The time-independent Schrödinger equation

Start from iħ ∂Ψ/∂t = −(ħ²/2m) ∂²Ψ/∂x² + U(x)Ψ, with a potential that depends on x alone. Try a product Ψ = ψ(x)φ(t), substitute, and divide by ψφ: iħ (dφ/dt)/φ = [−(ħ²/2m)ψ″ + Uψ]/ψ. The left side depends on t only and the right on x only, yet they agree for every x and t, so both equal one constant, E. The time equation integrates at once to φ = e^(−iEt/ħ). The space equation is −(ħ²/2m)ψ″ + Uψ = Eψ. Written Ĥψ = Eψ, with Ĥ = p̂²/2m + U and p̂ = −iħ d/dx, it says the Hamiltonian acting on ψ returns ψ scaled by a number: an eigenvalue equation. A clamped string has the same structure, T y″ = −μω²y, with acceptable solutions only at its harmonics. The constant E is the energy: in such a state ⟨Ĥ⟩ = E and the spread of energy is zero.

The factor e^(−iEt/ħ) is a unit phasor turning at E/ħ; for E = 1.00 eV that is 1.52 × 10¹⁵ rad s⁻¹. Its modulus is 1 at every instant, so |Ψ(x, t)|² = |ψ(x)|²: the probability density, and every expectation value of an operator with no explicit t, is constant. That is all “stationary” means. The phase alone is unobservable. Add a constant U_c to the potential and every E shifts by U_c, every Ψ gains a factor e^{(−iU_c t/ħ)}, and no measurement changes. Differences are observable. For Ψ = c₁ψ₁e^{(−iE₁t/ħ)} + c₂ψ₂e^{(−iE₂t/ħ)} with real ψ₁ and ψ₂, |Ψ|² contains the cross term 2Re[c₁*c₂ e^{(−i(E₂ − E₁)t/ħ)}]ψ₁ψ₂, which oscillates at ω₂₁ = (E₂ − E₁)/ħ. A density that moves needs at least two energies, and a spectral line measures exactly such a difference.

Rearranged, the equation reads ψ″ = (2m/ħ²)(U − E)ψ, and that is enough to sketch ψ before solving anything. Where E > U, the classically allowed region, ψ″ has the opposite sign to ψ: the curve bends back toward the axis and oscillates, locally like sin kx with k = √(2m(E − U))/ħ. That is de Broglie’s λ = h/p with the classical momentum √(2mK). Where E < U, the forbidden region, ψ″ has the same sign as ψ: the curve bends away from the axis, and the solutions are e^(±κx) with κ = √(2m(U − E))/ħ. At a turning point, E = U and ψ has an inflection. For an electron, 1.00 eV of kinetic energy gives k = 5.12 nm⁻¹ and λ = 1.23 nm; a 1.00 eV shortfall gives a decay length 1/κ = 0.195 nm. Where U varies slowly, the wavelength is short where a classical particle would be fast and the amplitude large where it would be slow: the classical dwell-time density, recovered.

ψ is continuous everywhere: a jump would put a delta spike in ψ′ and its derivative in ψ″, which no finite Uψ or Eψ can balance. For ψ′, integrate the equation across a point a, from a − ε to a + ε: ψ′(a + ε) − ψ′(a − ε) = (2m/ħ²)∫(U − E)ψ dx. If U is finite at a, even where it steps, the integrand is bounded and the integral vanishes as ε → 0, so ψ′ is continuous too. Two potentials evade the argument: at an infinite wall ψ must vanish and ψ′ may jump, and a spike S δ(x − a) leaves a finite integral, so ψ′ jumps by (2mS/ħ²)ψ(a). In practice, match ψ and ψ′/ψ at each finite step; the log-derivative is blind to normalisation. An electron with E = 1.00 eV meeting a step up to 3.00 eV has A cos(kx + φ) before it and B e^(−κx) after; matching −k tan φ = −κ gives tan φ = κ/k = √2, so φ = 54.7° and B = A cos φ = 0.577A.

At any E the equation has two independent solutions; nothing in it picks an energy. Count conditions instead. Left of a well, where E < U, the solution is Ae^(κx) + Be^(−κx), and normalisability as x → −∞ forces B = 0. Carry that one solution across the well with the matching rules, which fix it up to scale. It reaches the far side as Ce^(κx) + De^(−κx), with C generically not zero, so ψ blows up. Demanding C(E) = 0 is one equation for one unknown, with discrete roots: for an electron in a well 1.00 nm wide and 3.00 eV deep they are 0.249, 0.978 and 2.107 eV. Sweep E through a root and the divergence flips sign as ψ gains a node, so the nth level has n − 1 nodes. Normalisation comes afterwards and fixes only the scale. If E exceeds U on both sides, neither solution blows up, nothing is left to impose, and the energies form a continuum of scattering states.

The kinetic term is a low-speed limit: √(p²c² + m²c⁴) − mc² = p²/2m − p⁴/(8m³c²) + …, so p²/2m is off by about K/(2mc²). For a 3 eV electron in a well that is 3 × 10⁻⁶; for a 50 keV electron in a microscope it is 5%, and the equation itself must change. The rest energy is not lost: it multiplies every Ψ by the global phase e^{(−imc²t/ħ)}, which drops out of |Ψ|² just as a constant added to U does. The equation is also for one particle, with every other particle frozen into U. In hydrogen that means a fixed nucleus, and restoring the reduced mass shifts every level by 1 part in 1837. And the potential must be fixed: a U that depends on t, such as a laser field or a moving wall, spoils the separation, and the particle is driven between stationary states instead of sitting in one. Spin and magnetic fields lie outside the equation as well.