Measurement · Quantum Mechanics (Phillips)

A classical measurement can always be made gentler. A quantum one cannot, and the floor is not set by the apparatus — it is set by the wavelength you had to use to look.

In classical physics, measuring something need not disturb it. You can always imagine using a gentler probe — dimmer light, a lighter pointer — and in the limit the disturbance goes to zero. That is why a classical object can be described without ever mentioning how you looked at it.

Quantum mechanics does not allow the limit. The probe is made of quanta, and there is no such thing as half a quantum. This section works out what follows, in three steps of increasing strangeness: a limit on what can be known, then a limit on what can be inferred, and finally the discovery that a measurement here can settle a matter over there.

The uncertainty principle

Heisenberg’s argument is a thought experiment: look at a particle through a microscope and see how well you can do.

Physics you need first — why a lens has a resolution limit

Equation (1.12) below is not quantum mechanics; it is ordinary wave optics, and it is worth thirty seconds because the whole argument leans on it.

Light entering a lens of finite width diffracts. A point source therefore does not image to a point but to a small blurred disc, and two points closer together than roughly that disc size cannot be told apart. The size of the blur scales as

\Delta x \sim \frac{\lambda}{\text{(angular size of the lens)}}

so you get a sharper image with shorter wavelengths or a wider lens — which is why electron microscopes beat optical ones (§1.2: a 100 eV electron has $\lambda \approx 0.12$ nm) and why big telescopes resolve better than small ones.

If you have done signal processing, this is the same statement as: a finite aperture is a finite window on the spatial-frequency content of the image, and truncating a spectrum blurs the thing it describes. The lens is a low-pass filter, and $\lambda/\sin\alpha$ is its impulse-response width.

wavelength λ = 500 nmaperture half-angle α = 30°

Δx = λ/sin α: 1000 nm
Δp = (h/λ) sin α: 0.66 × 10⁻²⁷
Δx · Δp: 6.63 × 10⁻³⁴ J s
compare h: 6.63 × 10⁻³⁴ J s

Drag either slider. Δx and Δp always move in opposite directions and the product never budges — because λ and sin α cancel exactly when you multiply (1.12) by (1.13). There is no setting of the apparatus that beats it.

Two facts fight each other.

Light is a wave, so the microscope has finite resolving power. You cannot locate the particle better than about one wavelength, improved slightly by a wide lens:

\Delta x \approx \frac{\lambda}{\sin\alpha}\tag{1.12}

Short wavelengths are better: X-rays beat visible light, visible light beats microwaves. So far this is ordinary optics, and it says use the shortest wavelength you can.

But light is also particles, and seeing means scattering one off the target. For the photon to reach the lens at all, it must leave with a sideways momentum somewhere between $-(h/\lambda)\sin\alpha$ and $+(h/\lambda)\sin\alpha$ — and we cannot tell which, or we would know exactly where in the lens it landed. Momentum is conserved, so the particle absorbed an equal and opposite kick that we likewise cannot pin down:

\Delta p \approx \frac{h}{\lambda}\sin\alpha\tag{1.13}

And now the trap. This says use the longest wavelength you can — the exact opposite. Multiply the two together and both knobs cancel:

\Delta x\,\Delta p \approx h\tag{1.14}

The precise statement of the Heisenberg uncertainty principle , which chapter 7 §7.4 derives properly, replaces the hand-waving $\approx h$ with an exact inequality:

\Delta x\,\Delta p \ge \frac{\hbar}{2}, \qquad \text{where } \hbar = \frac{h}{2\pi}\tag{1.15}

Engineer's bridge — you already proved this in a DSP course

Equation (1.15) is not new mathematics. It is the time–bandwidth product, and you have met it as a theorem about Fourier transforms:

\sigma_t\,\sigma_f \ge \frac{1}{4\pi}

A signal cannot be both brief and narrow-band. A click has a broad spectrum; a pure tone lasts forever; a Gaussian pulse achieves the minimum and nothing beats it. This is a mathematical fact about any function and its transform, with no physics in it at all.

Now recall from §1.2 that $p = \hbar k$ — momentum is spatial frequency, with $\hbar$ as the conversion factor. Substitute it into the Fourier theorem for the pair $(x, k)$ :

\sigma_x\,\sigma_k \ge \tfrac{1}{2} \quad\Longrightarrow\quad \sigma_x\,(\sigma_p/\hbar) \ge \tfrac{1}{2} \quad\Longrightarrow\quad \Delta x\,\Delta p \ge \frac{\hbar}{2}

The uncertainty principle is the time–bandwidth theorem wearing physics units. All the physics is in the single assumption $p = \hbar k$ ; the inequality itself is free.

