The Schrödinger Equation & the Born Rule

De Broglie said matter is a wave; the double slit agreed. But a wave needs a wave equation, and in 1926 Schrödinger wrote it down. This lesson builds the equation the honest way — motivated, not derived — then confronts the question the equation cannot answer: what is $\Psi$? Born's answer, that $|\Psi|^2$ is a probability density, is the interpretive heart of quantum mechanics. Like a cat in a closed box, $\Psi$ is never seen directly — only its statistical paw prints.

Learning Objectives

After this lesson you will be able to:

1. Reconstruct the motivation for the time-dependent Schrödinger equation from the free-particle plane wave and the de Broglie relations, and explain why it is a postulate, not a theorem.
2. Run the alternative construction — Helmholtz equation $\nabla^2\psi = -k^2\psi$ plus $k = p/\hbar$ — to reach the time-independent Schrödinger equation, and say which route needs a complex $\psi$ and why.
3. Argue why the wavefunction must be complex, and why linearity implies the superposition principle.
4. State the Born rule, impose normalization, and identify the conditions an admissible wavefunction must satisfy.
5. Derive the continuity equation with the probability current $j$, and prove that normalization is conserved in time.
6. Compute expectation values from $|\Psi|^2$ and explain the ensemble interpretation, connecting $\Psi(x,t) = \langle x|\Psi(t)\rangle$ to the state ket of Term 1.
7. Show that square-integrable functions form the Hilbert space $L^2$: closure under superposition, a finite inner product, and completeness; and say why plane waves lie outside it.

Intuition

By the end of Course P.3 the verdict is in: an electron sent through two slits builds an interference pattern one dot at a time (P.3.2). Something wavelike, with $\lambda = h/p$, guides where each dot may land — yet each electron arrives as a single localized dot.

So we need two things. First, a wave equation for the guiding wave $\Psi(x,t)$ — the quantum analogue of Newton's second law, turning an initial condition $\Psi(x,0)$ into $\Psi(x,t)$ for all later times. Second, a rule connecting $\Psi$ to the dots: the wave is spread out, the detection is pointlike, so the connection must be statistical. Schrödinger supplied the equation, Born the rule. Neither can be deduced from classical physics — they were inspired guesses, and a century of experiments says the guesses were right.

Theory

Constructing the equation (motivated, not derived)

Be clear up front: the Schrödinger equation cannot be derived from anything more fundamental. What follows is the plausibility argument that led to it — a postulate whose only justification is that its predictions have never failed.

Start with the simplest matter wave, a free particle of definite momentum (P.3.2):

and impose the Planck–Einstein and de Broglie relations $E = \hbar\omega$, $p = \hbar k$, plus the nonrelativistic dispersion $E = p^2/2m$. Differentiate the plane wave and watch the relations become operator statements:

For the free particle $E = p^2/2m$ forces these to be equal. To include forces, note that classically $E = p^2/2m + V(x)$; the natural generalization — the leap of faith — adds the potential as a multiplicative term:

This is the time-dependent Schrödinger equation. Every step was suggestive: we assumed the plane-wave form and the dispersion, then declared the result valid for all $\Psi$ and all $V$. That declaration is the postulate; electron diffraction, atomic spectra (Course P.6), and tunneling (Course P.5) are its validation.

A second route: de Broglie → Helmholtz → Schrödinger

The construction above goes through time. There is a shorter road that never mentions $t$ at all, and it is worth walking because it reaches the time-independent equation directly, using only classical wave machinery and one substitution.

Restrict attention to waves of a single wavelength $\lambda$ — monochromatic waves. Any such wave in one dimension satisfies the Helmholtz equation

whose solutions are exactly the wave shapes we already know work: $\sin kz$, $\cos kz$, $e^{ikz}$ (and their negative-$k$ partners). It is nothing but "curvature proportional to minus the function" — the defining property of a sinusoid of wavelength $\lambda$. In three dimensions the same statement uses the Laplacian of P.1.2:

with solutions $\sin(\mathbf k\cdot\mathbf r)$, $\cos(\mathbf k\cdot\mathbf r)$, $e^{i\mathbf k\cdot\mathbf r}$. So far this is classical optics: no quantum mechanics has entered, and $k$ is just a number describing how tightly the wave ripples.

