Step 1: Write the normalized ground-state wavefunction of a quantum harmonic oscillator of mass m and angular frequency ω. Step 2: The ground state is n = 0, and H_0(x) = 1. With α = √(mω/ℏ), the eigenfunction formula ψ_n = N_n·H_n(αx)·e^(−α²x²/2) gives ψ_0 = N_0·e^(−α²x²/2). Step 3: Normalize: ∫|ψ_0|²dx = 1 requires N_0 = (α/√π)^(1/2), i.e. N_0 = (mω/πℏ)^(1/4). Step 4: So ψ_0(x) = (mω/πℏ)^(1/4)·exp(−mωx²/2ℏ). Step 5: The probability density |ψ_0|² is a Gaussian centred at x = 0 — the particle is most likely found at the equilibrium position, with a spread Δx = √(ℏ/2mω). Result: The ground state is a pure Gaussian because H_0 = 1; the first excited state (n = 1) gets the factor H_1 = 2αx, giving a node at the origin.