F. Revolutions in Modern Physics: Quantum Mechanics and Special Relativity · Quantum MechanicsThe Uncertainty Principle and the Probability Interpretation
F1.1 F2.1 F3.2About 70 minutes
Learning goals- I can state Heisenberg's uncertainty principle and use Δx Δp ≥ h/(4π) to estimate uncertainties.
- I can explain the uncertainty principle using single-slit diffraction and Heisenberg's microscope thought experiment.
- I can describe the wave function and the probability interpretation of quantum mechanics, including tunnelling.
- I can analyse how thought experiments and debates (Bohr, Einstein, Schrödinger) changed scientific thought.
Words to know
uncertainty principle
Δx Δp ≥ h/(4π): the position and momentum of a particle cannot both be known with unlimited precision at the same time.
wave function (ψ)
The mathematical wave that describes a quantum particle; its square |ψ|² gives the probability of finding the particle at each place.
probability density
|ψ|², the probability per unit length (or volume) of finding the particle at a location.
orbital
A region around a nucleus where an electron is likely to be found, described by a wave function.
quantum tunnelling
A particle passing through a barrier that classical physics says it does not have enough energy to cross.
thought experiment
An imagined experiment, reasoned through carefully, used to test the logic or consequences of a theory.
entanglement
A link between quantum particles so that measurements on them are correlated, however far apart they are.
You can't pin it all down
In everyday life, we assume we can know exactly where something is and how fast it is moving: a hockey puck, a car, a planet. In 1927, the German physicist Werner Heisenberg showed that for quantum objects this is impossible, not because our instruments are clumsy, but because of the wave nature of matter itself.
Heisenberg's uncertainty principle:
Δx Δp ≥ h/(4π)
Δx is the uncertainty in position and Δp is the uncertainty in momentum (along the same direction). h/(4π) = 5.28 × 10⁻³⁵ J·s. The more precisely you know one, the less precisely you can know the other.
Seeing it with a single slit
Send electrons (or photons) through a slit of width Δy. Passing through tells you each particle's sideways position to within about Δy. But the slit diffracts the wave: particles spread out sideways, so their sideways momentum becomes uncertain.
- A wide slit: position not well known, but little diffraction, so the sideways momentum is well known.
- A narrow slit: position well known, but the wave spreads widely, so the sideways momentum is very uncertain.
From single-slit diffraction (sin θ ≈ λ/Δy for the first minimum), the sideways momentum spread is about Δp ≈ p sin θ = (h/λ)(λ/Δy) = h/Δy. So Δy Δp ≈ h, which is consistent with the uncertainty principle. Squeezing one quantity always spreads the other.
Heisenberg's microscope (a thought experiment)
Imagine trying to see where an electron is by bouncing light off it. To locate it precisely you need a short wavelength (fine detail needs short waves, as you learned with diffraction). But a short-wavelength photon carries a large momentum h/λ, and in the collision (a Compton kick) it changes the electron's momentum by an unknown amount. Longer wavelengths disturb it less but blur its position. There is no way to win both. Heisenberg used this imagined experiment to make his principle concrete; the modern view is that the uncertainty is built into the wave description itself, not just caused by the measurement.
Worked exampleExample 1: An electron inside an atom
An electron is confined within an atom, so Δx ≈ 1.0 × 10⁻¹⁰ m. Find the minimum uncertainty in its momentum and speed.
- Δp ≥ h/(4πΔx) = 5.28 × 10⁻³⁵ ÷ 1.0 × 10⁻¹⁰ = 5.3 × 10⁻²⁵ kg·m/s.
- Δv = Δp/m = 5.3 × 10⁻²⁵ ÷ 9.11 × 10⁻³¹ = 5.8 × 10⁵ m/s.
Δv ≥ 5.8 × 10⁵ m/s, a large fraction of the speeds electrons have in atoms (around 10⁶ m/s). We cannot picture an electron as a dot on a definite orbit: this is why the Bohr model's sharp orbits had to go.
Worked exampleExample 2: A baseball
A 0.145 kg baseball's position is measured to within 1.0 × 10⁻⁶ m (a micrometre). Find the minimum uncertainty in its speed.
- Δp ≥ 5.28 × 10⁻³⁵ ÷ 1.0 × 10⁻⁶ = 5.3 × 10⁻²⁹ kg·m/s.
- Δv = 5.3 × 10⁻²⁹ ÷ 0.145 = 3.6 × 10⁻²⁸ m/s.
