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7.5 — Operators, Measurement and the Uncertainty Principle
Chapter 7.4 solved three problems by writing an equation and applying boundary conditions. That works and it hides the structure. This chapter sets out what the structure is: what an observable is, why measurement gives definite answers from indefinite states, and why some pairs of quantities can never both be known exactly.
The uncertainty principle is the famous result here, and it is almost always explained badly — as though a clumsy microscope knocks the electron. It is not about clumsiness. It is a mathematical property of waves, and it would hold for a perfect measuring device.
Observables are operators
The rule: every measurable quantity corresponds to an operator — a mathematical instruction that acts on the wavefunction.
Chapter 7.3 already produced two:
\hat{x} = x \quad(\text{multiply by }x), \qquad \hat{p} = -i\hbar\frac{\partial}{\partial x}
and everything else is built from them by taking the classical expression and substituting.
\hat{T} = \frac{\hat{p}^2}{2m} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}
\hat{H} = \hat{T}+\hat{V} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x)
\hat{H} is the Hamiltonian, the total energy operator, named after William Rowan Hamilton's reformulation of classical mechanics. The Schrödinger equation is then just:
i\hbar\frac{\partial\psi}{\partial t} = \hat{H}\psi
Angular momentum:
\hat{L}_z = -i\hbar\frac{\partial}{\partial\phi}
which is what Chapter 7.6 needs for the hydrogen atom.
Eigenvalues are the possible results
An eigenvalue equation looks like:
\hat{A}\psi = a\psi
Read it: applying the operator to this particular function gives back the same function, multiplied by a number. The function is an eigenfunction and the number is an eigenvalue.
The measurement postulate:
The only possible results of measuring A are the eigenvalues of \hat{A}.
This is where discreteness comes from in general. The time-independent Schrödinger equation of Chapter 7.3:
\hat{H}\psi = E\psi
is an eigenvalue equation, and the allowed energies are its eigenvalues. In the box of Chapter 7.4 the boundary conditions permitted only certain ones — that is why the energy came out quantised.
Worked check. Is \psi = e^{ikx} an eigenfunction of momentum?
\hat{p}\psi = -i\hbar\frac{d}{dx}e^{ikx} = -i\hbar(ik)e^{ikx} = \hbar k\,e^{ikx}
Yes, with eigenvalue \hbar k = p. So a plane wave has a perfectly definite momentum — and it is spread over all space, which is the uncertainty principle appearing before it has been derived.
Now try position on the same function. \hat{x}e^{ikx} = xe^{ikx}, which is not a constant times e^{ikx}. Not an eigenfunction. A plane wave has no definite position, which is exactly right.
Why operators must be Hermitian
Measurement results are real numbers, so the eigenvalues must be real. The condition that guarantees this is that the operator be Hermitian:
\int\psi^*\hat{A}\phi\,dx = \int(\hat{A}\psi)^*\phi\,dx
Two properties follow, and both are used constantly.
Eigenvalues are real. Good, since you cannot measure 3+2i volts.
Eigenfunctions with different eigenvalues are orthogonal, meaning \int\psi_m^*\psi_n\,dx = 0 for m \neq n. They behave like perpendicular directions in space, which is what lets any state be decomposed uniquely into them.
Superposition and the measurement rule
A general state is a combination of eigenfunctions:
\psi = \sum_n c_n\psi_n
The full measurement postulate:
- The result will be one of the eigenvalues a_n.
- The probability of getting a_n is |c_n|^2.
- Immediately after, the state is \psi_n — the superposition is gone.
Point 3 is collapse, and it is the most disputed statement in physics. Chapter 7.8 covers what the interpretations make of it.
Extracting the coefficients uses orthogonality:
c_n = \int\psi_n^*\psi\,dx
And the average over many measurements is the expectation value from Chapter 7.3:
\langle A\rangle = \sum_n|c_n|^2a_n = \int\psi^*\hat{A}\psi\,dx
Commutators
Order matters for operators. Compute \hat{x}\hat{p}\psi and \hat{p}\hat{x}\psi and compare.
\hat{x}\hat{p}\psi = x\left(-i\hbar\frac{\partial\psi}{\partial x}\right) = -i\hbar x\frac{\partial\psi}{\partial x}
\hat{p}\hat{x}\psi = -i\hbar\frac{\partial}{\partial x}(x\psi) = -i\hbar\left(\psi + x\frac{\partial\psi}{\partial x}\right)
using the product rule. Subtract:
(\hat{x}\hat{p}-\hat{p}\hat{x})\psi = -i\hbar x\frac{\partial\psi}{\partial x}+i\hbar\psi+i\hbar x\frac{\partial\psi}{\partial x} = i\hbar\psi
The derivative terms cancel and:
\boxed{[\hat{x},\hat{p}] = \hat{x}\hat{p}-\hat{p}\hat{x} = i\hbar}
This is the canonical commutation relation, and it is arguably the single most important equation in quantum mechanics. Everything else can be built from it.
