1.7 Where the uncertainty principle came from

The previous section concluded that “which slit the electron went through” has no answer until the measurement is arranged.

This section tightens that into a calculable relation — and, just as importantly, picks out the most widely repeated wrong reason for it.

Heisenberg’s microscope

In 1927 the 26-year-old Heisenberg wrote the paper in Copenhagen. His argument is a thought experiment.

The γ-ray microscope

To measure an electron’s position you have to see it, which means a photon must scatter off it into your microscope.

Position accuracy is limited by wavelength. A microscope resolves down to about

\Delta x\sim\frac{\lambda}{\sin\varepsilon}\tag{1.7.1}

where $\varepsilon$ is the half-angle of the objective. To measure precisely you need short wavelengths — hence a γ-ray microscope.

The price is paid in momentum. The shorter the wavelength, the larger the photon momentum $p=h/\lambda$ . And a photon scattering into the objective could have come from anywhere within $\pm\varepsilon$ ; you cannot know which. So the recoil delivered to the electron is uncertain by

\Delta p\sim\frac{h}{\lambda}\sin\varepsilon\tag{1.7.2}

Multiply the two and both $\lambda$ and $\sin\varepsilon$ cancel:

\Delta x\,\Delta p\sim h\tag{1.7.3}

Measuring position precisely requires short wavelengths, and short wavelengths necessarily kick the electron harder — the trade cannot be avoided.

The argument is enormously persuasive: it turns an abstract restriction into a picture you can run in your head. And it does give the right order of magnitude.

But the reason is wrong

The uncertainty principle is not about measurement disturbance

Heisenberg’s microscope says: the act of measuring position disturbs the momentum.

That sounds unarguable. The problem is that on this account the electron has a definite position and a definite momentum before measurement, and our clumsy apparatus merely spoils one of them.

That picture is wrong, for two independent reasons.

Reason one: $\sigma_x$ and $\sigma_p$ are two properties of the same wavefunction.

Given a state $\Psi$ , you can compute both $\sigma_x$ and $\sigma_p$ with pen and paper, without performing any measurement at all. Do so and you find their product is never less than $\hbar/2$ . With no measurement anywhere in sight, there is nothing for “measurement disturbance” to refer to. Section 2.6 carries this calculation out.

Reason two: disturbance can be measured separately, and it is not the same thing.

Later work (Ozawa 2003, experimentally tested from 2012) separated “measurement error” and “measurement-induced disturbance” into two independently measurable quantities and found that the Heisenberg-form error–disturbance relation does not always hold — measurement schemes exist whose error–disturbance product is below $\hbar/2$ .

Whereas $\sigma_x\sigma_p\ge\hbar/2$ has never been violated.

These are different statements. The first is about instruments and can be improved on; the second is about the state itself and cannot be touched.

So what is uncertain

The correct statement is surprisingly plain — and we already glimpsed it in the previous section.

The picture

It is a theorem about waves.

An infinitely long sine wave has a perfectly definite wavelength, but it fills all space and you cannot say where it is.

Conversely, a very short pulse has a well-defined position, but you cannot say what its wavelength is — building an edge that sharp requires superposing a great many different wavelengths.

The more tightly you fix the position, the more wavelengths you need. This is a theorem of Fourier analysis and holds equally for sound, light pulses and radio signals. Audio engineers have always known it: a very short drum hit occupies a wide band.

The mathematics

Fourier analysis gives

\sigma_x\,\sigma_k\ge\frac12\tag{1.7.4}

Pure mathematics, nothing quantum about it.

Quantum mechanics adds one sentence, the de Broglie relation

p=\hbar k\tag{1.7.5}

Substituting gives immediately

\sigma_x\,\sigma_p\ge\frac{\hbar}{2}\tag{1.7.6}

The uncertainty principle = a theorem of Fourier analysis + the de Broglie relation. There is nothing else in it.

Back to the slits — you have already seen this

Replace the double slit with a single slit of adjustable width and send electrons through it onto a screen.

The narrower the slit, the better you know the electron’s transverse position as it passes ( $\Delta x$ small). And what is observed is: the narrower the slit, the wider the diffraction pattern — a larger spread of transverse momentum ( $\Delta p$ large).

The first minimum of single-slit diffraction, $a\sin\theta=\lambda$ , gives a transverse momentum spread $\Delta p_y\approx p\sin\theta=h/a$ , while $\Delta y\approx a$ , so

\Delta y\,\Delta p_y\approx h\tag{1.7.7}

This is not a new experiment; it is the diffraction formula of 1801. The uncertainty principle had been lying inside Young’s experiment for a hundred and twenty-six years, unrecognised.

