The Time-Independent Schrodinger Equation

Every electronic structure method used in modern computational chemistry — Hartree–Fock, density functional theory, coupled cluster, configuration interaction — is, at bottom, a strategy for approximately solving a many-electron time-independent Schrödinger equation, because molecular Hamiltonians used to compute ground-state geometries, bond energies, and reaction barriers are overwhelmingly time-independent (the Born–Oppenheimer electronic Hamiltonian at fixed nuclear positions). Drug-design software that predicts binding energies of a candidate molecule to a protein pocket is, underneath the graphical interface, solving variants of $\hat H\psi = E\psi$ for the relevant electronic states. Vibrational spectroscopy (IR, Raman) assigns absorption bands by computing TISE eigenvalues of nuclear motion on a potential energy surface, exactly as in the CO oscillator worked above, and anharmonic corrections to the pure harmonic TISE solution explain overtone and combination bands used to fingerprint functional groups. Semiconductor and photovoltaic device design relies on solving the TISE for electrons in periodic crystal potentials (Bloch's theorem), producing the band structures that determine whether a material is a conductor, insulator, or useful solar-cell absorber. Scanning tunneling microscopy, which images individual atoms on a surface, functions entirely because of the nonzero tunneling transmission computed in Example 3 — an effect with zero classical analog, derivable only from stationary scattering solutions of the TISE. In magnetic resonance and quantum computing, engineered systems are deliberately placed into superpositions of just two TISE eigenstates (qubits), and their controlled time evolution — governed by the Bohr-frequency beating derived above — is the operational basis of the technology.

Ask the tutor: How does the choice of boundary conditions on ψ(x) determine whether solving the TISE gives a discrete spectrum or a continuous one?

Start from the full machinery: a quantum particle's state is a wavefunction $\Psi(x,t)$ that evolves according to the The Time-Dependent Schrodinger Equation,

where $\hat{H} = -\dfrac{\hbar^2}{2m}\dfrac{\partial^2}{\partial x^2} + V(x,t)$ in one dimension. In general $V$ can depend on time — a molecule sitting in a laser pulse, say — and then $\Psi$ has no simple factored form. But an enormous number of chemically important situations have a potential that is static: the Coulomb attraction in a hydrogen atom, the harmonic restoring force of a bond, the walls of a "particle in a box" model of a conjugated chain. For these, $V = V(x)$ only, and the Hamiltonian operator itself does not depend on time. When the operator on the right-hand side has no explicit time dependence, it becomes possible to look for special solutions in which the space and time dependence multiply rather than mix: $\Psi(x,t) = \psi(x) f(t)$. Substituting this ansatz into the time-dependent equation and dividing through by $\psi(x) f(t)$ splits the equation into a piece that depends only on $t$ and a piece that depends only on $x$. Because these two sides are equal for all values of $x$ and $t$ independently, each side must equal the same constant — call it $E$, anticipating that it will turn out to be the energy. That is the entire logical content of separation of variables here: two independent variables, one equation, forces both sides to a shared constant. The $t$-side gives an ordinary differential equation whose solution is a pure phase factor, $f(t) = e^{-iEt/\hbar}$, identical for every system regardless of what $V(x)$ looks like. The $x$-side gives the spatial eigenvalue equation

the time-independent Schrödinger equation. This equation by itself says nothing about dynamics — it is a boundary-value problem, exactly like finding the normal modes of a vibrating string. Acceptable solutions $\psi(x)$ (finite, single-valued, continuous with continuous first derivative wherever $V$ is finite, and normalizable — the Conditions on Acceptable Wavefunctions) typically exist only for a discrete ladder of $E$ values when the particle is confined, and this discreteness is the mathematical seed of quantization. Each pair $(\psi_n, E_n)$ obtained this way defines a stationary state $\Psi_n(x,t) = \psi_n(x)e^{-iE_n t/\hbar}$. The word "stationary" refers precisely to the probability density, not to the complex wavefunction:

The phase factors cancel exactly, leaving a probability density frozen in time. Any single stationary state therefore describes a genuinely static physical picture: an electron in an atomic orbital does not "orbit," its charge cloud simply sits there, unchanging, forever (until perturbed). Time dependence in observable quantities only appears when a wavefunction is a superposition of stationary states with different energies, because then cross terms carry a relative phase $e^{-i(E_n - E_m)t/\hbar}$ that does not cancel — this is the microscopic origin of spectroscopic transition frequencies, $\omega_{nm} = (E_n - E_m)/\hbar$.

