# 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](../c03-wave-particle-duality/02-matter-as-waves.md)). 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](../c03-wave-particle-duality/02-matter-as-waves.md)):
$$\Psi(x,t) = e^{i(kx - \omega t)},$$
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:
$$i\hbar\frac{\partial\Psi}{\partial t} = i\hbar(-i\omega)\Psi = \hbar\omega\,\Psi = E\,\Psi, \qquad
-\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} = -\frac{\hbar^2}{2m}(ik)^2\Psi = \frac{p^2}{2m}\,\Psi.$$
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:
$$\boxed{\;i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V(x)\,\Psi\;}$$
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**
$$
\frac{d^2\psi}{dz^2} = -k^2\psi, \qquad k = \frac{2\pi}{\lambda},
$$
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](../c01-theoretical-mechanics/02-wave-propagation-diffraction.md):
$$
\nabla^2\psi \equiv \frac{\partial^2\psi}{\partial x^2} + \frac{\partial^2\psi}{\partial y^2} + \frac{\partial^2\psi}{\partial z^2} = -k^2\psi ,
$$
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](../c03-wave-particle-duality/02-matter-as-waves.md)) converts the *geometric* quantity $k$
into a *mechanical* one:
$$
k = \frac{2\pi}{\lambda} = \frac{2\pi p}{h} = \frac{p}{\hbar},
\qquad \text{where } \boxed{\ \hbar \equiv \frac{h}{2\pi}\ }
\qquad\Longrightarrow\qquad k^2 = \frac{p^2}{\hbar^2}.
$$
That single substitution is the entire quantum content. Feed it back into Helmholtz and multiply by
$-\hbar^2$:
$$
\nabla^2\psi = -\frac{p^2}{\hbar^2}\,\psi
\qquad\Longleftrightarrow\qquad
-\hbar^2\nabla^2\psi = p^2\psi .
$$
The right-hand side is begging to be an energy. Divide by twice the particle's mass:
$$
-\frac{\hbar^2}{2m}\nabla^2\psi = \frac{p^2}{2m}\,\psi ,
$$
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
$$
\frac{p^2}{2m} = E - V(\mathbf r)
\qquad\Longrightarrow\qquad
-\frac{\hbar^2}{2m}\nabla^2\psi = \big(E - V(\mathbf r)\big)\psi ,
$$
or, collecting the operator on the left,
$$
\boxed{\;\left(-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf r)\right)\psi = E\,\psi\;}
$$
— 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](02-stationary-states-superposition.md), and Courses P.5–P.6 are nothing but this equation with
different $V$.
```mermaid
flowchart TD
A["Free plane wave in space and time"] --> B["E = ℏω, p = ℏk
plus dispersion E = p²/2m"]
B --> C["Time-dependent SE
iℏ ∂Ψ/∂t = ĤΨ"]
D["Monochromatic wave, one λ"] --> E["Helmholtz: ∇²ψ = −k²ψ"]
E --> F["de Broglie: k = p/ℏ
so −ℏ²∇²ψ = p²ψ"]
F --> G["÷2m, then p²/2m = E − V"]
G --> H["Time-independent SE
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](../c01-theoretical-mechanics/01-waves-and-the-wave-equation.md) 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](../../term-1-quantum-mechanics/c01-postulates/01-state-postulate.md).
### 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](../c01-theoretical-mechanics/01-waves-and-the-wave-equation.md), and the same principle
elevated to Postulate 1 in [1.1.1](../../term-1-quantum-mechanics/c01-postulates/01-state-postulate.md).
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 **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 $P(-a= a) & (x <= b)
print(np.trapz(np.abs(psi[m_ab])**2, x[m_ab])) # ≈ 0.6827 (1σ Gaussian mass)
# Born rule as a sampling recipe: draw synthetic position measurements
rho = np.abs(psi)**2
samples = np.random.default_rng(0).choice(x, size=20000, p=rho*dx/np.sum(rho*dx))
plt.hist(samples, bins=80, density=True, alpha=0.5, label="measured positions")
plt.plot(x, rho, "k", label=r"$|\Psi|^2$"); plt.xlim(-6, 6); plt.legend(); plt.show()
# Expected: histogram hugs the density — dots build the wave's statistics.
```
```python
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 $P(x>a)$.
Solution
$1 = |A|^2\int_{-\infty}^{\infty}e^{-2|x|/a}dx = |A|^2\cdot 2\cdot\tfrac a2 = |A|^2a$, so
$A = 1/\sqrt a$. Then $P(x>a) = \tfrac1a\int_a^\infty e^{-2x/a}dx = \tfrac12 e^{-2} \approx 0.0677$.
Solution
With $\hat H = -\frac{\hbar^2}{2m}\partial_x^2 + V$:
$i\hbar\partial_t(c_1\Psi_1 + c_2\Psi_2) = c_1 i\hbar\partial_t\Psi_1 + c_2 i\hbar\partial_t\Psi_2 = c_1\hat H\Psi_1 + c_2\hat H\Psi_2 = \hat H(c_1\Psi_1 + c_2\Psi_2)$,
because differentiation and
multiplication by $V(x)$ are linear. The property is **linearity** — the superposition principle.
