The Hamiltonian Operator

Introduction

Every quantum-chemical calculation, from a particle in a box to a protein modelled on a supercomputer, begins by writing down one object: the Hamiltonian operator , Ĥ. It is the operator that corresponds to total energy, and it appears on the left-hand side of the time-independent Schrödinger equation, Ĥψ = Eψ. If you can write the Hamiltonian correctly, you have fully specified the physical problem; everything else is mathematics. This page shows how the Hamiltonian is assembled from two parts — kinetic energy and potential energy — and how to build it for the systems met in this unit.

Core explanation

Energy in classical mechanics. In classical physics the total energy of a particle is the sum of its kinetic and potential energies. Written in terms of momentum p rather than velocity, the kinetic energy is p²/2m, so the classical Hamiltonian function is

H = p²/2m + V(x)

Replacing quantities by operators. Quantum mechanics keeps the same structure but replaces each observable by its operator. Position x becomes "multiply by x", and the linear momentum becomes p̂ₓ = −iħ d/dx. Squaring the momentum operator means applying it twice:

p̂ₓ² = (−iħ d/dx)(−iħ d/dx) = −ħ² d²/dx²

Dividing by 2m gives the kinetic energy operator in one dimension:

T̂ = −(ħ²/2m) d²/dx²

The potential energy V(x) depends only on position, so its operator simply multiplies the wavefunction by V(x). The one-dimensional Hamiltonian is therefore

Ĥ = −(ħ²/2m) d²/dx² + V(x)

Three dimensions. In three dimensions the kinetic energy has contributions from motion along x, y and z, and the three second derivatives combine into the Laplacian , ∇² = ∂²/∂x² + ∂²/∂y² + ∂²/∂z². Then

Ĥ = −(ħ²/2m)∇² + V(x, y, z)

What the kinetic term measures. The second derivative measures the curvature of the wavefunction. A wavefunction that bends sharply — a short wavelength or a rapid rise and fall — has a large kinetic energy. This is the quantum-mechanical link between confinement and energy: squeezing a particle into a small region forces ψ to curve sharply, which raises T.

What the potential term contains. V carries all the chemistry. For a free particle V = 0; for a particle in a box V = 0 inside and infinite outside; for a harmonic oscillator V = ½kx²; for the electron in a hydrogen atom V = −e²/4πε₀r, the Coulomb attraction to the nucleus. For many particles, the Hamiltonian contains a kinetic term for each particle and a potential term for every pair of interacting charges.

Hermitian character. The Hamiltonian is a Hermitian operator, which guarantees that its eigenvalues — the allowed energies — are real numbers, as any measured energy must be.

Formulae

One dimension: Ĥ = −(ħ²/2m) d²/dx² + V(x). Three dimensions: Ĥ = −(ħ²/2m)∇² + V(x, y, z). Momentum operator: p̂ₓ = −iħ d/dx. ħ = h/2π ≈ 1.055 × 10⁻³⁴ J s.

Step-by-step reasoning

To build the Hamiltonian for any system:

1. Identify every moving particle and write a kinetic energy term −(ħ²/2mᵢ)∇ᵢ² for each one. 2. Identify every interaction: attractions to nuclei, repulsions between electrons, external fields or walls. 3. Write each interaction as a potential energy function of the coordinates. 4. Add all the terms together to form Ĥ. 5. Check that the Hamiltonian has units of energy and that each term depends on the right coordinates.

Visual explanation

Imagine a graph of a wavefunction drawn above a graph of the potential V(x). The kinetic operator reads the shape of the upper curve — how sharply it bends at each point — while the potential operator reads the height of the lower curve at each point. Ĥψ combines the two, point by point, across the whole of space.

Real-world analogy

A Hamiltonian is like a household energy bill with two lines on it: one line charges for motion (the kinetic term, set by how "wiggly" the wavefunction is) and the other charges for location (the potential term, set by where the particle is likely to be). The total bill is the sum, and the eigenstates are the arrangements with a steady, well-defined total.