Now insert de Broglie. His hypothesis $\lambda = h/p$ (P.3.2) converts the geometric quantity $k$ into a mechanical one:

That single substitution is the entire quantum content. Feed it back into Helmholtz and multiply by $-\hbar^2$:

The right-hand side is begging to be an energy. Divide by twice the particle's mass:

and recognize $p^2/2m$ from classical mechanics as the kinetic energy. The last step is the same leap of faith as before, in a cleaner disguise: classically the total energy splits as $E = \text{kinetic} + \text{potential}$, so

or, collecting the operator on the left,

— the time-independent Schrödinger equation, valid (we now postulate) for a particle of any mass $m$ in any potential $V(\mathbf r)$. Its solutions and their dynamics are the whole subject of P.4.2, and Courses P.5–P.6 are nothing but this equation with different $V$.

flowchart TD
    A["Free plane wave in space and time"] --> B["E = ℏω, p = ℏk<br/>plus dispersion E = p²/2m"]
    B --> C["Time-dependent SE<br/>iℏ ∂Ψ/∂t = ĤΨ"]
    D["Monochromatic wave, one λ"] --> E["Helmholtz: ∇²ψ = −k²ψ"]
    E --> F["de Broglie: k = p/ℏ<br/>so −ℏ²∇²ψ = p²ψ"]
    F --> G["÷2m, then p²/2m = E − V"]
    G --> H["Time-independent SE<br/>Ĥψ = Eψ"]
    C --> |"separate variables (P.4.2)"| H
    H --> |"× exp(−iEt/ℏ)"| C

Two honest observations about this second route. First, it is still a postulate: the substitution $k = p/\hbar$ was justified only for a free particle of definite momentum, and writing $E - V(\mathbf r)$ with a position-dependent $V$ silently assumes a local wavelength that varies from place to place — plausible, unproven, and correct. Second, notice what is missing: no $i$, no time derivative. The Helmholtz route never forces $\psi$ to be complex, because $\sin$ and $\cos$ solve it perfectly well. Complexity is forced only by the demand for time evolution that is first order in $t$ — the argument of the next section. The two routes are complementary: one gives the dynamics and pays for it with $i$; the other gives the spectrum for free.

Why $\Psi$ must be complex

The $i$ is forced, not decorative. Suppose we wanted a real equation, first order in time (first order, so $\Psi(x,0)$ alone fixes the future): $\partial_t\Psi = \alpha\,\partial_x^2\Psi$ with real $\alpha$. Try a traveling wave $\Psi = \cos(kx - \omega t)$: the left side is $\omega\sin(kx-\omega t)$, the right is $-\alpha k^2\cos(kx-\omega t)$, and sine and cosine of the same argument are linearly independent — no choice of constants works. A real first-order-in-time equation cannot carry a traveling wave (with real $\alpha$ it is the diffusion equation: spreading and decay, never propagation). The classical wave equation of P.1.1 escapes by being second order in time — but then $E = \hbar\omega$ would require $E^2 \propto p^2$, the wrong (relativistic) dispersion. The only way to be first order in time and propagate is to let $\Psi$ be complex, so $e^{i(kx-\omega t)}$ — packaging $\cos$ and $\sin$ together — is a solution. Complex amplitudes are structural, just as in 1.1.1 The State Postulate.

Linearity and superposition

The equation is linear: if $\Psi_1, \Psi_2$ are solutions, so is $c_1\Psi_1 + c_2\Psi_2$ for any complex $c_1, c_2$ (every term acts linearly; substitute and check). This is the superposition principle — the same structure that made classical waves interfere in P.1.1, and the same principle elevated to Postulate 1 in 1.1.1. Matter-wave interference, and every quantum algorithm, lives in this linearity.

The Born rule

The equation evolves $\Psi$; it does not say what $\Psi$ means. Born's statistical interpretation (1926) supplies the link to experiment:

Born rule. $|\Psi(x,t)|^2\,dx$ is the probability of finding the particle in $[x, x+dx]$ if position is measured at time $t$: $\rho = |\Psi|^2$ is a probability density, and $P(a<x<b) = \int_a^b |\Psi(x,t)|^2\,dx$.

Total probability must be one, which imposes normalization:

By linearity, $A\Psi$ solves the equation whenever $\Psi$ does; normalization fixes $|A|$ (the phase of $A$ is the unobservable global phase of 1.1.1). This selects the admissible wavefunctions:

• Square-integrable: $\int|\Psi|^2dx < \infty$; in particular $\Psi \to 0$ as $x \to \pm\infty$.
• Continuous everywhere, with $\partial\Psi/\partial x$ continuous wherever $V$ is finite (integrating the equation across a point shows a kink in $\Psi'$ needs an infinite $V$, as in the idealized wells of P.5.2).
• Non-normalizable solutions like $e^{ikx}$ are not physical states — they are idealized basis functions, handled honestly in P.4.3 and P.5.1.

Conservation of probability: the continuity equation

Normalizing at $t=0$ would be useless if evolution destroyed it. It doesn't. Take the Schrödinger equation and its complex conjugate (with $V$ real):

Differentiate the density $\rho = \Psi^*\Psi$; the potential terms cancel identically:

where the last step is checked by expanding the derivative (the $\Psi'^*\Psi'$ cross terms cancel). Hence the continuity equation

probability is locally conserved, flowing with current density $j$ like charge in electromagnetism. Integrating over all space,

since $\Psi \to 0$ at $\pm\infty$ kills $j$ there. Normalize once, stay normalized. In Term 1 language: evolution is unitary — this is the wave-mechanics face of 1.1.3 The Evolution Postulate.

Expectation values and the ensemble interpretation

Since $|\Psi|^2$ is a probability density, averages follow ordinary probability theory (0.2.2):

Read $\langle x\rangle$ carefully: it is not the average of repeated measurements on one particle — the first measurement collapses the state, and re-measuring just repeats the same value. It is the average over an ensemble: many systems, all identically prepared in $\Psi$, each measured once. This is the operational reading formalized in 1.1.2 Observables & the Measurement Postulate.

Wavefunctions form a Hilbert space

Square-integrability is more than a technical condition. It gives the admissible wavefunctions exactly the structure Term 1 takes as its starting point. Let $L^2(\mathbb R)$ be the set of complex functions $f$ with $\int_{-\infty}^{\infty}|f|^2\,dx < \infty$.

It is a vector space. Multiplying by a constant keeps a function in $L^2$, and so does adding two of them, because of the pointwise inequality $|f + g|^2 \le 2|f|^2 + 2|g|^2$ (expand $|f - g|^2 \ge 0$ to get $2\,\mathrm{Re}(f^*g) \le |f|^2 + |g|^2$). Superpositions of admissible wavefunctions are admissible: the linearity of the equation and the structure of the state space fit together.

It carries an inner product.

This is finite for every $f, g \in L^2$. The product $f^*g$ need not itself be square-integrable, but it is integrable, since $|f^*g| \le \tfrac12\big(|f|^2 + |g|^2\big)$; more sharply, $|\langle f, g\rangle| \le \lVert f\rVert\,\lVert g\rVert$ with $\lVert f\rVert = \sqrt{\langle f, f\rangle}$, the Cauchy–Schwarz inequality of 0.1.2. It has the three defining properties: conjugate symmetry $\langle f, g\rangle = \langle g, f\rangle^*$; linearity in the second slot, $\langle f, \alpha g_1 + \beta g_2\rangle = \alpha\langle f, g_1\rangle + \beta\langle f, g_2\rangle$ (so it is antilinear in the first, $\langle\alpha f, g\rangle = \alpha^*\langle f, g\rangle$); and positivity, $\langle f, f\rangle \ge 0$ with equality only for $f = 0$. (Strictly, "$f = 0$ except on a set of zero total length": functions that differ only there are identified, since no measurement can tell them apart.)

It is complete. Every Cauchy sequence of square-integrable functions converges in norm to a square-integrable function (the Riesz–Fischer theorem). A complete inner-product space is a Hilbert space, and $L^2$ is the prototype.

Everything in this lesson is now a statement in that space. Normalization is $\lVert\Psi\rVert = 1$, conserved because evolution preserves the norm; the overlaps $\langle\psi_n, \Psi\rangle$ that P.4.2 uses to expand a state in energy eigenfunctions are inner products; in Dirac notation (0.1.3) $\langle f, g\rangle$ is written $\langle f|g\rangle$. The idealizations live outside the space: neither the plane wave $e^{ikx}$ nor the position eigenfunction $\delta(x - x_0)$ is square-integrable. They are indispensable as limits and as labels for continuous expansions, where sums become integrals, but making them rigorous needs an enlarged ("rigged") Hilbert space, beyond this program. For the finite-dimensional systems of Term 1 none of these subtleties arise: every $\mathbb C^n$ is a Hilbert space as it stands.

The bridge to Term 1: $\Psi(x,t) = \langle x|\Psi(t)\rangle$

Everything in this term connects to the main program through one identity. Term 1 describes the state as an abstract ket $\lvert\Psi(t)\rangle$ (1.1.1); the wavefunction is its expansion in the continuum basis of position states:

The wavefunction is the position representation of the ket — the continuous analogue of listing a qubit's components $\langle 0|\psi\rangle, \langle 1|\psi\rangle$. Normalization is $\langle\Psi|\Psi\rangle = 1$; the position Born rule is the general Born rule in the position basis. Schrödinger's wave mechanics and Heisenberg's matrix mechanics looked like rival theories in 1926; they are the same theory in different bases — a change of basis in the sense of 0.1.4. Momentum space gives a third face of the same object (P.4.3).

Caution. $\Psi$ itself is not measurable and is not a physical wave in ordinary space. No instrument reads off $\Psi(x,t)$; only Born-rule quantities — $|\Psi|^2$, probabilities, expectation values — touch experiment. And for $N$ particles, $\Psi(\mathbf r_1,\dots,\mathbf r_N,t)$ lives in $3N$-dimensional configuration space, not in the 3D space around you: it cannot be a ripple in any ordinary medium. Treating $\Psi$ as a classical field in real space is the most common misreading of wave mechanics.

Worked Examples

Example 1 — Normalizing a Gaussian and using the Born rule

Let $\Psi(x,0) = A\,e^{-x^2/4a^2}$, $a>0$. Normalize, then find $P(-a<x<a)$.

Using $\int_{-\infty}^{\infty} e^{-\lambda x^2}dx = \sqrt{\pi/\lambda}$:

Then $|\Psi|^2 = \frac{1}{\sqrt{2\pi}\,a}e^{-x^2/2a^2}$ — a normal density with standard deviation $a$ — so $P(-a<x<a) = \mathrm{erf}(1/\sqrt2) \approx 0.6827$, the familiar "one sigma" mass. By symmetry $\langle x\rangle = 0$, and $\langle x^2\rangle = a^2$.

Example 2 — Probability current of a plane wave and of a real wavefunction

(a) For $\Psi = A\,e^{i(kx-\omega t)}$ (idealized, non-normalizable):

density times the classical velocity $p/m$ — a steady rightward flow, consistent with $\partial_t\rho = 0 = \partial_x j$.

(b) If $\Psi = \psi(x)e^{-iEt/\hbar}$ with $\psi$ real, then $\Psi^*\partial_x\Psi = \psi\psi'$ is real and $j = 0$ everywhere: real-up-to-a-phase wavefunctions carry no probability flow. Bound states are of this type — nothing flows in a stationary state (P.4.2).

Hands-on (Python)

import numpy as np
import matplotlib.pyplot as plt

# --- Normalize a packet, apply the Born rule, then SAMPLE from it ------------
x = np.linspace(-20, 20, 4001); dx = x[1] - x[0]
psi = np.exp(-x**2 / 4) * np.exp(2j * x)      # unnormalized Gaussian packet, k0 = 2

psi /= np.sqrt(np.trapz(np.abs(psi)**2, x))   # normalize with np.trapz
print(np.trapz(np.abs(psi)**2, x))            # 1.0 (normalized)

a, b = -1.0, 1.0                              # Born rule: P(a < x < b)
m_ab = (x >= a) & (x <= b)
print(np.trapz(np.abs(psi[m_ab])**2, x[m_ab]))  # ≈ 0.6827 (1σ Gaussian mass)

# Born rule as a sampling recipe: draw synthetic position measurements
rho = np.abs(psi)**2
samples = np.random.default_rng(0).choice(x, size=20000, p=rho*dx/np.sum(rho*dx))
plt.hist(samples, bins=80, density=True, alpha=0.5, label="measured positions")
plt.plot(x, rho, "k", label=r"$|\Psi|^2$"); plt.xlim(-6, 6); plt.legend(); plt.show()
# Expected: histogram hugs the density — dots build the wave's statistics.

