Postulates of Quantum Mechanics, Wave Functions and the Born Interpretation

The previous chapter ended with de Broglie's claim that a particle has a wave. This chapter turns that claim into a working framework: a particle is described by a complex wave function $\Psi(x,t)$, its modulus squared is a probability density, and every physical prediction is a statistical statement extracted from $\Psi$ by a fixed set of rules. Nothing here is derived from classical mechanics; the postulates are justified only by the fact that everything built on them agrees with experiment. We work in one dimension throughout, which contains every conceptual point without the bookkeeping of three.

The organising idea is the statistical interpretation, due to Born. Quantum mechanics does not predict where a particle is; it predicts the probability of each possible outcome of a measurement. Between measurements the state evolves smoothly and deterministically (by the Schrödinger equation, Chapter 3); a measurement is a separate kind of process that returns one eigenvalue at random, with probabilities set by $\Psi$, and leaves the system in the corresponding eigenstate. This chapter states the postulates, defines normalisation and probability current, and shows how expectation values and the momentum operator follow.

How to read this chapter. The three things to carry forward are: (i) $|\Psi|^2$ is a probability density, so $\Psi$ must be normalised and normalisation must be preserved in time; (ii) observables are represented by operators, and $\langle Q\rangle = \int \Psi^* \hat Q\,\Psi\,dx$; (iii) the momentum operator is $\hat p = -i\hbar\,\partial/\partial x$, which is what makes $[\hat x,\hat p]=i\hbar$ (Chapter 5) and the uncertainty principle unavoidable.

1. The Wave Function and the Born Rule

Postulate 1 (state). The state of a particle at time $t$ is a complex-valued function $\Psi(x,t)$, square-integrable over all space.

Postulate 2 (Born rule). If the position of the particle is measured at time $t$, the probability of finding it between $x$ and $x+dx$ is

The quantity $\rho(x,t) = |\Psi|^2$ is the position probability density. Only $|\Psi|^2$ has direct physical meaning; $\Psi$ itself is a probability amplitude, and an overall constant phase $e^{i\alpha}$ multiplying $\Psi$ changes no prediction. Amplitudes, not probabilities, are what superpose — this is the origin of interference.

Born's rule replaces the classical idea of a definite trajectory. Before the measurement the particle does not have a position that $\Psi$ merely fails to reveal; the indefiniteness is a property of the state, not of our knowledge. This is the orthodox (Copenhagen) position; experiments rule out local hidden-variable alternatives, though other interpretations reproduce the same predictions.

2. Normalisation and Its Preservation

For $|\Psi|^2$ to be a probability density it must integrate to one:

A solution of the Schrödinger equation can always be scaled to satisfy this, provided the integral is finite (the function is square-integrable, $\Psi \in L^2$). Functions that are not normalisable — a pure plane wave $e^{ikx}$, for instance — are not physical states on their own, though they are useful building blocks.

Normalisation is preserved in time. If $\Psi$ is normalised at $t=0$, it stays normalised. Using the Schrödinger equation one shows

because a normalisable $\Psi$ and its derivative vanish at infinity. If the equation did not have this property it could not support a probability interpretation at all — so this is a consistency check on the dynamics, not an extra assumption.

3. The Probability Current and the Continuity Equation

Define the probability current

Then probability obeys a local conservation law, exactly like charge in electrodynamics:

Integrating over a finite interval $[a,b]$ gives $\dfrac{d}{dt}\displaystyle\int_a^b \rho\,dx = j(a,t) - j(b,t)$: the probability in a region changes only by flowing across its boundaries. Probability is not created or destroyed, only transported. For a stationary state $j$ is independent of $x$; for a real $\Psi$ (bound states can be chosen real) $j = 0$ everywhere — the particle is "standing still" in the probabilistic sense.

4. Operators and Expectation Values

Postulate 3 (observables). Every measurable quantity $Q(x,p)$ is represented by a linear Hermitian operator $\hat Q$, obtained from the classical expression by the substitutions

Postulate 4 (expectation value). The mean of many measurements of $Q$ on identically prepared systems in state $\Psi$ is

For position this reduces to the obvious $\langle x\rangle = \int x\,|\Psi|^2\,dx$. For momentum it gives $\langle p\rangle = \int \Psi^*(-i\hbar\,\partial_x)\Psi\,dx$, which is not an average of any pre-existing values — there is no "momentum at $x$". The operator ordering matters: $\widehat{xp}$ is ambiguous classically, and the correct quantum operator for such products is the symmetrised (Hermitian) form $\tfrac12(\hat x\hat p + \hat p\hat x)$.

Worked example — expectation values in the ground state of a box. For $\Psi(x) = \sqrt{2/a}\,\sin(\pi x/a)$ on $[0,a]$: by symmetry $\langle x\rangle = a/2$; a short integration gives $\langle x^2\rangle = a^2\left(\tfrac13 - \tfrac{1}{2\pi^2}\right)$; and $\langle p\rangle = 0$ (real $\Psi$), while $\langle p^2\rangle = (\pi\hbar/a)^2$, consistent with $E = \langle p^2\rangle/2m = \pi^2\hbar^2/2ma^2$.

5. Why the Momentum Operator Is $-i\hbar\,\partial_x$

The form of $\hat p$ is fixed by requiring $\dfrac{d\langle x\rangle}{dt} = \dfrac{\langle p\rangle}{m}$, the quantum echo of $v = p/m$. Differentiating $\langle x\rangle = \int x|\Psi|^2\,dx$ and using the Schrödinger equation to replace $\partial_t\Psi$:

(the third step is integration by parts). The last integral is what we must call $\langle p\rangle/m$, which identifies $\hat p = -i\hbar\,\partial_x$. Momentum eigenfunctions are then $e^{ipx/\hbar}$: $\hat p\,e^{ipx/\hbar} = p\,e^{ipx/\hbar}$. Expanding a state in these is the Fourier transform, and $|\tilde\Psi(p)|^2$ is the probability density in momentum space.

