The Schrödinger Equation

What the Schrödinger equation says

Quantum mechanics needs a rule that tells you how a system changes from one instant to the next. In Newtonian physics that rule is F = ma; in quantum mechanics it is the Schrödinger equation. Instead of tracking a particle's position and momentum, it tracks the wavefunction ψ — a complex-valued field that encodes every predictable property of the system. The equation is:

iℏ ∂ψ/∂t = Ĥψ

Reading it symbol by symbol:

• i = √(−1), the imaginary unit — the wavefunction is genuinely complex.
• ℏ (h-bar) = h / 2π = 1.054571817 × 10⁻³⁴ J·s, the reduced Planck constant.
• ∂ψ/∂t = the rate of change of the wavefunction in time.
• ψ(x, t) = the wavefunction, units of m^−3/2 in three dimensions so that |ψ|² integrates to a pure number.
• Ĥ = the Hamiltonian operator, representing the system's total energy (kinetic + potential).

The left side is "how fast the state is changing," the right side is "the energy operator acting on the state." Because the equation is first order in time, knowing ψ at one instant fixes ψ for all later times — the evolution is completely deterministic. Probability only enters when you measure, through Born's rule.

Why it matters

The Schrödinger equation is the mathematical engine behind essentially all of chemistry and much of modern technology:

• Atomic structure. Solving it for the Coulomb potential −ke²/r reproduces the hydrogen spectrum exactly — the 13.6 eV ionization energy, the Rydberg formula, orbital shapes (s, p, d) — without any ad-hoc rules.
• The periodic table. Combined with the Pauli exclusion principle, its solutions explain electron shells and hence why elements bond the way they do.
• Chemical bonding. Molecular orbitals, reaction barriers, and spectroscopy all come from solving Ĥψ = Eψ for many-electron systems.
• Semiconductors and lasers. Band structure, tunnelling diodes, and stimulated emission are Schrödinger-equation solutions in periodic or engineered potentials.
• Quantum tunnelling. The equation predicts nonzero ψ inside classically forbidden regions — the basis of the scanning tunnelling microscope, flash memory, and alpha decay.

It is fair to say the Schrödinger equation is to the microscopic world what Newton's laws are to the macroscopic one.

How it works, step by step

1. Build the Hamiltonian. Start from the classical energy E = p²/2m + V(x). Promote momentum and energy to operators — p̂ = −iℏ∇ and Ê = iℏ ∂/∂t — and you get the single-particle Hamiltonian:

Ĥ = −(ℏ²/2m)∇² + V(x)

The first term is kinetic energy (∇² is the Laplacian, the sum of second spatial derivatives), the second is potential energy.

2. Write the full (time-dependent) equation. Substituting Ĥ gives the wave equation:

iℏ ∂ψ/∂t = −(ℏ²/2m)∇²ψ + V(x)ψ

3. Separate variables for a static potential. If V does not depend on time, try ψ(x, t) = φ(x)·f(t). The equation splits into two, forcing f(t) = e^−iEt/ℏ and leaving the spatial part as the time-independent Schrödinger equation:

Ĥφ = Eφ    ⇔    −(ℏ²/2m)∇²φ + V(x)φ = Eφ

4. Apply boundary conditions. Demanding that φ be finite, single-valued, and normalizable selects only special discrete energies E for bound states. These are the energy eigenvalues; the corresponding φ are the stationary states.

5. Build the general solution. Any state is a superposition of eigenstates: ψ(x, t) = Σ_n c_n φ_n(x) e^−iE_nt/ℏ. Each term rotates in phase at its own frequency ω_n = E_n/ℏ — the whole of quantum dynamics is these phases beating against one another.

Stationary states and energy eigenvalues

A stationary state is a solution ψ_n(x, t) = φ_n(x) e^−iE_nt/ℏ of definite energy E_n. It is called "stationary" because its probability density |ψ_n|² = |φ_n|² is constant in time — the time dependence is a pure global phase that cancels when you square. Nothing observable about a single eigenstate changes; only superpositions of different energies produce time-varying probabilities (with beat frequency (E_m − E_n)/ℏ, which is exactly a spectral line).

The energies E_n are the eigenvalues of Ĥ. Because energy is a real, measurable quantity, Ĥ must be Hermitian, which guarantees three things: the eigenvalues are real, eigenstates of different energy are orthogonal, and together they form a complete basis. That completeness is why "expand in energy eigenstates" always works.

System	Potential V(x)	Energy eigenvalues En
Infinite square well (box), width L	0 inside, ∞ at walls	n²π²ℏ² / (2mL²), n = 1, 2, 3…
Quantum harmonic oscillator	½mω²x²	(n + ½)ℏω, n = 0, 1, 2…
Hydrogen atom	−ke²/r	−13.6 eV / n², n = 1, 2, 3…
Free particle	0 everywhere	ℏ²k²/2m (continuous, not quantized)
Finite square well	−V₀ inside, 0 outside	Discrete bound states + continuum above 0

Note the pattern: confinement produces discreteness. A free particle (no confinement) has a continuous spectrum; a bound particle has a discrete ladder of levels, exactly like the discrete standing-wave modes of a string clamped at both ends.