About 3.6 × 10⁻²⁸ m/s, unimaginably small. At that rate the ball would take about 9 billion years (most of the age of the universe) to drift the width of an atom. For everyday objects, the uncertainty principle is irrelevant; Newton's laws work perfectly.
Waves of probability: the wave function
In 1926 the Austrian physicist Erwin Schrödinger wrote an equation for how matter waves behave. Its solution, the wave function ψ (psi), describes the particle completely. Max Born then proposed the interpretation used today:
- |ψ|² at a point gives the probability of finding the particle there when you look.
- Before measurement, the particle does not have a single definite position; it has a spread of possibilities.
- When you measure, you find it in one place, with odds given by |ψ|².
For electrons in atoms, the wave functions give the orbitals of chemistry: fuzzy clouds showing where the electron is likely to be, not paths it follows. The energies from Schrödinger's equation match Bohr's for hydrogen, but it also works for many-electron atoms and molecules, which made it the foundation of modern chemistry and materials science.
Tunnelling
One striking result: a wave function does not stop dead at a barrier. If a particle meets a barrier it does not have enough energy to climb over, its wave decays inside the barrier but, if the barrier is thin enough, a small part comes out the other side. So there is a small probability that the particle simply appears on the far side. This is quantum tunnelling.
Tunnelling explains alpha decay of nuclei (which you studied in Grade 11): the alpha particle tunnels out through the nuclear energy barrier. It is also used in technology: the scanning tunnelling microscope (STM), invented by Gerd Binnig and Heinrich Rohrer (Nobel Prize 1986), measures the tunnelling current between a sharp tip and a surface so sensitively that it can map individual atoms. Flash memory in USB drives and phones stores data by moving electrons through thin insulating layers, using tunnelling.
Debates and thought experiments (F1.1)
Many physicists, including Einstein, were deeply uneasy with a theory that gives only probabilities. Einstein argued with Bohr for years using clever thought experiments. In 1935 Einstein, Podolsky and Rosen argued that quantum mechanics must be incomplete, pointing to the strange correlations of entangled particles. The same year, Schrödinger described his famous cat in a sealed box, linked to a random atomic decay, to show how odd it would be to apply quantum uncertainty to everyday objects. Decades later, experiments with entangled photons settled much of the argument in quantum theory's favour; Alain Aspect, John Clauser and Anton Zeilinger shared the 2022 Nobel Prize in Physics for this work. Entanglement is now a resource for quantum computing and quantum encryption (lesson 13).
Just for funHeisenberg is pulled over for speeding. The officer asks, "Do you know how fast you were going?" Heisenberg replies, "No, but I know exactly where I am." (A physics classic. As Example 2 shows, a real car's uncertainty is far too small for this excuse to work.)
Hands-on activityA ruler, a slit and your own diffraction
You need: two pencils (or two straight edges like credit cards), a lamp with a straight bright edge or a long LED strip, and a dim room.
- Hold the two pencils side by side, close to one eye, so there is a thin gap between them. Look through the gap at the lamp a couple of metres away, with the gap parallel to the bright edge. Do not look at the Sun.
- Slowly squeeze the pencils together so the gap narrows.
- Watch the bright line: it spreads out and faint light-and-dark bands appear as the gap gets very thin.
- Explain what you saw using the uncertainty principle: as the gap (Δy) gets smaller, what happens to the spread of directions (Δp) of the light?
What to look for: a narrower gap gives a wider smear of light with faint fringes. Passing through a narrow gap fixes the photons' sideways position more precisely, so their sideways momentum (direction) becomes more uncertain. That is Δy Δp ≥ h/(4π) with your own eyes.
Your turn
1State the uncertainty principle in words.
It is impossible to know both the position and the momentum of a particle (in the same direction) at the same time with unlimited precision; the product of their uncertainties is at least h/(4π).
2An electron's position is known to within 0.50 nm. Find the minimum uncertainty in its speed.
Δp ≥ 5.28 × 10⁻³⁵ ÷ 5.0 × 10⁻¹⁰ = 1.06 × 10⁻²⁵ kg·m/s; Δv = 1.06 × 10⁻²⁵ ÷ 9.11 × 10⁻³¹ = 1.2 × 10⁵ m/s.
3An electron's speed (2.0 × 10⁶ m/s) is known to within 5%. What is the smallest possible uncertainty in its position?
Δv = 1.0 × 10⁵ m/s; Δp = 9.11 × 10⁻²⁶ kg·m/s. Δx ≥ 5.28 × 10⁻³⁵ ÷ 9.11 × 10⁻²⁶ = 5.8 × 10⁻¹⁰ m (0.58 nm).
