Q4(b). Schrödinger wave equation in three dimensions; meaning of ψ and ψ²

Setting up the equation

1. A standing wave of amplitude ψ(x,y,z) and wavelength λ obeys the classical wave equation: ∇²ψ+4π²λ²ψ=0, where ∇²=∂²∂ x²+∂²∂ y²+∂²∂ z²
2. The particle has a de Broglie wavelength λ=h/mv, so 1λ²=m²v²h².
3. Its kinetic energy is total minus potential energy: 12mv²=E-V, so m²v²=2m(E-V).
4. Substituting: ∇²ψ+8π²mh²(E-V)ψ=0 or [-ħ²2m∇²+V]ψ=Eψ, i.e. Hψ=Eψ.

Nature of the equation and its terms

1. It is a second-order, linear, homogeneous partial differential equation in x, y, z; it is time-independent (stationary states). Any multiple or sum of solutions of equal E is also a solution.
2. It is an eigenvalue equation: H (the Hamiltonian, energy operator) acting on ψ returns a number, E, times ψ. Acceptable solutions exist only for certain values of E: quantisation follows from the boundary conditions, not from an extra assumption.
3. Terms: m mass of the particle; E total energy (eigenvalue); V(x,y,z) potential energy (for the H atom -e²/4πε₀r); h Planck's constant; -ħ²2m∇² the kinetic-energy operator; ψ the wave function (eigenfunction).

Significance of ψ and ψ²

1. ψ is the amplitude of the matter wave and contains all that can be known about the state. It has no direct physical meaning: it can be negative (as beyond the node of the 2s orbital, panel) or complex. Its sign matters in overlap and bonding.
2. An acceptable ψ must be single-valued, continuous, finite and have a continuous first derivative; it must vanish at infinity so that it can be normalised.
3. Born interpretation: ψ² (strictly ψ*ψ=|ψ|²) is the probability density; ψ² dτ is the probability of finding the particle in the volume element dτ. It is never negative.
4. Normalisation: ∫ψ² dτ=1 over all space (the particle is somewhere). Where ψ=0 (a node) the particle is never found; regions of large ψ² define the shape of an orbital.

Application. the same equation with the Coulomb potential gives the hydrogen orbitals, the quantum numbers n, l, mₗ and the energies Eₙ=-13.6/n² eV; with V=0 inside a box it gives the particle-in-a-box levels used for conjugated dyes.