The Time-Dependent Schrodinger Equation

The TDSE is not a historical curiosity confined to textbooks — it is the working equation behind entire modern subfields. In ultrafast spectroscopy, femtosecond and attosecond laser pulses prepare exactly the kind of coherent superpositions worked through above, and pump–probe experiments directly measure the beat frequencies $(E_j-E_k)/\hbar$ as oscillating signals in real time, watching chemical bonds vibrate and electrons redistribute on their natural timescales. In magnetic resonance (NMR and MRI), nuclear spins precess under a time-dependent Hamiltonian containing static and radiofrequency magnetic fields; the Rabi-flopping mathematics of the second worked example is literally the equation solved to design the radiofrequency pulses ($\pi/2$- and $\pi$-pulses) that make every NMR and MRI pulse sequence work. In photosynthesis research, the TDSE governs how excitonic coherences move energy through light-harvesting antenna complexes with remarkable efficiency, a topic of intense study in quantum biology. In quantum computing, qubits are literally two-level systems whose state vector is propagated by the TDSE (or its extension including decoherence), and gate operations are engineered pulse sequences that drive precisely controlled Rabi oscillations. In computational chemistry, real-time TDSE propagation (real-time TDDFT) is used to simulate absorption spectra, nonlinear optical response, and photochemical reaction dynamics directly in the time domain, complementing the energy-domain picture from the Time-Independent Schrodinger Equation. And in coherent control, chemists shape laser pulses to steer superpositions of vibrational and electronic states — quantum "steering wheels" built entirely from solving the TDSE for a chosen molecular Hamiltonian to select which bond breaks.

Ask the tutor: How does the time-dependent Schrödinger equation reduce to the time-independent one, and why do chemists usually start with the latter?

Consider a free particle described, as motivated by the de Broglie Relation, by a plane wave $\Psi(x,t) = A e^{i(kx - \omega t)}$, with $p = \hbar k$ and $E = \hbar \omega$. This wave is not a static snapshot; it is explicitly a function of time, oscillating in place as it also propagates. If quantum mechanics is to describe how any wavefunction changes from one instant to the next, we need a differential equation whose solutions include this plane wave and, more generally, any physically allowed $\Psi$. Schrödinger's insight (1925–1926) was to demand that this equation be linear — so that superpositions of solutions are also solutions, which is essential for interference — and that it link the time derivative of $\Psi$ to the energy of the particle, since $E=\hbar\omega$ links frequency to energy. Differentiating the plane wave with respect to time brings down a factor of $-i\omega$, and differentiating twice with respect to position brings down $-k^2$. Since $E = p^2/2m$ for a free particle, and $p = \hbar k$, matching the two derivative relations forces a very specific structure: a first time derivative must equal $i\hbar$ times a second spatial derivative (up to sign and constants), because $\hbar\omega$ is linear in $E$ but $E \propto k^2 \propto (\text{second spatial derivative})$. This is why the Schrödinger equation is first-order in time (unlike the classical wave equation, which is second-order) and why it necessarily involves the imaginary unit $i$ — a real equation of this structure could not support traveling, oscillating solutions with the right dispersion relation. Once a potential energy $V(x,t)$ is present, the total energy operator — the Hamiltonian — is kinetic plus potential, and the same structural equation holds with $\hat H$ in place of the free-particle kinetic operator alone. The resulting equation is the master law: $\hat H$ acting on $\Psi$ equals $i\hbar$ times the time derivative of $\Psi$. Crucially, this equation is exact — it is not an approximation, a rule of thumb, or a model. Every non-relativistic quantum system, from an electron in a Coulomb field to a nuclear spin in an oscillating magnetic field, obeys exactly this equation, differing only in what $\hat H$ is. The picture that emerges is one of a wavefunction as a vector rotating (in an abstract, infinite-dimensional space) under a "rotation generator" $\hat H$. If $\Psi$ is initially an eigenstate of $\hat H$ (a stationary state), it does not change its shape at all — only a uniform, physically unobservable overall phase factor accumulates. If $\Psi$ is a superposition of eigenstates with different energies, the relative phases between components drift apart at rates proportional to their energy differences, and this relative dephasing is what produces all observable time dependence — oscillating dipole moments, spectral line shapes, wavepacket motion.

