Time-Independent Schrödinger Equation in One Dimension

The TISE is arguably the most-solved differential equation in modern science. Every electronic-structure code — the density-functional machinery behind drug design, battery materials and semiconductor road maps — spends its inner loop solving a self-consistent cousin of $\hat H\psi = E\psi$ millions of times over. The "sketch by curvature, then shoot numerically" craft you learned here is the shooting method those solvers use in one dimension: integrate from both sides, tune $E$ until the tails match. The stationary-state clock is also the most accurate thing humanity has ever built. An atomic clock locks an oscillator to the Bohr frequency $(E_2 - E_1)/\hbar$ of two atomic eigenstates — the caesium hyperfine beat at $9.19\ \mathrm{GHz}$ defines the SI second, and optical lattice clocks track eigenstate beats to nineteen digits. Attaching $e^{-iEt/\hbar}$ to an eigenfunction writes down the tick of the best clock there is.

A struck guitar string has normal modes — the fundamental, the octave, the fifth — each a fixed shape oscillating at its own frequency without changing form. Pluck the string any way you like and the motion looks complicated, but it is only those rigid shapes superposed, each ticking at its own rate. Energy eigenstates are the normal modes of a quantum system: fixed spatial shapes $\psi_n(x)$, each multiplied by a phase clock $e^{-iE_nt/\hbar}$ spinning at frequency $E_n/\hbar$. One mode alone looks static, because the spinning phase cancels in $|\Psi|^2$. Two or more modes interfere, and their beat is what quantum motion is. The second habit to build is reading the TISE as a statement about curvature. Rewrite it as $\psi'' = \tfrac{2m}{\hbar^2}\,[V(x) - E]\,\psi$: the second derivative of the wavefunction — how it bends — is proportional to $\psi$ itself, with a sign switch controlled by whether the energy sits above or below the local potential. Where $E > V(x)$ (the classically allowed region), $\psi$ bends toward the axis, which is what oscillation is. Where $E < V(x)$ (classically forbidden), $\psi$ bends away from the axis, which is what exponential growth or decay is. Every eigenfunction in this chapter — box, well, oscillator, delta — is just these two behaviours stitched together smoothly at the classical turning points.

Read the figure the way you will read every well problem from now on. The dashed amber line is the energy; where it rides above the grey potential the blue wavefunction oscillates, and where it dips below — past the red turning points — oscillation hands over to smooth exponential tails: the quantum leak into classically forbidden territory, decaying faster the deeper the shortfall $V - E$. Curvature reading has a hard consequence: the spectrum has a floor. Give the particle an energy below the minimum of the potential and $V(x) - E$ is positive everywhere, so $\psi$ bends away from the axis everywhere — no turning points, nowhere to turn around. Such a function runs off to infinity and can never be normalized. No allowed energy sits at or below $\min V$:

The full dynamical law, met in Time-Dependent Schrödinger Equation, is first-order in time: give it $\Psi$ now and it delivers $\Psi$ forever after. In one dimension with a static potential it reads

where: $\Psi(x,t)$ — the complex wavefunction; $m$ — the particle's mass; $V(x)$ — a potential energy that depends on position only. That last restriction is the whole game: because no term in the equation carries an explicit $t$, the equation cannot tell one instant from another, and that symmetry is what permits solutions with trivially simple time dependence. Hunting for such solutions with a product ansatz turns (1) into a pair of far easier equations — the derivation below. The spatial half is the protagonist of this entire chapter, the time-independent Schrödinger equation:

where: $\psi(x)$ — the spatial profile of the mode; $E$ — the separation constant, which the derivation identifies as the mode's definite energy; $\hat H = \hat p^2/2m + V(\hat x)$ — the Hamiltonian. Note what kind of problem this is: not "given $\psi$ at one place, march it along" but "find the special pairs $(E, \psi)$ that satisfy the equation and the boundary conditions." It is an eigenvalue problem, the same mathematics that picks out the resonant frequencies of an organ pipe.

The product solutions earn their name — stationary states — from what happens when you compute anything observable. The phase factor meets its conjugate and vanishes:

where: $\psi_n, E_n$ — the $n$-th eigenfunction and eigenvalue. The density is frozen, and by the same cancellation so is $\langle \hat A\rangle$ for every operator $\hat A$ with no explicit time dependence — position, momentum, kinetic energy, all of them constants. This is the frozen hydrogen atom of the overview, and it resolves a classical paradox: a classical orbiting electron accelerates, radiates, and crashes; a quantum ground state has literally nothing about it that changes in time, so there is nothing to radiate. Stillness is not the whole story. The TISE delivers a family $\{\psi_n\}$, and because $\hat H$ is Hermitian the family is orthonormal — $\int \psi_m^*\psi_n\,dx = \delta_{mn}$ — and complete, making the general solution of (1) a superposition of modes, each with its own clock:

where: $c_n$ — the (complex, time-independent) amplitude of mode $n$, obtained by projecting the initial state onto the basis; $|c_n|^2$ — the probability that an energy measurement returns $E_n$, a distribution that never changes as the state evolves. Equation (4) is the master recipe of wave mechanics: solve the eigenvalue problem once, expand the initial state, attach clocks. When the spectrum has a continuous part the sum gains an integral — the subject of Bound States versus Scattering States. Interference between two clocks is the simplest possible motion. Keep two terms of (4) and compute any observable: the cross terms carry $e^{-i(E_2-E_1)t/\hbar}$, so everything measurable oscillates at the Bohr frequency $\omega_{21} = (E_2 - E_1)/\hbar$ — the frequency such a system absorbs or emits.