What a commutator means physically:
- If [\hat{A},\hat{B}] = 0, the operators share a complete set of eigenfunctions, so a state can have definite values of both. They are simultaneously measurable.
- If [\hat{A},\hat{B}] \neq 0, no such shared set exists. Some uncertainty in one or both is unavoidable.
Worked examples.
[\hat{x},\hat{y}] = 0 — you can know both coordinates exactly.
[\hat{p}_x,\hat{p}_y] = 0 — you can know both momentum components.
[\hat{x},\hat{p}_y] = 0 — position in x and momentum in y are compatible. Only the matching pair is restricted.
[\hat{L}_x,\hat{L}_y] = i\hbar\hat{L}_z — the components of angular momentum do not commute with each other, so you can know the total angular momentum and one component, never two components. Chapter 7.6 shows what this does to atomic orbitals, and it is why an orbital has a definite z-projection and an indefinite direction in the xy plane.
Deriving the uncertainty principle
Define the uncertainty of a measurement as the standard deviation:
\Delta A = \sqrt{\langle A^2\rangle-\langle A\rangle^2}
The general theorem, proved by Howard Robertson in 1929:
\boxed{\Delta A\,\Delta B \geq \frac{1}{2}\left|\langle[\hat{A},\hat{B}]\rangle\right|}
The proof
Take the two operators with their means subtracted, so \hat{A}' = \hat{A}-\langle A\rangle and likewise for B. Define f = \hat{A}'\psi and g = \hat{B}'\psi.
The Cauchy–Schwarz inequality — the same one from ordinary vector geometry, which says |\vec{u}\cdot\vec{v}| \leq |u||v| — applied to these functions:
\left(\int|f|^2\right)\left(\int|g|^2\right) \geq \left|\int f^*g\right|^2
The left side is (\Delta A)^2(\Delta B)^2.
For the right side, write the complex number z = \int f^*g and use |z|^2 \geq [\text{Im}(z)]^2. The imaginary part is:
\text{Im}(z) = \frac{z-z^*}{2i} = \frac{1}{2i}\langle[\hat{A},\hat{B}]\rangle
Putting it together:
(\Delta A)^2(\Delta B)^2 \geq \left|\frac{1}{2i}\langle[\hat{A},\hat{B}]\rangle\right|^2
Take square roots and the theorem follows.
Now insert [\hat{x},\hat{p}] = i\hbar:
\Delta x\,\Delta p \geq \frac{1}{2}|i\hbar| = \frac{\hbar}{2}
\boxed{\Delta x\,\Delta p \geq \frac{\hbar}{2}}
Heisenberg's uncertainty principle, derived — not postulated, not an argument about microscopes. It follows from the commutator, which follows from the operator definitions, which follow from de Broglie's relation.
What it actually says, and what it does not
It is a property of waves. Any wave theory has this. A pulse of sound short in time contains a broad spread of frequencies; a pure tone must last a long time. Sound engineers have known this forever — it is why a very short click has no pitch. In Fourier terms (Volume II, Chapter 9), a function narrow in x has a transform broad in k, and vice versa. Quantum mechanics inherits it because p = \hbar k.
It is not about disturbance. Heisenberg's original 1927 argument used a gamma-ray microscope and did involve the photon kicking the electron. That argument gives the right answer for the wrong reason, and Heisenberg himself moved away from it. The modern statement is about the preparation of the state: there is no state in which both spreads are small. No cleverness in the measuring apparatus helps, because the state itself does not have the property.
Bohr corrected Heisenberg on precisely this point, and the disagreement was sharp enough that Heisenberg is reported to have been reduced to tears.
It is not about our knowledge of a definite underlying value. Chapter 7.8's discussion of Bell's theorem rules out, experimentally, the idea that the particle "really has" a definite position and momentum that we merely fail to know.
A useful reading: a particle does not have a position and a momentum simultaneously in the sense classical physics assumes. It has a wavefunction, and position and momentum are two different questions you can ask of it, whose answers cannot both be sharp.
The minimum-uncertainty state
Equality holds only for a Gaussian wave packet:
\psi(x) = (2\pi\sigma^2)^{-1/4}e^{-x^2/4\sigma^2}
for which \Delta x\,\Delta p = \hbar/2 exactly. Every other shape does worse. This is why Gaussian beams are the ideal in optics and why coherent states of light — the closest quantum analogue of a classical wave — are Gaussian.
Consequences worth computing
Why atoms do not collapse
Chapter 7.1 noted that classical physics gives an atom a life of 10^{-11} s. The uncertainty principle explains why it does not happen.
Confine an electron to radius r. Then \Delta x \sim r, so \Delta p \sim \hbar/r, and since the average momentum is zero, the typical momentum magnitude is p \sim \hbar/r. The energy:
E = \frac{p^2}{2m}-\frac{ke^2}{r} \approx \frac{\hbar^2}{2mr^2}-\frac{ke^2}{r}
The competition is the whole story. Squeezing the electron in makes the potential term more negative as 1/r, which is favourable — but makes the kinetic term positive as 1/r^2, which grows faster. There is a minimum.