That is why the microscope argument feels like it is about clumsy apparatus and yet arrives at a universal bound. The clumsiness is a story about how the limit shows up in one experiment. The limit itself is a property of describing a particle by a wave at all.

Where the analogy breaks. In DSP, $\sigma_t$ and $\sigma_f$ describe one signal you possess; you can compute both to arbitrary precision from the same data, and nothing stops you knowing both. Here the claim is stronger and stranger: it is not that you cannot know $x$ and $p$ together, but that a particle with both simultaneously well-defined does not exist to be known. Phillips is explicit about this, and it is worth taking seriously.

The microscope story is useful and it is also misleading

Phillips warns about this directly, and the warning is easy to skim past:

readers should resist the temptation to believe that a particle can really have a definite position and momentum, which, because of the clumsy nature of the observation, cannot be measured. In fact, there is no evidence for the existence of particles with definite position and momentum.

The microscope makes it sound like a disturbance story: the particle had a position and a momentum, and our photon knocked one of them askew. That reading is natural, wrong, and will actively mislead you in chapters 3 and 7.

The correct reading is the Fourier one from the bridge above. A wave packet narrow in $x$ is made of a wide band of $p$ — not “has an unknown $p$ ”, but is a superposition of many. There is nothing hidden to be disturbed.

Phillips offers a better way to hold it: the uncertainty principle is a danger signal telling you how far you can push the classical concepts of position and momentum before they stop corresponding to anything real.

There is one more twist. If you could use $\lambda \to 0$ you would get $\Delta x \to 0$ . But the Compton effect (§1.1) says a photon scattering off a particle of mass $m$ comes back longer by $\Delta\lambda = (h/mc)(1-\cos\theta)$ — so however short the light you send in, what reaches the lens has a wavelength of order $h/mc$ . The resolution therefore bottoms out:

\Delta x \approx \frac{\lambda}{\sin\alpha} \gtrsim \frac{h}{mc\,\sin\alpha}\tag{1.16}

Put a number on it — and the uncertainty principle rebuilds the atom

How well can you locate an electron with light? The floor is the Compton wavelength:

\frac{h}{m_ec} = 2.43\times10^{-12}\ \mathrm{m} = 2.43\ \mathrm{pm}

About a fiftieth of an atom. No optical trick gets below it.

Now run the principle backwards. Confine an electron to an atom, $\Delta x \approx a_0$ . Then $\Delta p \gtrsim \hbar/a_0$ , and since a particle’s momentum is at least as big as its uncertainty, its kinetic energy is at least

E \approx \frac{(\Delta p)^2}{2m_e} = \frac{\hbar^2}{2m_ea_0^2} = 13.6\ \mathrm{eV}

That is the Rydberg energy — the same 13.6 eV that §1.3 obtained by dimensional analysis and that Eq. (1.9) asserted. Squeezing a wave into a small box costs kinetic energy, and that is what holds the electron up against the Coulomb attraction. This is the quantitative version of “there is nowhere lower to fall”, and it is problem 6 at the end of the chapter.

And why you have never noticed any of this. Locate a 1 gram ball to within a micrometre and the principle demands only $\Delta v \ge 5\times10^{-26}\ \mathrm{m\,s^{-1}}$ — a velocity you could not detect if you watched for the age of the universe.

Reproduce it in NumPy

Both arguments of this section in one script. The last three lines are the punchline of the movable-screen experiment below: the blur you are forced to accept is exactly the fringe spacing you were trying to see.

import numpy as np

h, hbar, me, c, eV = 6.626e-34, 1.055e-34, 9.109e-31, 2.998e8, 1.602e-19
a0 = 0.5292e-10

print(f"hbar/2 = {hbar/2:.3e} J s   -- the floor on Delta_x * Delta_p")

# smallest patch you can pin an electron into with light (Eq 1.16)
print(f"minimum Delta_x ~ h/(me c) = {h/(me*c):.3e} m = {h/(me*c)*1e12:.2f} pm")

# confine an electron to an atom: the uncertainty principle alone gives E_R
p = hbar / a0
KE = p**2 / (2*me)
print(f"electron confined to a0: Delta_p ~ hbar/a0 = {p:.3e} kg m/s")
print(f"  -> kinetic energy ~ hbar^2/(2 me a0^2) = {KE/eV:.2f} eV   (E_R = 13.6 eV)")