The full time-dependent equation for a static, one-dimensional potential:

where:

• $\Psi(x,t)$ — the full wavefunction, a complex function of position and time,
• $m$ — the particle's mass,
• $V(x)$ — the potential energy, assumed independent of time,
• $\hbar = h/2\pi$ — the reduced Planck constant.

Reading: this is Newton's second law rewritten for a wave amplitude; the left side encodes how fast the state changes in time, the right side (the Hamiltonian $\hat H$) encodes kinetic plus potential energy acting on it. Separating $\Psi(x,t) = \psi(x)f(t)$ and dividing by $\psi(x) f(t)$ gives two decoupled equations sharing the constant $E$:

where:

• $E$ — the separation constant, later identified as the total energy,
• $f(t)$ — the universal time factor, the same functional form for every static-potential problem.

Reading: whatever the potential is, the time dependence of any stationary state is always this one oscillating phase; all system-specific physics is pushed entirely into the spatial equation.

where:

• $\psi(x)$ — the spatial eigenfunction (orbital, vibrational wavefunction, etc.),
• $E$ — the eigenvalue: the energy associated with that particular $\psi$.

Reading: this is the time-independent Schrödinger equation, an eigenvalue equation $\hat H\psi = E\psi$; acceptable solutions exist only for particular $E$, giving discrete (or continuous, for unbound states) energy levels.

where:

• $n$ — quantum number labeling distinct stationary states,
• $|\Psi_n(x,t)|^2$ — probability density, shown here to be time-independent for a single stationary state.

Reading: any observable computed from a single stationary state — position expectation value, momentum expectation value, energy — is constant in time; only coherent superpositions of stationary states with different $E_n$ display genuine dynamics.

where:

• $c_n$ — expansion (superposition) coefficients set by the initial condition $\Psi(x,0) = \sum_n c_n \psi_n(x)$.

Reading: the general solution of the full time-dependent equation, once the stationary states and energies are known, is just this sum — solving the TISE once solves the dynamics for any initial condition built from those eigenfunctions.

Example 1 — Foundational: particle in a one-dimensional box, energies and the ground/first-excited stationary states. Consider an electron confined to a box of length $L = 0.30\ \text{nm}$ (a crude but standard model for a short conjugated system), with $V(x) = 0$ for $0 < x < L$ and $V = \infty$ outside. Inside the box the TISE reads

whose general solution is $\psi(x) = A\sin(kx) + B\cos(kx)$ with $k = \sqrt{2mE}/\hbar$. The boundary condition $\psi(0)=0$ (continuity with the zero wavefunction outside the infinite wall) forces $B=0$. The condition $\psi(L)=0$ forces $\sin(kL) = 0$, so $kL = n\pi$ for $n=1,2,3,\dots$ — quantization has appeared directly from the boundary conditions, exactly as promised. Solving for the energy,

Plug in numbers for $n=1$ and $n=2$ using $m = 9.109\times10^{-31}\ \text{kg}$, $h = 6.626\times10^{-34}\ \text{J}\,\cdot\,\text{s}$, $L = 3.00\times10^{-10}\ \text{m}$:

