(a) State how for spin-half particles, the spin (σ) can be expressed by its three components σ_x, σ_y and σ_z. 20 marks (b) By…

(a) Spin-half particles and Pauli matrices

For a spin-half particle the intrinsic spin is represented by the dimensionless Pauli vector σ = σₓ î + σ_y ĵ + σ_z k̂, where σₓ = [[0,1],[1,0]], σ_y = [[0,-i],[i,0]], σ_z = [[1,0],[0,-1]]. In the standard spinor basis, σ_z is diagonal, σₓ changes |↑> into |↓> and vice versa, and σ_y does the same with a relative phase i. The physical spin operator is S = (ℏ/2)σ, so Sᵢ=(ℏ/2)σᵢ. Each σᵢ has eigenvalues ±1, hence Sᵢ eigenvalues ±ℏ/2. The components satisfy [σᵢ,σⱼ]=2iε_ijkσₖ, σᵢ,σⱼ=2δ_ij I, and σᵢ²=I; equivalently σᵢσⱼ=δ_ij I+iε_ijkσₖ. Since they do not commute, only one component can be sharply defined at a time. Therefore [Sᵢ,Sⱼ]=iℏ ε_ijk Sₖ. For any unit direction n, n·σ = nₓσₓ+n_yσ_y+n_zσ_z is also Hermitian with eigenvalues ±1, so the three matrices encode all possible spin projections. The total spin magnitude is S²=Sₓ²+S_y²+S_z²=s(s+1)ℏ²=3ℏ²/4 for s=1/2. This is the spin-1/2 representation of SU(2), the double cover of the rotation group, and it is the complete statement of how spin is expressed through its three components.

(b) Hydrogen ground state and zero-point energy

For an electron of reduced mass μ in the Coulomb field of a proton, the time-independent Schrödinger equation is [-ℏ²/(2μ)∇² - e²/(4πε₀r)]ψ=Eψ. Separation in spherical coordinates gives angular spherical harmonics and a radial equation. For the ground state n=1, l=0, Y00 is constant and the radial equation reduces to -ℏ²/(2μ)(1/r² d/dr(r² dR/dr)) - e²/(4πε₀r)R=ER. Its solution is R(r)∝e^-r/a₀, with a₀=4πε₀ℏ²/(μe²); the normalized wavefunction is ψ100=(πa₀³)^-1/2e^-r/a₀. Substitution gives E₁=-μe⁴/(32π²ε₀²ℏ²)=-13.6 eV. Using μ≈mₑ gives the familiar 13.6 eV; the reduced-mass correction is small. This total ground-state energy is the zero-point energy of the hydrogen atom; its magnitude, 13.6 eV, is the binding or ionization energy. The expectation values obey the virial theorem: ⟨T⟩=+13.6 eV and ⟨V⟩=-27.2 eV, so ⟨T⟩+⟨V⟩=-13.6 eV. The positive kinetic term reflects confinement expressed by ΔrΔp≥ℏ/2; minimizing T~ℏ²/(2μa₀²) and V~-e²/(4πε₀a₀) gives the same a₀ and energy, showing the ground state is a balance of kinetic and potential terms. The zero-point energy is here taken as the total ground-state energy, −13.6 eV, the lowest energy the atom can have. Some texts instead call the kinetic energy that survives in the lowest state, ⟨T⟩ = +13.6 eV, the zero-point energy; either way, the electron cannot sit at rest at the nucleus, because confinement costs kinetic energy.

(c) Fluorescence and phosphorescence

In a Jablonski diagram, absorption promotes a molecule from the singlet ground state S₀ to an excited singlet S₁. The diagram thus contains both radiative transitions, S₁→S₀ and T₁→S₀, and non-radiative transitions, vibrational relaxation, internal conversion, and ISC. Fluorescence is prompt radiative decay S₁→S₀ with no change in spin multiplicity; it is spin-allowed, has lifetimes of about 10⁻⁹–10⁻⁷ s, and follows vibrational relaxation and internal conversion. The emitted fluorescence is usually at longer wavelength than the absorbed light because of this relaxation, producing a Stokes shift. Phosphorescence begins with intersystem crossing (ISC), a non-radiative S₁→T₁ transition mediated by spin-orbit coupling, followed by delayed radiative decay T₁→S₀. Because this is a singlet-triplet, spin-forbidden transition, phosphorescence has much longer lifetimes, typically 10⁻³–10³ s, and can persist after the excitation source is removed; the forbidden transition is made weakly allowed by spin-orbit coupling, especially in heavier atoms. Fluorescence is therefore useful for real-time imaging, while phosphorescence is useful where afterglow or long-lived emission is needed.

Applications follow these mechanisms. In biochemistry, fluorescence is used in green fluorescent protein (GFP) tagging, flow cytometry, DNA sequencing, and live-cell fluorescence microscopy. In materials science, phosphorescence is exploited in organic light-emitting diodes (OLEDs), persistent-luminescent (glow-in-the-dark) materials, safety signage, and security inks. Fluorescence resonance energy transfer (FRET) measures molecular distances of 1–10 nm, and phosphorescent iridium and platinum complexes allow OLEDs to harvest triplet excitons, raising internal quantum efficiency towards 100%.