Solution
The potential terms in $\partial_t\rho$ no longer cancel:
$-\tfrac{i}{\hbar}(V - V^*)|\Psi|^2 = -\tfrac{2\Gamma}{\hbar}|\Psi|^2$. The current term still
integrates to zero, so $\frac{d}{dt}\int|\Psi|^2dx = -\frac{2\Gamma}{\hbar}\int|\Psi|^2dx$, giving
exponential decay with lifetime $\tau = \hbar/2\Gamma$. It models **absorption or decay** (an
unstable particle, a beam eaten by a detector). Real potentials — Hermitian Hamiltonians — are
exactly what probability conservation requires.
Solution
With $\psi = Ae^{ikx} + Be^{-ikx}$:
$\psi^*\partial_x\psi = ik\big(|A|^2 - |B|^2\big) + ik\big(A^*Be^{-2ikx} - AB^*e^{2ikx}\big)$. The
second bracket is $2i\,\mathrm{Im}(A^*Be^{-2ikx})$, purely imaginary, so times $ik$ it is real and
drops from $\mathrm{Im}$. Hence $j = \frac{\hbar k}{m}(|A|^2 - |B|^2)$: rightward flow minus leftward
flow, with no interference in the *net* current — the calculation behind reflection and transmission
coefficients in [P.5.2](../c05-bound-states/02-step-well-barrier.md).
Solution
$\frac{d\langle x\rangle}{dt} = \int x\,\partial_t\rho\,dx = -\int x\,\partial_xj\,dx = -[xj]_{-\infty}^{\infty} + \int j\,dx = \int j\,dx$,
the boundary term vanishing because
square-integrability makes $j \to 0$ faster than $1/x$. For the boxed plane wave,
$\int_0^L j\,dx = \frac1L\cdot\frac{\hbar k}{m}\cdot L = \frac{\hbar k}{m} = \frac pm$: the mean
position moves at the classical velocity — foreshadowing the group velocity and Ehrenfest's theorem
of [P.4.3](03-matter-waves-uncertainty-momentum.md).
Answers
1. It is a postulate, not deducible from classical physics. Its form was motivated by requiring the
plane wave $e^{i(kx-\omega t)}$ with $E = \hbar\omega$, $p = \hbar k$ to satisfy $E = p^2/2m$,
then generalizing $E \to p^2/2m + V$ and promoting the result to all $\Psi$, all $V$. Experiment
is its only justification.
2. $k = 2\pi/\lambda$ with $\lambda = h/p$ gives $k = p/\hbar$, so $\nabla^2\psi = -(p^2/\hbar^2)\psi$,
i.e. $-\hbar^2\nabla^2\psi = p^2\psi$; dividing by $2m$ makes the right side the kinetic energy,
and substituting $p^2/2m = E - V(\mathbf r)$ gives
$\big({-}\tfrac{\hbar^2}{2m}\nabla^2 + V\big)\psi = E\psi$. The postulate enters at that last
substitution: $k = p/\hbar$ was justified only for a free particle of definite momentum, yet we
apply it where $V$ — and hence the local wavelength — varies with position, then declare the
result valid for every $V$ and every $\psi$. Nothing here needs complex numbers because
$\sin$ and $\cos$ solve Helmholtz perfectly well; only the demand for first-order-in-time
*evolution* forces $\Psi \in \mathbb{C}$ (next question).
3. A single real function first order in time gives the diffusion equation, which spreads but cannot
propagate; traveling waves need $\cos$ and $\sin$ exchanging roles, i.e. a complex exponential.
Hence $\Psi \in \mathbb{C}$ is forced.
4. $|\Psi(x,t)|^2dx$ = probability of finding the particle in $[x,x+dx]$. Admissible: square-
integrable ($\Psi \to 0$ at $\pm\infty$), continuous, $\Psi'$ continuous wherever $V$ is finite.
5. $\partial_t\rho + \partial_xj = 0$ with $j = \frac{\hbar}{m}\mathrm{Im}(\Psi^*\partial_x\Psi)$.
Integrating over all space turns the current term into a vanishing boundary term, so
$\frac{d}{dt}\int|\Psi|^2dx = 0$: the norm is a constant of the motion.
6. The mean of single position measurements over an *ensemble* of identically prepared systems — not
the average of repeated measurements on one system, which the first measurement's collapse would
spoil.
7. Pointwise, $|f^*g| \le \tfrac12(|f|^2 + |g|^2)$, so $f^*g$ is integrable (Cauchy–Schwarz gives
the sharper bound $\lVert f\rVert\lVert g\rVert$). $L^2$ is a vector space with this inner
product and is complete (Riesz–Fischer), which is the definition of a Hilbert space. $e^{ikx}$ is
not in it: $\int|e^{ikx}|^2dx = \int dx$ diverges.