The Time-Independent Schrödinger Equation

Introduction

The time-independent Schrödinger equation, Ĥψ = Eψ, is the central energy problem of quantum chemistry. Given a system's Hamiltonian and boundary conditions, solve for wavefunctions ψ whose energies E are definite. “Time-independent” does not mean the electron is frozen in place; it refers to a spatial eigenvalue equation derived when the Hamiltonian has no explicit time dependence and a stationary state can be separated from its time factor.

Core explanation

For one particle moving in one dimension with potential V(x), Ĥ = −(ħ²/2m)d²/dx² + V(x). The equation is −(ħ²/2m)d²ψ/dx² + V(x)ψ = Eψ. The second derivative represents kinetic-energy behavior, while V(x) multiplies the wavefunction. Different potentials and boundary conditions produce different allowed ψ and E. In an infinite box, ψ must vanish at the walls; this selects discrete standing-wave energies. For a hydrogen electron, the attractive Coulomb potential produces bound atomic orbitals and discrete negative energies.

The equation is an eigenvalue equation: the Hamiltonian acting on an energy eigenfunction returns the same function multiplied by E. A physical bound-state wavefunction must be finite, single-valued, suitably continuous and normalizable under the problem's conditions. Mathematical solutions that violate boundaries or cannot represent a normalizable bound state are rejected. The allowed energies arise from these physical restrictions, not from inserting a “quantization rule” after solving.

The full time-dependent equation for a time-independent Hamiltonian allows a separated stationary solution Ψ(x,t) = ψ(x)e^(−iEt/ħ). The time factor changes the complex phase, but Ψ(x,t) ² = ψ(x) ², so the spatial probability density is constant in time. A stationary state can still have nonzero kinetic energy and nontrivial momentum distribution. An electron in a hydrogen 1s orbital is not a tiny object sitting motionless at one point.

A superposition of different energy eigenstates generally has a time-dependent probability density because the components acquire different phases e^(−iEᵢt/ħ) and interfere. Therefore “stationary” is a property of one energy eigenstate or special degenerate combinations, not every solution of the full time-dependent equation. Measurement of energy for a superposition can return different allowed E values. The equation determines idealized isolated-system energies; real atoms may require additional terms for spin, external fields or relativity.

MIT OpenCourseWare's quantum and physical chemistry lectures at https://ocw.mit.edu/courses/5-61-physical-chemistry-fall-2007/pages/lecture-notes/ develop this separation of energy eigenstates from time evolution. The distinction makes later topics such as spectroscopy and approximate molecular calculations coherent.

Step-by-step reasoning

1. Specify the system, potential V and any time-independent assumptions. 2. Build Ĥ from kinetic and potential terms, including coordinates and masses. 3. Solve Ĥψ = Eψ subject to physical boundary and normalization conditions. 4. Identify allowed energies and interpret ψ ² as spatial probability density. 5. Add e^(−iEt/ħ) for time evolution of one stationary state and compare with a superposition.

Visual explanation

Draw a potential well with several horizontal allowed energy levels and wavefunctions that fit its boundaries. Beside one level draw a rotating phase arrow for e^(−iEt/ħ), while a probability-density curve stays unchanged. For two levels together draw two phase arrows turning at different rates and a changing interference pattern.

Real-world analogy

A standing wave on a string has a shape selected by boundary conditions. Its nodes and allowed frequencies arise because only certain patterns fit. A quantum stationary state similarly satisfies a boundary problem, although its wavefunction is a probability amplitude and its complex time phase has no direct classical-string equivalent.

Real-world example

In atomic spectroscopy, photon energies match differences between allowed stationary-state energies. A hydrogen atom can absorb a photon that promotes its electron from one bound energy state to another when the transition is allowed; arbitrary photon energies do not correspond to every possible bound-to-bound transition.

Why?

Why can probability density stay constant while the wavefunction changes with time? The energy-state time factor is a complex phase of unit magnitude. Multiplying by it rotates phase but does not change Ψ ². The electron's kinetic energy can still be nonzero because stationary density is not a statement of zero motion.

Common misconception

“Time-independent Schrödinger equation means nothing changes with time” is too strong. The stationary state's overall phase evolves, and superpositions can have time-dependent densities. Another error is treating ψ itself as a probability; probability density is ψ ², integrated over a region with the appropriate volume element.

Worked example

Inside a one-dimensional infinite well from x = 0 to L, V = 0 and the wavefunction must vanish at both walls. A trial ψ = A sin(kx) satisfies ψ(0) = 0; ψ(L) = 0 requires kL = nπ for positive integer n. Applying Ĥ gives E = ħ²k²/(2m) = n²π²ħ²/(2mL²). Thus quantized energies follow from the eigenvalue equation plus boundaries, not from a separate arbitrary restriction.

Quick check

1. Does a stationary energy state have zero kinetic energy because ψ ² is time independent? Answer: No. Time-independent probability density does not imply zero kinetic energy. Its wavefunction can have spatial curvature and a positive kinetic-energy expectation.

Exam focus

Write Ĥψ = Eψ and identify Ĥ, ψ and E with units. State boundary and normalization conditions. Show why a single energy eigenstate has constant Ψ ² but different-energy superpositions generally do not. Avoid classical orbit language when interpreting atomic orbitals.

Advanced insight

The time-independent equation is especially useful because spectral expansions of the Hamiltonian provide a basis for evolving more general initial states. In molecular chemistry, exact solutions are rare, so variational approximations build finite basis representations of Ĥ and solve matrix eigenvalue problems. The same eigenvalue logic survives the approximation.

Summary

The time-independent Schrödinger equation is an energy eigenvalue problem for a specified Hamiltonian. Physical boundary conditions select allowed wavefunctions and energies. An energy eigenstate's probability density is stationary even while its phase evolves; superpositions can vary in time. This framework underlies atomic spectra and molecular computation.

Practice questions

1. What condition on ψ makes a one-dimensional infinite-well solution quantized? Answer: ψ must vanish at both walls; this permits only standing-wave k values nπ/L and corresponding discrete energies.

2. What is the probability density associated with a wavefunction Ψ? Answer: Ψ ², interpreted with the appropriate coordinate volume element and normalization.

3. Why can a superposition of states with different E values change its density over time? Answer: Components acquire different phase rates, so their interference terms vary with time even though each individual eigenstate's density is constant.