While we are here: matrix mechanics and wave mechanics

Between 1925 and 1926 quantum mechanics was invented twice, independently.

The picture

Matrix mechanics (Heisenberg, 1925)

Heisenberg decided to build the theory out of observable quantities only — spectral line frequencies and intensities — abandoning unobservable things like electron orbits altogether.

The objects he obtained obeyed a strange multiplication: $AB\ne BA$ . He did not know at the time that these were matrices; Born recognised them.

The key relation:

\hat x\hat p-\hat p\hat x=\ii\hbar\tag{1.7.8}

The mathematics

Wave mechanics (Schrödinger, 1926)

Schrödinger started from de Broglie’s matter waves and looked for the equation such a wave obeys. He got a partial differential equation — mathematics physicists of the day already knew.

\ii\hbar\frac{\partial\Psi}{\partial t}=\hat H\Psi\tag{1.7.9}

The reception could not have been more different: physicists could solve PDEs and could not handle matrices. Schrödinger’s version took over almost at once.

Neither camp thought much of the other

Heisenberg wrote to Pauli that he found Schrödinger’s theory “repulsive” (abscheulich). Schrödinger wrote in a paper that he had been “discouraged, if not repelled” by the transcendental algebra of matrix mechanics.

In 1926 Schrödinger proved the two are mathematically equivalent, and Dirac and Jordan soon supplied a unified framework.

They are two ways of writing one theory, like describing the same circle in Cartesian or polar coordinates. Heisenberg’s commutator and the uncertainty principle correspond, in Schrödinger’s language, to properties of the Fourier transform — one fact, two accounts.

Chapter 3 builds that unified framework, after which you will find that most of what was argued about in this chapter turns out to be two projections of the same equation.

The energy–time relation is a different animal

You will often see $\Delta E\,\Delta t\ge\hbar/2$ , which looks just like the position–momentum relation.

Its meaning is quite different. Quantum mechanics has a position operator $\hat x$ and a momentum operator $\hat p$ , but no time operator — time is a parameter, not an observable.

The correct reading is: an unstable state with lifetime $\tau$ necessarily has an energy spread $\Delta E\sim\hbar/\tau$ . This explains the natural linewidth of atomic spectral lines, and why short-lived particle resonances are broad.

It is not a licence to borrow energy. Using it to explain tunnelling (“the particle borrows energy from the vacuum”) is a widespread but untenable story, discussed in section 2.9.

Key formulas

Heisenberg's microscope estimate
$\Delta x\,\Delta p\sim h$
Right order of magnitude, but "measurement disturbance" is not why
Fourier inequality
$\sigma_x\,\sigma_k\ge\frac12$
Pure mathematics, true of every wave
Uncertainty principle
$\sigma_x\,\sigma_p\ge\frac{\hbar}{2}$
= the Fourier theorem + de Broglie’s p = ħk
Canonical commutation relation
$\hat x\hat p-\hat p\hat x=\ii\hbar$
The core of matrix mechanics; chapter 3 derives the inequality from it rigorously

Self-check4 questions

End of the chapter

Look back at the road this chapter travelled:

Three dead ends all point at one assumption — that energy can be divided continuously.
Planck replaced it with “one unit at a time”, and the ultraviolet catastrophe and the specific-heat anomaly fell together.
Einstein moved “one unit at a time” into light itself, and the four oddities of the photoelectric effect fell together.
Bohr moved it onto electron orbits, and the hydrogen spectrum came out to four significant figures.
De Broglie reversed the logic: quantisation is a standing-wave condition, because particles are waves too.
The double slit showed that this “wave” is about probability, not about some substance vibrating.
The uncertainty principle turns out to be a general property of waves, plus $p=\hbar k$ .

All the phenomena are now on the table, but what we hold is a set of disconnected rules: $E=h\nu$ , $\lambda=h/p$ , $L=n\hbar$ … each governing a small patch, none able to compute the next new problem.

What physics needs is an equation: given a particle’s surroundings, work out how its state changes in time. Schrödinger wrote it down in 1926.

The price is a new object — a complex-valued function $\Psi$ . And what it represents is the first question to answer.

Chapter 2 starts there.

Where the uncertainty principle came from

Recommended first

After this section you should be able to

Heisenberg’s microscope

But the reason is wrong

So what is uncertain

The picture

The mathematics

While we are here: matrix mechanics and wave mechanics

The picture

The mathematics

Key formulas

Self-check4 questions

End of the chapter

Heisenberg’s microscope

But the reason is wrong

So what is uncertain

◑The picture

∑The mathematics

While we are here: matrix mechanics and wave mechanics

◑The picture

∑The mathematics

∎Key formulas

End of the chapter

The picture

The mathematics

The picture

The mathematics

Key formulas