# why nobody notices: a 1 g ball located to 1 micron
m_ball, dx_ball = 1e-3, 1e-6
print(f"1 g ball located to 1 um: Delta_v >= {hbar/(2*m_ball*dx_ball):.2e} m/s")

# the movable-screen argument: Delta_x of the screen equals the fringe spacing
lam, D, d = 500e-9, 1.0, 100e-6
p_part = h/lam
dp = p_part*d/D
dx = h/dp
print(f"\nmovable screen, lambda={lam*1e9:.0f} nm, d={d*1e6:.0f} um, D={D} m:")
print(f"  fringe spacing lambda D/d = {lam*D/d*1e3:.2f} mm")
print(f"  screen Delta_p ~ pd/D     = {dp:.3e} kg m/s")
print(f"  screen Delta_x ~ h/Delta_p = {dx*1e3:.2f} mm   <- same as the fringe spacing")

printshbar/2 = 5.275e-35 J s -- the floor on Delta_x * Delta_p minimum Delta_x ~ h/(me c) = 2.426e-12 m = 2.43 pm electron confined to a0: Delta_p ~ hbar/a0 = 1.994e-24 kg m/s -> kinetic energy ~ hbar^2/(2 me a0^2) = 13.62 eV (E_R = 13.6 eV) 1 g ball located to 1 um: Delta_v >= 5.27e-26 m/s movable screen, lambda=500 nm, d=100 um, D=1.0 m: fringe spacing lambda D/d = 5.00 mm screen Delta_p ~ pd/D = 1.325e-31 kg m/s screen Delta_x ~ h/Delta_p = 5.00 mm <- same as the fringe spacing

Measurement and wave–particle duality

Now the second step, and it is the best argument in the chapter.

Wave–particle duality is, operationally, a statement about what you observe versus what you infer. In a two-slit experiment you observe particle-like behaviour — a dot on the screen — and you infer wave-like behaviour from the pattern the dots make. Phillips’ question is whether you could catch the wave in the act: arrange to learn which slit each particle went through and still see the fringes.

The apparatus is Fig. 1.7. A pin can either clamp the screen or release it. When released, the screen recoils when a particle hits, and by measuring that recoil you learn the particle’s vertical momentum — which tells you which slit it came from.

Fig. 1.7 — the screen can be clamped or released. Released, it becomes a which-path detector; clamped, it is an ordinary screen.

The scheme looks watertight, and the reason it fails is beautiful: the screen is a quantum object too, and the uncertainty principle applies to it.

step 1 of 4

No new physics is used here. Only Eq. (1.15), applied to the SCREEN rather than to the particle.

1Work out how different the two slits’ momenta actually are. Near the centre of the screen, a particle from the upper slit arrives heading slightly down and one from the lower slit slightly up.
$Δ p \approx \frac{p d}{D}$
Equation (1.17). To tell the slits apart, you must measure the screen’s recoil at least this precisely.
2Apply the uncertainty principle TO THE SCREEN. Knowing its momentum that well costs you knowledge of where it is.
$Δ x_{screen} \approx \frac{h}{Δ p} \approx \frac{h D}{p d}$
Equation (1.18). The screen is no longer at a definite height — and a screen whose position is uncertain cannot record a pattern sharper than that uncertainty.
3Rewrite it in terms of the particle’s wavelength, using the de Broglie relation p = h/λ from §1.2.
$Δ x_{screen} \approx \frac{λ D}{d}$
Equation (1.19).
4Compare with what you were trying to see. The fringe spacing in a two-slit experiment is — exactly this.
$fringe spacing = \frac{λ D}{d} = Δ x_{screen}$
The blur you are forced to accept is precisely one fringe. The pattern is washed out at the exact moment the which-path information becomes available. Not approximately: exactly.

What this really says — the disturbance is not clumsiness, and nothing was added by hand

Nothing in that argument is about the apparatus being crude, and nothing is added by hand. Quantum mechanics does not need a separate rule saying “you may not have both interference and which-path information”. The rule falls out of the uncertainty principle applied consistently to everything involved, including the measuring device.

The margin is exactly zero. Measure the recoil slightly less precisely and you cannot quite tell the slits apart, but the fringes survive slightly. Measure it slightly better and you learn the path, but the fringes are gone. Nature does not merely forbid having both — it prices them against each other so that you can trade, and never profit.

That is why Phillips says the wave-like behaviour of quantum particles is always covert. The wave is never observed; it is only ever inferred from a pattern, and any arrangement that would let you watch it directly destroys the pattern that was the evidence.

You can do the experiment here. The detector toggle is exactly the pin in Fig. 1.7:

Wheeler’s delayed choice

Phillips ends the argument with a variation from John Wheeler (1978) that most people find genuinely disturbing.

Wait until the particle has passed the slits. Then decide whether to insert the pin. If you insert it, an interference pattern builds up — behaviour of something that went through both slits. If you withdraw it, you learn the path — behaviour of something that went through one.

The choice is made after the particle has supposedly already “decided” what to do. Phillips’ resolution is worth quoting because it is the whole epistemological position of the book in two sentences:

the history created in this experiment is not classical history. The particles concerned are not classical particles which pass through one slit or the other, nor are they classical waves which pass through both slits. They are quantum particles which have the capability to behave in both of these ways, but only one of these ways may be inferred in a particular experimental arrangement.

The paradox comes entirely from assuming there was a fact about which slit the particle went through, waiting to be revealed. There was not. There was a quantum state, and different experiments extract different inferences from it.

Measurement and non-locality

The third and strangest step. Phillips states the conclusion baldly:

quantum mechanics only describes what we can know about the world.

Because we cannot know an electron’s position and momentum together, we cannot describe a world in which it has both. In the standard interpretation, a precise position is brought into existence by the measurement — and the theory makes no attempt to explain how.

That would be merely philosophical if it applied only to single particles. It does not.

Consider an atom that emits two photons in opposite directions with the same circular polarization. Measure the eastbound one: if it is right-handed, the westbound one is certain to be right-handed too. If left, then left.

The correlation is perfect and it is not carried by anything travelling between the photons.

That correlation on its own would be unremarkable — it is what you would expect if the handedness were fixed at emission, like posting one glove of a pair to each of two cities. But that is not what is happening. At the moment of emission, an entangled state is created in which the pair is simultaneously right- and left-handed, and only one of those alternatives is brought into existence by the later measurement.

Phillips’ image is a pair of ambidextrous gloves: neither is a left or a right glove until one is looked at, and the instant one becomes right-handed, so does the other, however far away. The reasons, he says, are that (1) the initial state is a superposition, exactly as a particle’s state can be a superposition of two slits, and (2) a measurement not only disturbs what is measured but brings into existence what is measured.

This is exactly where the book stops, and where the real argument starts

Everything above is consistent with a much more boring story: the photons agreed on a handedness at emission and simply carried it with them. Correlated socks. Phillips asserts that this is not what happens, and at this point in the book he cannot show you why.

The reason it is not — and it is one of the great results of twentieth-century physics — is Bell’s theorem: the correlations predicted by quantum mechanics for measurements along different axes are stronger than any pre-agreed-value story can produce, whatever the agreement. It is not that we have failed to find the hidden agreement; it is that the numbers rule it out. Alain Aspect’s experiments, which Phillips cites, measured those numbers.

The reason to flag this now is that “the photons agreed in advance” is the single most natural way to misread this passage, and nothing in §1.4 closes it off. Non-locality is a real claim about the world, not a way of speaking. Chapter 10 develops the entanglement machinery that makes the argument possible.

What chapter 1 has established

The chapter closes by listing what any theory of quantum particles must now deliver:

It must handle both characters at once, particle and wave, and it must contain the constant that links them, $h$ .
It must treat measurement as active — not a passive reading of properties that were already there, but a process that creates the property it reports.
It must be a theory of what can be known, since that is all the experiments give access to.

Notice what is not on the list: any picture of what a quantum particle is between measurements. Phillips does not supply one, and the theory the next three chapters build does not contain one.

He also fixes the scope: the book is about non-relativistic particles. Relativistic ones — photons included — require handling the creation and destruction of particles, which is quantum field theory and is not this book. That is worth remembering, given that chapter 1 has been arguing largely with photons.

The machinery starts in chapter 2, with a wave equation whose solutions carry exactly the properties chapter 1 has just demonstrated.

Check yourself

0 / 6 answered

Drag the λ slider and watch all three readouts.
1.In the microscope figure, you halve the wavelength. What happens to $Δ x$ , $Δ p$ and their product?
2.What is the most accurate reading of $Δ x Δ p \geq ℏ/2$ ?
3.An electron confined to roughly a Bohr radius must have kinetic energy of at least about 13.6 eV. Why does that matter?
4.In the movable-screen experiment, why do the fringes disappear when you measure the screen's recoil precisely enough to identify the slit?
5.Wheeler's delayed-choice experiment seems to let a later decision change an earlier history. What dissolves the paradox?
6.Two entangled photons fly apart; measuring one right-handed makes the other right-handed. What does this section NOT establish?