The general, exact statement, valid for any potential (even time-dependent ones, as in a molecule driven by a laser field):

where:

• $\Psi(\mathbf r, t)$ — the time-dependent wavefunction, whose modulus-squared gives the probability density (Born interpretation, now time-dependent).
• $\hat H(t) = \hat T + \hat V(\mathbf r, t) = -\dfrac{\hbar^2}{2m}\nabla^2 + V(\mathbf r, t)$ — the Hamiltonian operator, kinetic plus potential energy.
• $\hbar = h/2\pi$ — the reduced Planck constant.
• $i = \sqrt{-1}$ — the imaginary unit; the wavefunction is intrinsically complex.

Reading: the rate of change of the state is proportional to the Hamiltonian acting on that state; $\hat H$ literally is the time-evolution generator, not merely an energy bookkeeping device. When $\hat H$ has no explicit time dependence (no external field turning on and off, no moving nuclei in the Born–Oppenheimer sense), separation of variables gives the single most important special solution:

where:

• $\psi(\mathbf r)$ — a time-independent spatial function satisfying $\hat H \psi = E\psi$, the eigenvalue equation that becomes the Time-Independent Schrodinger Equation.
• $E$ — the (real, conserved) energy eigenvalue associated with that eigenstate.
• $e^{-iEt/\hbar}$ — a pure phase factor of unit modulus; it never affects $|\Psi|^2$.

Reading: for a stationary state, all time dependence is an unobservable global phase — the probability density, and every expectation value of a time-independent operator, is frozen in time. This is precisely why we can speak of "the 2p orbital" as a fixed spatial shape. For a general initial state expanded in energy eigenstates,

where:

• $c_n$ — fixed (time-independent) expansion coefficients, determined entirely by the initial wavefunction at $t=0$.
• $\psi_n$, $E_n$ — the complete set of orthonormal eigenfunctions and eigenvalues of $\hat H$.
• the sum runs over all stationary states (a discrete sum here; more generally an integral over a continuum too).

Reading: this is the fully general solution to the TDSE whenever $\hat H$ is time-independent — you only ever need to solve the time-independent eigenvalue problem once, and then time evolution is just attaching phase factors $e^{-iE_nt/\hbar}$ to each piece and re-summing. This is the essential content of Superposition of States: physically distinct stationary states can be combined, and the combination evolves by each component spinning at its own rate.

Example 1 — Foundational: a superposition in the infinite square well. Take a particle of mass $m = 9.109\times10^{-31}\,\text{kg}$ (an electron) confined to an infinite square well of width $L = 0.500\,\text{nm}$ — a crude but standard model for a delocalized $\pi$-electron. The normalized eigenstates and energies are

At $t=0$ the electron is prepared in the asymmetric (non-special, deliberately unequal-weight) superposition

so $c_1 = \sqrt{0.30} = 0.5477$, $c_2=\sqrt{0.70}=0.8367$ (chosen unequal on purpose, and already normalized since $0.30+0.70=1$). Step 1: energies. With $h = 6.626\times10^{-34}\,\text{J s}$, $m=9.109\times10^{-31}\,\text{kg}$, $L=5.00\times10^{-10}\,\text{m}$:

Step 2: beat frequency.

This lies in the near-ultraviolet (corresponding wavelength $\lambda = c/\nu_{21} = 275\,\text{nm}$), showing that the electron's charge density genuinely sloshes back and forth at a UV-range frequency — the same frequency a photon would need to drive this transition. Step 3: the time-dependent density. Using the beat-formula derived above with $c_1c_2 = 0.5477\times0.8367 = 0.4583$,

with $\omega_{21} = 2\pi\nu_{21} = 6.856\times10^{15}\,\text{rad/s}$. At $x = L/4$ (an asymmetric, non-central point), $\psi_1(L/4) = \sqrt{2/L}\sin(\pi/4) = \sqrt{2/L}(0.7071)$ and $\psi_2(L/4)=\sqrt{2/L}\sin(\pi/2)=\sqrt{2/L}(1)$, so with $2/L = 4.00\times10^9\,\mathrm{m^{-1}}$, $\psi_1(L/4)=4.472\times10^4\,\mathrm{m^{-1/2}}$ and $\psi_2(L/4)=6.325\times10^4\,\mathrm{m^{-1/2}}$; the probability density genuinely oscillates in time between roughly $1.07\times10^9\,\mathrm{m^{-1}}$ and $3.87\times10^9\,\mathrm{m^{-1}}$ over one beat period $T = 1/\nu_{21} = 0.917\,\text{fs}$. Limiting-case check: if instead $c_2 \to 0$ (pure $\psi_1$), the cross term vanishes and $|\Psi|^2 = \psi_1^2(x)$ is exactly static — recovering the stationary-state result derived above, as it must.

