  The eigenstates of H are |+> and |->,
H|+> = (ħω0/2)|+> and H|-> = -(ħω0/2)|->.
|+>x can be written as a linear combination of eigenstates of H.
(a)  Find the probability of measuring Sx = ħ/2 at t = T.
(b)  What is the mean value of Sx, <Sx>, at t = T?
(c)  Find the probability of measuring Sz = ħ/2 at t = T.

Solution:

Problem:

An elementary spin-½ particle with magnetic moment µB is in it's lower level state in a magnetic field B parallel to z-axis.  At time t = 0 the magnetic field B is flipped to point parallel to x-axis.
(a)  Find the time-dependent spin wave function of the particle for t > 0.
(b)  Find the rotation frequency for the magnetic moment of the particle.

Solution:

Problem:

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[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.

Solution: