Chemistry · Lesson

States, operators and the quantum description of molecules

Read a wavefunction as a probability amplitude, with a stated model and normalization. Distinguish allowed measurement outcomes from their average and a molecular state from a one-electron orbital.

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By the end, you can

Normalize complex coefficients in an orthonormal basis.
Distinguish position density, finite-region probability and expectation.
Interpret stationary states and molecular many-particle descriptions without assigning classical electron orbits.

An amplitude supplies probabilities through its modulus squared

For a normalized pure one-particle state, a position wavefunction ψ is a complex amplitude. In one dimension, |ψ(x)|² is a probability density and the probability of a region is its integral over that region. If x is in metres, ψ has units m^(−1/2) and the density has units m⁻¹. A density value is not itself the probability of one exact coordinate.

For a molecule, the full many-electron wavefunction depends on all electron coordinates and spin variables, with nuclear variables included or treated under specified approximations. A one-electron orbital is a component of an electronic-structure description. Its modulus squared is not generally the entire molecule’s many-electron probability distribution or total electron density.

Complex coefficients are amplitudes rather than probabilities

For orthonormal discrete basis states, a normalized expansion has Σ|cⱼ|² = 1. If a finite nonzero state norm is not one, divide the state by its norm. With raw coefficients 1 and 2i, the squared norm is 1 + 4 = 5, so normalized coefficients are 1/√5 and 2i/√5.

Multiplying the entire normalized state by a common constant phase changes no measurement predictions. Relative phases between components can matter for other observables, even when a particular energy measurement has the same probabilities. Nonorthogonal basis functions require an overlap matrix rather than a simple sum of coefficient squares.

An expectation need not be one allowed result

An observable is represented by an operator with the appropriate self-adjoint domain; a finite-dimensional example uses a Hermitian matrix. In an orthonormal basis of its nondegenerate eigenstates, the probability of eigenvalue aⱼ is |cⱼ|² and the expectation is Σ|cⱼ|²aⱼ. For a degenerate eigenvalue, sum the probability in its eigenspace. Continuous spectra require integrals.

In an invented two-state energy model with eigenvalues 1 eV and 4 eV, the normalized coefficients above give probabilities 1/5 and 4/5. The average is 3.4 eV. A single ideal energy measurement returns 1 or 4 eV, not their average. Repeated identically prepared systems provide the statistical meaning of that expectation.

Possible energies and their expectation are different quantities

Energy component	Normalized coefficient	Probability	Contribution to ⟨E⟩
Eigenvalue 1 eV	1/√5	1/5 = 0.200	0.200 eV
Eigenvalue 4 eV	2i/√5	4/5 = 0.800	3.20 eV
Complete normalized state	Squared norm = 1	Probability sum = 1	⟨E⟩ = 3.4 eV

Original two-state pure-state model with nondegenerate energies 1 eV and 4 eV and orthonormal eigenstates. The coefficient i is a relative complex phase. The total expectation 3.4 eV is a statistical mean, rather than a third possible ideal energy outcome.

A stationary state differs from a general superposition

The time-dependent Schrödinger equation is iħ∂Ψ/∂t = ĤΨ, with ħ = h/(2π) and h the Planck constant. For a time-independent Hamiltonian, an energy eigenstate evolves with a phase exp(−iEt/ħ). Its position probability density is stationary even though the state vector has a time-dependent phase.

A superposition of different energies acquires changing relative phases, so some measurement distributions can vary with time. This model evolution does not give a classical electron trajectory. Interactions with other systems or a changing Hamiltonian require their own description rather than a universal picture of electrons moving on fixed planetary paths.

Worked examples

Normalize amplitudes before averaging energies

For two orthonormal energy states at 1 and 4 eV, the raw coefficients are 1 and 2i. Find their probabilities and the mean energy.

The squared norm is |1|² + |2i|² = 5.
Divide each coefficient by √5.
Their squared moduli are 1/5 and 4/5.
Average the energy: (1/5) × 1 + (4/5) × 4 = 3.4 eV.

The probabilities are 20% and 80%, and ⟨E⟩ = 3.4 eV. The possible single energy outcomes remain 1 and 4 eV.

Try it yourself

In an orthonormal basis, a normalized coefficient has magnitude 1/2. What probability does it assign to that nondegenerate outcome?
Show the answer and reasoning
1/4.
Probability is the squared modulus, not the amplitude magnitude.
Does a one-dimensional density of 2 × 10⁹ m⁻¹ represent a probability greater than 1?
Show the answer and reasoning
No.
A finite-region probability integrates the dimensional density.
For the supplied two-state model, is 3.4 eV an allowed single ideal energy outcome?
Show the answer and reasoning
No.
It is the expectation of outcomes at 1 and 4 eV.
Is a single orbital generally the full interacting many-electron molecular wavefunction?
Show the answer and reasoning
No.
It is a one-electron function used within an explicitly chosen description.

What to carry forward

Normalize amplitudes on the correct basis.
Integrate a density to obtain a region probability.
Keep measurement outcomes distinct from their expectation.
A molecular state and a single orbital describe different objects.

References

Teaching resources and reference material supporting this lesson’s explanations and examples.