from scipy.linalg import solve_banded

# --- Crank–Nicolson: Gaussian in a harmonic trap; the norm stays 1 -----------
# Numerical values hbar = m = 1 (physics keeps ħ explicit; arrays don't care).
hbar, m, w = 1.0, 1.0, 1.0
xg = np.linspace(-10, 10, 1200); dxg = xg[1]-xg[0]; dt = 0.002
V = 0.5 * m * w**2 * xg**2
psi_t = ((1/np.pi)**0.25 * np.exp(-(xg-2.0)**2/2)).astype(complex)  # displaced Gaussian

# Tridiagonal H = -(ħ²/2m)d²/dx² + V; CN step: (1 + i dt H/2ħ) ψ' = (1 - i dt H/2ħ) ψ
main = hbar**2/(m*dxg**2) + V
off  = -hbar**2/(2*m*dxg**2) * np.ones(len(xg)-1)
ab = np.zeros((3, len(xg)), dtype=complex)    # banded form of (1 + i dt H/2ħ)
ab[0,1:], ab[1,:], ab[2,:-1] = (1j*dt/(2*hbar)*off, 1 + 1j*dt/(2*hbar)*main,
                                1j*dt/(2*hbar)*off)
norm0 = np.trapz(np.abs(psi_t)**2, xg)
for _ in range(2000):                         # evolve to t = 4
    Hpsi = (main*psi_t + np.concatenate(([0], off*psi_t[:-1]))
                       + np.concatenate((off*psi_t[1:], [0])))
    psi_t = solve_banded((1, 1), ab, psi_t - 1j*dt/(2*hbar)*Hpsi)
print(norm0, np.trapz(np.abs(psi_t)**2, xg))
# Expected: 1.0 1.0000000000000... — Crank–Nicolson is unitary, so the norm is
# conserved to machine precision, the discrete shadow of d/dt ∫|Ψ|²dx = 0.

Exercises

E1 (easy). Normalize $\Psi(x,0) = A e^{-|x|/a}$ ($a>0$) and compute $P(x>a)$.

E2 (easy). Show directly that if $\Psi_1, \Psi_2$ solve the time-dependent Schrödinger equation, so does $c_1\Psi_1 + c_2\Psi_2$. Which property of the equation is responsible?

E3 (medium). Redo the conservation calculation for a complex potential $V = V_0 - i\Gamma$ ($V_0, \Gamma$ real, $\Gamma>0$) and show $\frac{d}{dt}\int|\Psi|^2dx = -\frac{2\Gamma}{\hbar}\int|\Psi|^2dx$, so the norm decays as $e^{-2\Gamma t/\hbar}$. What could this model?

E4 (medium). For $\Psi = \big(Ae^{ikx} + Be^{-ikx}\big)e^{-i\hbar k^2t/2m}$, compute $j$ and interpret each term.

E5 (hard). Prove $\frac{d\langle x\rangle}{dt} = \int_{-\infty}^{\infty} j\,dx$ (use the continuity equation and integrate by parts). Evaluate it for the plane wave of Example 2(a) on a large box of length $L$ with $|A|^2 = 1/L$, and interpret.

Checkpoint

1. In what precise sense is the Schrödinger equation "not derived"? What inputs motivated its form?
2. Starting from the Helmholtz equation $\nabla^2\psi = -k^2\psi$, use de Broglie to reach the time-independent Schrödinger equation. Where exactly does the postulate enter, and why does this route never force $\psi$ to be complex?
3. Why can't the wavefunction be a real function obeying a first-order-in-time equation?
4. State the Born rule and the admissibility conditions on $\Psi$.
5. Write the continuity equation and the probability current, and explain why normalizing once suffices forever.
6. What does $\langle x\rangle$ mean operationally — and what does it not mean?
7. Why is the inner product of two square-integrable functions always finite, and what makes $L^2$ a Hilbert space? Is $e^{ikx}$ in it?

Further Reading

• [Gri] Griffiths & Schroeter, §1.1–1.4 — the equation, the statistical interpretation, normalization.
• [Gri] Griffiths & Schroeter, §1.5 — expectation values and conservation of probability.
• [ER] Eisberg & Resnick, Ch. 5 — the plausibility construction of the Schrödinger equation, in loving detail.
• [Sha] Shankar, §4.1–4.2 — the postulates of wave mechanics, bridging to the abstract formalism.
• [NC] Nielsen & Chuang, §2.2.3 — the Born rule in its general, basis-independent form.

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