(a) What is the spin wave function (for s = ½) if the spin component in the direction of unit vector η has a value of ½ℏ ? (15…

(a) For a spin-1/2 particle, let η=(sinθ cosφ, sinθ sinφ, cosθ). The spin component along η is S·η=(ℏ/2)(σₓ sinθ cosφ+σ_y sinθ sinφ+σ_z cosθ). The required state is the normalized eigenstate satisfying S·ηχ=(+ℏ/2)χ. Writing χ=(a,b)^T and using σ·η = [[cosθ, sinθ e^-iφ],[sinθ eⁱφ, -cosθ]], the eigenvalue equations are (cosθ−1)a+sinθ e^-iφb=0 and sinθ eⁱφa−(cosθ+1)b=0. Choosing a=cos(θ/2) and b=eⁱφsin(θ/2) satisfies them because cosθ cos(θ/2)+sinθ sin(θ/2)=cos(θ/2) and sinθ cos(θ/2)−cosθ sin(θ/2)=sin(θ/2). Hence χ₊(η)=cos(θ/2)χ₊ + eⁱφsin(θ/2)χ₋. The phase eⁱφ fixes the azimuthal direction, while the overall phase is arbitrary; the state reduces to χ₊ for θ=0 and to eⁱφχ₋ for θ=π. This is the required spin wave function for the +1/2ℏ component along η.

(b)(i) The Stern–Gerlach experiment is important because it gave the first direct beam-splitting evidence for space quantization of angular momentum. In 1922 the silver beam split into discrete components, showing that a magnetic moment cannot orient continuously in a field. This turned an abstract quantum postulate into a visible spatial separation. After the 1925 spin hypothesis, the same two-line result became a standard demonstration of the two projections of electron spin, and it now underlies quantum measurement theory and qubit realizations in quantum information.

(b)(ii) Schematic: Ag oven → collimating slits S1 and S2 → inhomogeneous magnet → detection screen; the beam travels along z and two spots appear at ±Δz on the screen. The slits define a narrow beam and select atoms with similar velocities. In the magnet, B_z(z) is strong and varies with z; atoms with different μ_z receive forces F_z=μ_z∂B_z/∂z and are deflected by different amounts. The pole pieces are not flat parallel plates; one is shaped as a knife-edge or wedge and the other as a broad curved/cylindrical surface. This geometry makes ∂B_z/∂z large and nearly linear over the beam while keeping the field direction approximately along z, so the force separates the beam instead of merely precessing the spin. A uniform field would produce no net transverse force.

(b)(iii) Silver was used because its ground state is 5s¹, ^2S₁/2: L=0, S=1/2, J=1/2. With L=0 there is no orbital angular momentum contribution to the magnetic moment, and J=1/2 gives only two Zeeman components, m_J=±1/2, so the beam splits into two clean spots. Silver also forms an intense neutral atomic beam when heated and is chemically stable. Natural silver isotopes have I=1/2, so a small hyperfine structure exists; it is small compared with the spin Zeeman splitting and was not resolved in the original experiment, so it does not obscure the main two-line separation.

(c) The Franck–Condon principle states that an electronic transition is much faster than nuclear motion, so the internuclear distance remains essentially fixed during the transition; on a potential-energy diagram the transition is vertical. The vertical transition picture is the causal link between nuclear geometry and band intensity. For a diatomic molecule, if the electronic transition moment is slowly varying over vibrational levels, the intensity of a band from v'' in the lower state to v' in the upper state is governed by the Franck–Condon factor |⟨ψᵥ'|ψᵥ''⟩|², the overlap of the two vibrational wavefunctions at the same nuclear coordinate. When the upper and lower potential curves have different equilibrium positions, vertical transitions from a given v'' intersect the upper curve at different vibrational levels. The largest overlap, and hence the strongest band, occurs where the vertical line crosses a region of high vibrational probability. As v' changes, the overlap varies smoothly, giving a progression of bands whose intensities follow a Condon parabola; for a large shift of the upper curve the maximum moves to higher v'.

The principle therefore explains why vibrational-electronic spectra show a sequence of bands with non-uniform intensities rather than a single line.