Example 2 — Applied: Rabi-type oscillation of a two-level chromophore in a resonant field. A two-level model of a chromophore (ground state $g$, excited state $e$) with transition energy $\Delta E = E_e - E_g = 2.50\,\text{eV}$ is driven by a resonant oscillating field so that its Hamiltonian in the $\{g,e\}$ basis (rotating-wave approximation) is

with Rabi frequency $\Omega_R = 3.00\times10^{13}\,\text{rad/s}$ (a realistic strong-field value), and the molecule starts entirely in the ground state, $\Psi(0) = |g\rangle$. Step 1: solve the 2×2 TDSE. Writing $\Psi(t) = c_g(t)e^{-iE_gt/\hbar}|g\rangle + c_e(t)e^{-iE_et/\hbar}|e\rangle$, substitution into $i\hbar\dot\Psi = \hat H\Psi$ gives the standard coupled equations

whose solution with $c_g(0)=1,\ c_e(0)=0$ is

Step 2: excited-state population.

Step 3: numeric evaluation at $t = 80.0\,\text{fs} = 8.00\times10^{-14}\,\text{s}$ (an off-round time, not a special multiple of the period).

So after 80 fs, 86.9% of the population has been transferred to the excited state — this is the microscopic mechanism behind Rabi flopping used in coherent control and in modeling how a laser pulse of a specific duration can be tuned to leave a molecule almost entirely excited ("a $\pi$-pulse") or in an exact 50/50 superposition ("a $\pi/2$-pulse"). Step 4: the full period. Complete population transfer back to the ground state occurs when $\Omega_R t/2 = \pi$, i.e. $t = 2\pi/\Omega_R = 2.094\times10^{-13}\,\text{s} = 209.4\,\text{fs}$ — the Rabi period, a directly measurable observable in pump–probe and Ramsey-fringe spectroscopy. Limiting-case check: as $\Omega_R \to 0$ (field switched off), $P_e(t) \to 0$ for all $t$ — the molecule simply sits in its stationary ground state forever, exactly reproducing the stationary-state result of the first derivation.

Example 3 — Exam/competition level: expectation value dynamics and Ehrenfest's theorem for a harmonic oscillator wavepacket. A diatomic molecule modeled as a harmonic oscillator (reduced mass $\mu$, force constant $k_f$, angular frequency $\omega_0=\sqrt{k_f/\mu}$) is prepared, at $t=0$, in a coherent superposition of its ground state $|0\rangle$ (energy $\tfrac12\hbar\omega_0$) and first excited state $|1\rangle$ (energy $\tfrac32\hbar\omega_0$), with unequal amplitudes $c_0 = \sqrt{0.65}=0.8062$ and $c_1=\sqrt{0.35}=0.5916$ (real, normalized: $0.65+0.35=1$). Step 1: use the general result derived above. The energy gap is $E_1-E_0 = \hbar\omega_0$, so the beat frequency is exactly $\omega_0$ itself — a special feature of the harmonic oscillator's evenly spaced ladder. Step 2: expectation value of position. Using the standard raising/lowering-operator matrix elements $\langle 0|\hat x|1\rangle = \langle1|\hat x|0\rangle = \sqrt{\hbar/2\mu\omega_0}$ and $\langle0|\hat x|0\rangle=\langle1|\hat x|1\rangle=0$, the same cross-term algebra as in the second derivation gives

With $c_0c_1 = 0.8062\times0.5916 = 0.4770$, the prefactor is $2(0.4770) = 0.9540$ times the zero-point length scale. Step 3: plug in numbers for HCl-like parameters. Take $\mu = 1.628\times10^{-27}\,\text{kg}$ (the H–Cl reduced mass) and $k_f = 480.0\,\text{N/m}$ (a representative diatomic force constant, deliberately not a round number), so

So

The bond length genuinely oscillates with amplitude 7.37 pm at angular frequency $5.427\times10^{14}\,\text{rad/s}$ (period $T=2\pi/\omega_0 = 11.58\,\text{fs}$) — the quantum-mechanical wavepacket literally swings back and forth like a classical vibrating bond, exactly as expected from Ehrenfest's theorem, which states

i.e., quantum expectation values obey Newton's laws for the harmonic potential exactly (a special feature of quadratic potentials — for an anharmonic potential this equivalence would only be approximate). Limiting-case check: as $c_1\to 0$ (pure ground state), the prefactor $2c_0c_1\to 0$ and $\langle\hat x\rangle(t)\to 0$ for all $t$ — the bond length sits motionless at its equilibrium value, exactly the stationary-state behavior established in the first derivation.