Real-world example

Computational chemistry programs used to design drugs and catalysts begin by setting up the molecular Hamiltonian: kinetic energy terms for every electron, electron–nucleus attractions, electron–electron repulsions and nucleus–nucleus repulsions. For a water molecule this already means ten electrons and three nuclei, which is why approximations are unavoidable in practice.

Why?

Why does the kinetic energy involve a second derivative? Because kinetic energy is p²/2m and the momentum operator contains one derivative; squaring it gives two. Physically, the de Broglie relation links momentum to wavelength, and the second derivative of a wave is largest when its wavelength is shortest, so higher momentum means higher curvature.

Common misconception

"The potential energy operator involves a derivative too." It does not. V̂ is purely multiplicative: acting on ψ(x) it gives V(x)ψ(x). Only the kinetic energy operator involves differentiation, because only momentum is represented by a derivative in the position representation.

Worked example

Question: Write the Hamiltonian for a particle of mass m moving in one dimension in a harmonic potential V = ½kx², and show that Ĥ acting on ψ = e^(−ax²) contains a term proportional to x²ψ.

Reasoning: Ĥ = −(ħ²/2m) d²/dx² + ½kx². The first derivative is dψ/dx = −2ax e^(−ax²); the second derivative is d²ψ/dx² = (4a²x² − 2a) e^(−ax²). So

Ĥψ = −(ħ²/2m)(4a²x² − 2a)ψ + ½kx²ψ = [(ħ²a/m) + (½k − 2ħ²a²/m)x²]ψ

Answer: The x²ψ term has coefficient ½k − 2ħ²a²/m. It vanishes if a² = mk/4ħ², in which case ψ is an eigenfunction with E = ħ²a/m.

Quick check

1. Which part of the Hamiltonian changes when a particle moves from free space into a hydrogen atom? Answer: Only the potential energy term changes, from zero to the Coulomb attraction −e²/4πε₀r; the kinetic term keeps the same form.

Exam focus

Be able to derive T̂ = −(ħ²/2m) d²/dx² from p̂ₓ = −iħ d/dx, and to write Ĥ for a free particle, a particle in a box, a harmonic oscillator and the hydrogen atom. Watch the sign of the kinetic term and the factor of 2m in the denominator.

Advanced insight

For atoms and molecules the full Hamiltonian includes the motion of the nuclei. The Born–Oppenheimer approximation separates nuclear and electronic motion because nuclei are thousands of times heavier than electrons, giving an electronic Hamiltonian in which the nuclei are fixed and their repulsion is a constant. Relativistic corrections and spin–orbit coupling add further terms that matter for heavy elements.

Summary

The Hamiltonian is the total-energy operator, Ĥ = T̂ + V̂. The kinetic term, −(ħ²/2m)∇², comes from squaring the momentum operator and measures the curvature of the wavefunction. The potential term multiplies ψ by V and carries the physics of the system. Its eigenvalues are the allowed energies, which are real because Ĥ is Hermitian.

Practice questions

1. Write the Hamiltonian for a free particle of mass m in one dimension. Answer: Ĥ = −(ħ²/2m) d²/dx², because V = 0 everywhere. 2. Show that applying p̂ₓ twice gives −ħ² d²/dx². Answer: (−iħ)(−iħ) = i²ħ² = −ħ², and two derivatives give d²/dx², so p̂ₓ² = −ħ² d²/dx². 3. Write the Hamiltonian for the electron in a hydrogen atom, treating the nucleus as fixed. Answer: Ĥ = −(ħ²/2mₑ)∇² − e²/(4πε₀r). 4. How many kinetic energy terms appear in the electronic Hamiltonian of a helium atom with a fixed nucleus, and what extra potential term appears compared with hydrogen? Answer: Two kinetic terms, one per electron; the extra term is the electron–electron repulsion e²/(4πε₀r₁₂), and the nuclear attraction uses charge 2e.