Time-dependent vs. time-independent — a comparison

	Time-dependent	Time-independent
Equation	iℏ ∂ψ/∂t = Ĥψ	Ĥφ = Eφ
Unknown	ψ(x, t)	φ(x) and eigenvalue E
Type	Evolution law (initial-value problem)	Eigenvalue problem
Applies when	Always	V is time-independent
Gives you	Full dynamics of any state	Stationary states and their energies
Relationship	—	ψ = φ·e−iEt/ℏ solves the TDSE

Worked example: the particle in a box

Take a particle confined to a 1-D box of width L with infinite walls (V = 0 inside, ∞ outside). Inside, the time-independent equation is:

−(ℏ²/2m) d²φ/dx² = Eφ    ⇒    d²φ/dx² = −k²φ,   k = √(2mE)/ℏ

The general solution is φ(x) = A sin(kx) + B cos(kx). The walls demand φ(0) = φ(L) = 0. The first condition kills the cosine (B = 0); the second forces sin(kL) = 0, so kL = nπ. Therefore:

φ_n(x) = √(2/L) · sin(nπx / L)
E_n    = n²π²ℏ² / (2mL²),   n = 1, 2, 3, …

The energies are quantized purely because the wave had to fit an integer number of half-wavelengths inside the box. The lowest energy E₁ is not zero — this zero-point energy is a direct consequence of the uncertainty principle: perfect confinement (Δx finite) forbids Δp = 0.

Numerically, an electron (m = 9.11 × 10⁻³¹ kg) in a box of atomic size L = 0.1 nm has ground-state energy E₁ = π²ℏ²/(2mL²) ≈ 6.0 × 10⁻¹⁸ J ≈ 38 eV — the correct order of magnitude for atomic binding energies. Shrink the box and the energy shoots up as 1/L²; this is why electrons "resist" being localized.

A little history

In late 1925, Erwin Schrödinger, then at the University of Zürich, took Louis de Broglie's 1924 idea that particles have a wavelength λ = h/p seriously and asked: if electrons are waves, what wave equation do they obey? Over the Christmas holidays of 1925–26 he found the answer, and in 1926 published a series of four papers ("Quantisierung als Eigenwertproblem" — "Quantization as an Eigenvalue Problem") in Annalen der Physik. His first triumph was deriving the hydrogen spectrum from the eigenvalues of Ĥ.

The same year, Max Born supplied the physical meaning of ψ: |ψ|² is a probability density (for which Born received the 1954 Nobel Prize). Schrödinger's wave mechanics was soon shown to be mathematically equivalent to Werner Heisenberg's matrix mechanics — two faces of the same theory. Schrödinger shared the 1933 Nobel Prize in Physics with Paul Dirac. Ironically, Schrödinger himself never fully accepted the probabilistic interpretation his equation demanded — his famous 1935 "cat" thought experiment was an attempt to show how strange that interpretation was.

Common misconceptions

• "ψ is a physical wave you could measure." No — ψ is complex and not directly observable. Only |ψ|² (a probability density) is measurable. The phase of ψ is physically real (it drives interference) but you can't read it off with a single detector.
• "The Schrödinger equation is random." The opposite. Its evolution is perfectly deterministic and reversible (unitary). Randomness enters only at measurement, which is not described by the Schrödinger equation itself.
• "The time-independent equation is a different, more approximate law." It is the exact same physics — just the special case of definite-energy solutions of the full equation when V is static.
• "Ĥ is always −(ℏ²/2m)∇² + V." That's the single non-relativistic particle case. In general Ĥ is whatever the total-energy operator is: it can include spin, magnetic-field (−qA) terms, many-body interactions, or be entirely abstract.
• "Energy quantization is an extra postulate." It isn't imposed — it emerges from requiring normalizable solutions with the right boundary conditions.
• "It's relativistic / it's the final word." The Schrödinger equation is explicitly non-relativistic (first order in ∂t, second order in ∂x). Near light speed you need the Dirac or Klein–Gordon equations; for variable particle number, quantum field theory.

Key symbols and units

Symbol	Meaning	Units (SI)
ψ(x, t)	Wavefunction (state)	m−3/2 (3-D)
|ψ|²	Probability density (Born rule)	m−3
Ĥ	Hamiltonian (total-energy operator)	J (energy)
E, En	Energy eigenvalue	J (often quoted in eV)
ℏ	Reduced Planck constant, h/2π	1.0546 × 10⁻³⁴ J·s
m	Particle mass	kg
V(x)	Potential energy	J
∇²	Laplacian (kinetic term)	m−2
p̂ = −iℏ∇	Momentum operator	kg·m/s