(a) A particle constrained to move along x-axis in the domain 0 ≤ x ≤ L has a wave function ψ(x) = sin(nπx/L), where n is an…

(a)(i) Normalization: Let ψ(x) = A sin(nπx/L). The particle is confined to 0 ≤ x ≤ L, so ψ(0) = ψ(L) = 0, which holds for integer n. The normalization condition is 1 = ∫₀^L |ψ(x)|² dx = |A|² ∫₀^L sin²(nπx/L) dx. Using sin²θ = (1 − cos2θ)/2, ∫₀^L sin²(nπx/L) dx = L/2, for any nonzero integer n. Thus |A|² L/2 = 1, so A = √(2/L). Hence ψ(x) = √(2/L) sin(nπx/L), n = ±1, ±2, … (If n = 0, ψ = 0 and cannot be normalized.)

(a)(ii) Expectation value of momentum: The momentum operator is pₓ = −iħ d/dx. dψ/dx = √(2/L)(nπ/L) cos(nπx/L). Therefore ⟨pₓ⟩ = ∫₀^L ψ* pₓ ψ dx = −iħ(2/L)(nπ/L) ∫₀^L sin(nπx/L) cos(nπx/L) dx. Using sinθ cosθ = (1/2) sin2θ, ∫₀^L sin(nπx/L) cos(nπx/L) dx = (1/2) ∫₀^L sin(2nπx/L) dx = (1/2)[−L/(2nπ) cos(2nπx/L)]₀^L = 0, since cos(2nπ) = cos0 = 1. Thus ⟨pₓ⟩ = 0. Condition: n is a nonzero integer; the wave function is real, so the two opposite momentum components cancel.

(b)(i) Most probable distance in hydrogen 2p state: For hydrogen, the 2p radial wave function is R₂₁(r) = [1/(2√6 a₀√a₀)] (r/a₀) exp(−r/(2a₀)), where a₀ is the Bohr radius. The radial probability density is P(r) = r² |R₂₁(r)|² = r² [1/(24 a₀³)] (r²/a₀²) exp(−r/a₀) = r⁴/(24 a₀⁵) exp(−r/a₀). For the most probable distance, set dP/dr = 0: dP/dr = [1/(24 a₀⁵)] [4r³ exp(−r/a₀) − (r⁴/a₀) exp(−r/a₀)] = [r³ exp(−r/a₀)/(24 a₀⁵)] (4 − r/a₀). Thus r = 0 or r = 4a₀. Since r = 0 and infinity give P = 0, the maximum occurs at r_max = 4a₀.

(b)(ii) Radial probability density at that distance: At r = 4a₀, P(4a₀) = (4a₀)⁴/(24 a₀⁵) e⁻⁴ = 256/(24 a₀) e⁻⁴ = 32/(3 a₀ e⁴). Numerically, P(4a₀) = 0.1954/a₀. If by “radial density” the volume density |R₂₁|² is meant, then |R₂₁(4a₀)|² = 2e⁻⁴/(3a₀³). This result is independent of the magnetic quantum number m.

(c)(i) What is nuclear magnetic resonance? Nuclear magnetic resonance (NMR) is the resonant absorption (or emission during relaxation) of radiofrequency radiation by atomic nuclei that possess nonzero spin and hence a magnetic moment, when placed in a static magnetic field B₀. Examples include ¹H, ¹³C, and ³¹P. In the absence of B₀, the nuclear spin states are degenerate; the static field lifts this degeneracy.

(c)(ii) Working principle: For a spin-1/2 nucleus, the magnetic moment is μ = γ S, where γ is the gyromagnetic ratio. In a static field B₀ along z, the energy is E = −μ·B = −γ S_z B₀. The eigenvalues of S_z are ±ħ/2, so the two energy levels are E_± = ∓(1/2)γħB₀. The energy splitting is ΔE = γħB₀. Resonance occurs when the photon energy matches this splitting: hν = γħB₀, so ν = γB₀/(2π). The corresponding angular Larmor frequency is ω₀ = γB₀. In thermal equilibrium, the population difference between the two levels produces a net magnetization M aligned along B₀. The nuclear moments precess about B₀ at ω₀. A radiofrequency pulse at the Larmor frequency tips M away from B₀; after the pulse, the transverse component of M precesses and induces an oscillating voltage in a receiver coil. This is the free induction decay (FID) signal. The return to equilibrium is described by the longitudinal relaxation time T₁ and the transverse relaxation time T₂. The local electronic environment shifts the effective field, giving chemical shift and J-coupling, which are the basis of NMR spectroscopy. Conditions: the nucleus must have nonzero spin, and a static magnetic field must be present.

(c)(iii) Use in magnetic resonance imaging: MRI applies NMR to protons in water and lipids in tissue. The patient is placed in a strong static field B₀. Gradient coils add a controlled position-dependent field, so that the local Larmor frequency is ω(r) = γ[B₀ + G·r]. A narrow-band RF pulse then excites only a selected slice where the resonance condition is met. Within the slice, phase-encoding and frequency-encoding gradients give the signal a spatial dependence. The received MR signal is a sum of contributions from different positions; Fourier transformation of the encoded data reconstructs the image. Contrast is governed by proton density, T₁, and T₂. By choosing different repetition times TR and echo times TE, one obtains T₁-weighted (TR short, TE short), T₂-weighted (TR long, TE long), or proton-density (TR long, TE short) images. MRI is non-ionizing, provides excellent soft-tissue contrast, and is widely used for imaging the brain, spine, joints, tumors, and for functional brain imaging.