So $E_1 = 1\times 6.694\times10^{-19}\ \text{J} = 4.179\ \text{eV}$, and $E_2 = 4\times 6.694\times10^{-19}\ \text{J} = 2.678\times10^{-18}\ \text{J} = 16.72\ \text{eV}$. The normalized eigenfunctions are $\psi_n(x) = \sqrt{2/L}\sin(n\pi x/L)$; the full stationary states are $\Psi_n(x,t) = \sqrt{2/L}\sin(n\pi x/L)\,e^{-iE_nt/\hbar}$. As a limiting-case check, letting $L \to \infty$ (removing confinement) sends every $E_n \to 0$ and the levels crowd together into a continuum — recovering the free particle, as they must, since an unconfined particle has no reason to have discrete energies. If the electron is prepared in the superposition $\Psi(x,0) = \frac{1}{\sqrt2}\psi_1(x) + \frac{1}{\sqrt2}\psi_2(x)$, its probability density oscillates with angular frequency

corresponding to a transition wavelength $\lambda = 2\pi c/\omega_{21} \approx 99$ nm (deep UV) — illustrating directly how superpositions of TISE solutions predict spectroscopic frequencies.

Example 2 — Applied: vibrational stationary states of a diatomic modeled as a harmonic oscillator. Take carbon monoxide, CO, with reduced mass $\mu = \dfrac{m_C m_O}{m_C+m_O}$. Using $m_C = 12.000$ u and $m_O = 15.995$ u,

The experimental force constant of the C≡O bond is $k = 1902\ \text{N/m}$. Modeling the bond as a harmonic oscillator, $V(x) = \frac12 kx^2$ with $x$ the bond-length displacement from equilibrium, the TISE

has the well-known solutions $\psi_v(x) = N_v H_v(\alpha x)e^{-\alpha^2x^2/2}$ (Hermite polynomials $H_v$ times a Gaussian), with energies

Compute the classical vibrational angular frequency:

The zero-point energy ($v=0$) is

The spacing between adjacent stationary states is uniform: $\Delta E = \hbar\omega = 4.312\times10^{-20}\ \text{J}$, corresponding to a fundamental vibrational wavenumber

in good agreement with the experimental CO stretch (2143 cm⁻¹); the small discrepancy is the expected signature of anharmonicity not captured by the pure harmonic TISE. As a limiting-case check, setting $v=0$ recovers the zero-point energy rather than zero — a direct, testable consequence of solving the TISE rather than assuming a classical particle at rest (which would incorrectly give $E=0$).

Example 3 — Exam-level: transmission through a rectangular barrier, using stationary scattering states of the TISE. An electron of kinetic energy $E = 2.00\ \text{eV}$, moving in one dimension, strikes a rectangular potential barrier of height $V_0 = 5.00\ \text{eV}$ and width $L = 0.80\ \text{nm}$, with $E < V_0$ (classically forbidden region). This is a stationary scattering problem: seek time-independent solutions $\psi(x)$ of the TISE piecewise in three regions (incoming/reflected wave, exponentially decaying/growing wave inside the barrier, transmitted wave), then match $\psi$ and $\psi'$ at both boundaries. Inside the barrier the TISE $-\frac{\hbar^2}{2m}\psi'' + V_0\psi = E\psi$ gives $\psi'' = \kappa^2\psi$ with

Compute $\kappa$ numerically: $V_0 - E = 3.00\ \text{eV} = 4.806\times10^{-19}\ \text{J}$,

Since the barrier is wide compared to $1/\kappa$ ($\kappa L = 8.870\times10^9 \times 8.00\times10^{-10} = 7.096$), the standard opaque-barrier (WKB-consistent) transmission formula applies:

Evaluate the prefactor: $\dfrac{4E(V_0-E)}{V_0^2} = \dfrac{4(2.00)(3.00)}{(5.00)^2} = \dfrac{24.0}{25.0} = 0.960$. Evaluate the exponent: $2\kappa L = 2(7.096) = 14.19$, so $e^{-14.19} = 6.86\times10^{-7}$. Multiplying,

Roughly 7 electrons in 10 million tunnel through — a small but sharply nonzero probability that a classical particle would never have, since classically $E<V_0$ means total reflection ($T=0$ exactly). This is a direct experimental discriminator of quantum mechanics, and the entire calculation rests on solving the same TISE, just now with unbound (scattering) boundary conditions instead of the bound-state normalizability condition used in Examples 1 and 2. As a limiting-case check, letting $L\to\infty$ sends $e^{-2\kappa L}\to 0$, correctly recovering zero transmission through an infinitely thick barrier.