hbar2

hbar2 is a common notation for the square of the reduced Planck constant, written as ħ^2. The reduced Planck constant ħ is defined as h/2π, so hbar2 = ħ^2 = h^2/(4π^2). In quantum mechanics, ħ appears in fundamental operator definitions and equations, making hbar2 a frequently used factor in kinetic and dynamical expressions.

In operator form, the momentum operator is p = -iħ∇, which leads to p^2 = -ħ^2∇^2. The kinetic energy operator is therefore T = p^2/(2m) = -(ħ^2/2m)∇^2. This shows how hbar2 enters the Schrödinger equation, iħ ∂ψ/∂t = [-(ħ^2/2m)∇^2 + V]ψ, where ħ^2 multiplies the Laplacian to give the correct quantum kinetic contribution and energy units.

Units and scale: ħ has units of action (J·s), so ħ^2 has units of J^2·s^2. When combined with spatial derivatives, hbar2 yields quantities with energy units. In many contexts, ħ^2 also appears in radial or momentum-space equations and in dimensionless forms. In natural units where ħ is set to 1, hbar2 effectively takes the value 1, simplifying expressions.

Numerical value: ħ ≈ 1.0545718×10^-34 J·s; hence hbar2 ≈ 1.112×10^-68 J^2·s^2. The term hbar2 is thus a standard factor in quantum-mechanical formulas, reflecting the fundamental scale set by the reduced Planck constant.