\frac{dE}{dr} = -\frac{\hbar^2}{mr^3}+\frac{ke^2}{r^2} = 0
r = \frac{\hbar^2}{mke^2} = a_0 = 5.29\times10^{-11}\ \text{m}
The Bohr radius, from the uncertainty principle alone. And the energy there:
E = \frac{\hbar^2}{2ma_0^2}-\frac{ke^2}{a_0} = -\frac{mk^2e^4}{2\hbar^2} = -13.6\ \text{eV}
The exact ground state energy of hydrogen, from a two-line estimate with no Schrödinger equation solved.
This is why matter is stable and why it takes up space. An atom has a size because squeezing an electron costs kinetic energy faster than it saves potential energy. Nothing else prevents collapse.
Zero-point energy, again
For the harmonic oscillator of Chapter 7.4, the same argument gives:
E \approx \frac{\hbar^2}{2m x^2}+\frac{1}{2}m\omega^2x^2
Minimising over x gives x^2 = \hbar/2m\omega and:
E_{\min} = \frac{1}{2}\hbar\omega
Exactly the ground state energy. The oscillator cannot sit still at the bottom because that would mean \Delta x = 0 and \Delta p = 0 together.
Nuclear sizes and why electrons are not in nuclei
Before the neutron was discovered in 1932, one model of the nucleus had electrons inside it, to account for the mass and charge. The uncertainty principle kills it.
Confine an electron to a nucleus, r \approx 5\times10^{-15} m:
p \sim \frac{\hbar}{r} = \frac{1.055\times10^{-34}}{5\times10^{-15}} = 2.1\times10^{-20}\ \text{kg m/s}
pc = (2.1\times10^{-20})(3\times10^{8}) = 6.3\times10^{-12}\ \text{J} = 39\ \text{MeV}
Since this vastly exceeds m_ec^2 = 0.511 MeV, the electron is ultra-relativistic and E \approx pc = 39 MeV.
Nuclear binding energies are about 8 MeV per nucleon. An electron with 39 MeV of kinetic energy cannot possibly be held in. So there are no electrons in nuclei, and the beta particles emitted in beta decay must be created at the moment of decay rather than stored there — which is exactly what happens (Chapter 9.2).
Energy–time uncertainty
\Delta E\,\Delta t \geq \frac{\hbar}{2}
This one is different in character, because time is not an operator in quantum mechanics — it is a parameter. The correct reading is: a state that exists for a time \Delta t has an energy spread of at least \hbar/2\Delta t.
Consequence 1: spectral lines have width. An excited state with lifetime \tau has an energy uncertainty \hbar/2\tau, so the emitted photon has a spread of frequencies. For a typical atomic lifetime of 10^{-8} s:
\Delta E = \frac{1.055\times10^{-34}}{2\times10^{-8}} = 5.3\times10^{-27}\ \text{J} = 3.3\times10^{-8}\ \text{eV}
\Delta f = \frac{\Delta E}{h} = 8\ \text{MHz}
This is the natural linewidth, and no spectroscopic technique can beat it. Metastable states with long lifetimes have very narrow lines, which is why they are used for atomic clocks — the caesium clock transition and the optical clock transitions of Chapter 6.6 are chosen precisely for their long lifetimes.
Consequence 2: unstable particle masses are fuzzy. The Z boson lives 3\times10^{-25} s and has a mass spread of 2.5 GeV out of 91 GeV. Measuring that width is how its lifetime is determined, and Chapter 8.6 explains why the width told physicists there are exactly three light neutrino species.
Consequence 3: virtual particles. Energy conservation can be violated by \Delta E for a time \hbar/\Delta E. This permits particles to appear and vanish in the vacuum, and it is what makes forces work in quantum field theory (Chapter 7.10) and what produces the Casimir effect and Hawking radiation.
Where this shows up in your life
Every atom in your body has a size determined by the balance computed above. Solidity is a quantum effect.
Semiconductor lasers and LEDs have linewidths set by the energy–time relation.
Atomic clocks are engineered around it: choose a transition with the longest possible lifetime to get the narrowest possible line.
MRI trades resolution against signal-to-noise using exactly this relation between measurement time and frequency precision.
LIGO's squeezed light (Chapter 6.10) manipulates which member of an uncertainty pair carries the noise. You cannot beat the product; you can choose where to put it.
And electron microscopes face a hard trade-off: shorter wavelength means better resolution and also more energy delivered to the sample, so biological specimens are destroyed by the very beam imaging them. The uncertainty principle sets the exchange rate between seeing a thing and damaging it.
What the next chapter fixes
The machinery is now complete: operators, eigenvalues, superposition, commutators and uncertainty. Chapter 7.6 turns it on the one problem that matters most and solves it exactly — the hydrogen atom in three dimensions. Out of it come three quantum numbers with no assumptions, the exact energy levels Bohr guessed at, the shapes of the s, p, d and f orbitals that every chemistry course draws, and an explanation of what those pictures actually show.