Every solvable model of this chapter is the TISE with a particular $V(x)$; the table is the chapter's road map, and each row is an article of its own.

1 · The Bohr clock of a quantum dot (confidence builder)

A colloidal quantum dot has its two lowest electron levels split by $E_2 - E_1 = 0.50\ \mathrm{eV}$. An electron is prepared in an equal superposition of the two. At what angular frequency does its probability density slosh, what is the period of one full slosh, and what photon wavelength does the same splitting correspond to?

Step 1. The only time dependence of any observable comes from the phase difference of the two clocks, at the Bohr frequency $\omega_{21} = (E_2 - E_1)/\hbar = \dfrac{0.50 \times 1.602\times10^{-19}\ \mathrm{J}}{1.055\times10^{-34}\ \mathrm{J\,s}} = \boxed{7.6\times10^{14}\ \mathrm{rad/s}}$.

Step 2. One full slosh takes $T = 2\pi/\omega_{21} = \boxed{8.3\ \mathrm{fs}}$ — the density rocks from one side of the dot to the other and back in about eight femtoseconds. This is why level splittings of electron-volt size mean optical-frequency electron motion.

Step 3. A photon bridging the same gap has $\lambda = hc/\Delta E = \dfrac{(6.626\times10^{-34})(3.00\times10^8)}{0.50 \times 1.602\times10^{-19}} = \boxed{2.5\ \mathrm{\mu m}}$, in the infrared. The oscillating charge of Step 2 and the radiated wavelength of Step 3 are two faces of the same beat — an emitting system is a superposition ticking at its Bohr frequency.

2 · Energy statistics that never age (exam level)

A particle in a well is prepared at $t = 0$ in $\Psi = \tfrac{1}{\sqrt{14}}\,(3\psi_1 + 2\psi_2 + \psi_3)$, where the $\psi_n$ are orthonormal eigenstates with $E_1 = 1.0\ \mathrm{eV}$, $E_2 = 2.5\ \mathrm{eV}$, $E_3 = 4.1\ \mathrm{eV}$. Find the possible outcomes of an energy measurement and their probabilities, the mean energy and its spread, and state what changes if the measurement is made a week later.

Step 1. Check normalization: $\tfrac{1}{14}(9 + 4 + 1) = 1$. The expansion coefficients are $c_1 = 3/\sqrt{14},\ c_2 = 2/\sqrt{14},\ c_3 = 1/\sqrt{14}$.

Step 2. An energy measurement returns only eigenvalues, with probabilities $|c_n|^2$: $P(E_1) = \boxed{9/14 \approx 0.643}$, $P(E_2) = 4/14 \approx 0.286$, $P(E_3) = 1/14 \approx 0.071$. Never the average, always one rung.

Step 3. Mean: $\langle H\rangle = \tfrac{1}{14}(9 \times 1.0 + 4 \times 2.5 + 1 \times 4.1) = \boxed{1.65\ \mathrm{eV}}$.

Step 4. Spread: $\langle H^2\rangle = \tfrac{1}{14}(9 \times 1.0 + 4 \times 6.25 + 1 \times 16.81) = 3.63\ \mathrm{eV^2}$, so $\sigma_H = \sqrt{3.63 - 1.65^2} = \boxed{0.95\ \mathrm{eV}}$.

Step 5. A week later each $c_n$ has only rotated in phase, so every $|c_n|^2$ — the whole distribution, mean and spread — is exactly unchanged. Energy statistics are constants of the motion; only the interference between modes (positions, currents, dipoles) evolves.

Texts. Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., §2.1 · Shankar, Principles of Quantum Mechanics, Ch. 5 · Cohen-Tannoudji, Diu & Laloë, Vol. I, Ch. I & III · Townsend, A Modern Approach to Quantum Mechanics, Ch. 6. All equations and numbers are standard results. Key formulas covered include: the separation of variables yielding the TISE and temporal phase $e^{-iEt/\hbar}$; the stationarity of $|\Psi_n|^2$ and expectation values; the eigenvalue spectra — infinite well $E_n = n^2\pi^2\hbar^2/2ma^2$, oscillator $E_n = (n + \tfrac{1}{2})\hbar\omega$, delta well $E = -m\alpha^2/2\hbar^2$; and the reality and node theorems. No non-standard numbers are used. Additional results include the energy-floor argument, the complex-$E$ norm-conservation argument, and the worked numbers $7.6\times10^{14}$ rad/s, $8.3$ fs, $2.5\ \mu$m, $1.65$ eV, and $0.95$ eV.

Prerequisites. Free particle and plane waves · wave packets. Leads to. Bound states versus scattering states · the infinite and finite square wells · the harmonic oscillator · the delta-function well.