6. Superposition and Measurement

Postulate 5 (superposition). If $\Psi_1$ and $\Psi_2$ are possible states, so is any linear combination $c_1\Psi_1 + c_2\Psi_2$. The space of states is a complex vector space (a Hilbert space, Chapter 4).

Postulate 6 (measurement / collapse). A measurement of an observable $\hat Q$ yields one of its eigenvalues $q_n$. If $\Psi = \sum_n c_n \psi_n$ with $\hat Q\psi_n = q_n\psi_n$ and the $\psi_n$ orthonormal, then

and immediately after the measurement the state is $\psi_n$ (collapse). Between measurements the evolution is the smooth, unitary Schrödinger flow; the collapse is the discontinuous, probabilistic part. The two together are the full dynamics of the orthodox theory. Note $\sum_n |c_n|^2 = 1$ is exactly normalisation expressed in the $\{\psi_n\}$ basis (Parseval's theorem).

Worked example — a two-state superposition. Let $\Psi = \tfrac{1}{\sqrt5}\psi_1 + \tfrac{2}{\sqrt5}\psi_2$ with energies $E_1,E_2$. An energy measurement gives $E_1$ with probability $1/5$ and $E_2$ with probability $4/5$; the mean is $\langle H\rangle = \tfrac15 E_1 + \tfrac45 E_2$. After a result of $E_2$, an immediate repeat measurement gives $E_2$ with certainty — the state has collapsed to $\psi_2$.

7. Admissibility Conditions on $\Psi$

For $\Psi$ to represent a physical state and for the operators above to act sensibly:

1. $\Psi$ is single-valued — one amplitude per point.
2. $\Psi$ is square-integrable — $\int|\Psi|^2 < \infty$, so it can be normalised.
3. $\Psi$ is continuous, and $\partial_x\Psi$ is continuous except where the potential has an infinite jump (an infinite wall, or a $\delta$-function, where $\partial_x\Psi$ may step).
4. $\Psi \to 0$ as $|x|\to\infty$ fast enough for (2); its derivative too.

These are not extra postulates but the minimal regularity that makes the probability interpretation and the Schrödinger equation consistent. A "wave function" violating them (e.g. $1/x$ near the origin, or a step discontinuity in a smooth potential) is unphysical.

8. What the Formalism Does Not Say

• It does not give trajectories. $\langle x\rangle(t)$ moves smoothly (Ehrenfest, Chapter 5), but individual runs give scattered results.
• It does not explain why a particular outcome occurs on a particular run — only the probabilities. This is the measurement problem, still debated; the orthodox theory simply posits collapse.
• It is not a hidden-variable theory in disguise: Bell's theorem (beyond this course) rules out local hidden variables reproducing all quantum predictions.
• $|\Psi|^2$ is a probability density in configuration space, not a physical matter density; the electron does not smear out into a charge cloud, though time-averaged it can be treated as one for some purposes.

9. Common Pitfalls

• Treating $\Psi$ as observable. Only $|\Psi|^2$ (and relative phases in superpositions) are physical; a global phase is irrelevant.
• Forgetting to normalise before computing probabilities. $|c_n|^2$ are probabilities only if both $\Psi$ and the $\psi_n$ are normalised.
• Writing $\langle p\rangle = \int p\,|\Psi(x)|^2\,dx$. There is no $p$-value at each $x$; use $\langle p\rangle = \int \Psi^*(-i\hbar\partial_x)\Psi\,dx$, or work in momentum space with $|\tilde\Psi(p)|^2$.
• Putting the operator in the wrong place. $\langle Q\rangle = \int \Psi^* (\hat Q\Psi)\,dx$; $\hat Q$ acts on $\Psi$, then multiply by $\Psi^*$, then integrate.
• Demanding $\partial_x\Psi$ be continuous at an infinite wall. Only $\Psi$ is continuous there; its slope is allowed to jump because the potential is infinite.
• Using a plane wave as a state. $e^{ikx}$ is not square-integrable; physical states are wave packets (normalisable superpositions of plane waves).
• Confusing collapse with time evolution. The Schrödinger equation is deterministic and unitary; the random jump happens only at measurement.

Summary

• State: a normalised complex $\Psi(x,t)$; Born rule: $P(x)\,dx = |\Psi|^2\,dx$. Global phase is unphysical; amplitudes superpose.
• Normalisation $\int|\Psi|^2\,dx = 1$ is preserved by the dynamics, guaranteed by the continuity equation $\partial_t\rho + \partial_x j = 0$ with $j = \dfrac{\hbar}{m}\mathrm{Im}(\Psi^*\partial_x\Psi)$.
• Observables are Hermitian operators: $\hat x = x$, $\hat p = -i\hbar\,\partial_x$; expectation value $\langle Q\rangle = \int \Psi^*\hat Q\Psi\,dx$.
• $\hat p = -i\hbar\,\partial_x$ is forced by $d\langle x\rangle/dt = \langle p\rangle/m$; momentum eigenstates are $e^{ipx/\hbar}$ and $|\tilde\Psi(p)|^2$ is the momentum density.
• Superposition $\Psi = \sum_n c_n\psi_n$; measurement returns $q_n$ with probability $|c_n|^2$, $c_n = \int\psi_n^*\Psi\,dx$, then collapses the state to $\psi_n$.
• Admissible $\Psi$: single-valued, square-integrable, continuous, with continuous slope except at infinite potential features; $\Psi\to0$ at infinity.
• The formalism is statistical: no trajectories, no explanation of individual outcomes, no local hidden variables.