4A proton is confined inside a nucleus, Δx ≈ 1.0 × 10⁻¹⁴ m. Find the minimum uncertainty in its speed.
Δp ≥ 5.28 × 10⁻³⁵ ÷ 1.0 × 10⁻¹⁴ = 5.3 × 10⁻²¹ kg·m/s; Δv = 5.3 × 10⁻²¹ ÷ 1.67 × 10⁻²⁷ = 3.2 × 10⁶ m/s.
The probability (density) of finding the particle at each location when a measurement is made.
6Why does the uncertainty principle not affect a hockey game?
For a puck's large mass, h/(4πΔx) gives an uncertainty in speed that is many, many orders of magnitude too small to measure. Quantum uncertainty matters only for tiny masses like electrons.
7In the single-slit picture, what happens to the diffraction pattern as the slit is narrowed, and why does that match the uncertainty principle?
The pattern spreads out. Narrowing the slit reduces Δy, so Δp (the spread of sideways momentum, seen as the spread of directions) must increase.
More practice: warm-up
1Who proposed the uncertainty principle, and when?
2Write the uncertainty principle as an inequality.
A region where an electron is likely to be found, described by its wave function.
5Name one device that uses quantum tunnelling.
For example, the scanning tunnelling microscope or flash memory.
More practice: on your own
1A 1.0 × 10⁻¹² kg dust grain's position is known to 1.0 μm. Find the minimum speed uncertainty. Would you notice it?
Δp ≥ 5.28 × 10⁻³⁵ ÷ 1.0 × 10⁻⁶ = 5.3 × 10⁻²⁹; Δv = 5.3 × 10⁻²⁹ ÷ 1.0 × 10⁻¹² = 5.3 × 10⁻¹⁷ m/s. Far too small to notice.
2Explain Heisenberg's microscope thought experiment in your own words.
To locate an electron you must hit it with light. Short wavelengths give a sharp position but give the electron a big, uncertain momentum kick (h/λ is large). Long wavelengths kick it gently but give a blurry position. You cannot get both a precise position and a precise momentum.
3Why does the uncertainty principle mean electrons in atoms cannot follow sharp Bohr orbits?
A sharp orbit would mean knowing both the electron's position (on the circle) and momentum (its velocity along the circle) precisely. With Δx about the size of the atom, Δv is about 10⁵ to 10⁶ m/s, as large as the speed itself, so a definite path is meaningless. We use probability clouds (orbitals) instead.
4A barrier is made thicker. What happens to the probability of tunnelling? Explain using the wave picture.
It decreases (sharply). The wave decays inside the barrier, so a thicker barrier leaves a much smaller wave on the far side, and probability ∝ |ψ|².
5What point was Schrödinger making with his cat thought experiment?
That applying quantum uncertainty (a superposition of possibilities) to everyday objects like a cat seems absurd, challenging how and when a quantum system's possibilities become one definite outcome. It highlighted a deep puzzle in interpreting quantum mechanics.
Challenge
1Use the uncertainty principle to show why an electron cannot be confined inside a nucleus (Δx ≈ 1.0 × 10⁻¹⁵ m). Find Δp and the corresponding energy (use E ≈ pc, since the electron would be relativistic), and compare with the few MeV typical of nuclear energies.
Δp ≥ 5.28 × 10⁻³⁵ ÷ 1.0 × 10⁻¹⁵ = 5.3 × 10⁻²⁰ kg·m/s. E ≈ pc = 5.3 × 10⁻²⁰ × 3.00 × 10⁸ = 1.6 × 10⁻¹¹ J ≈ 99 MeV (about 100 MeV). That is far more than the few MeV that bind nuclear particles, so an electron could not be held in a nucleus. (This is part of why beta-decay electrons are created at the moment of decay, not stored in the nucleus.)
2A photon of 500 nm has its position along the beam known to within 500 nm. Compare Δp with p. What does this say about trying to localise a photon to within one wavelength?
p = h/λ = 1.33 × 10⁻²⁷ kg·m/s. Δp ≥ 5.28 × 10⁻³⁵ ÷ 5.0 × 10⁻⁷ = 1.06 × 10⁻²⁸ kg·m/s, about 8% of p. Localising a photon to about one wavelength leaves its momentum (and so its wavelength and colour) uncertain by several percent; a perfectly pure colour would need an infinitely long wave.
3Classical physics was deterministic; quantum mechanics is probabilistic. Explain the difference and give one reason scientists accepted the probabilistic theory despite their discomfort.
Deterministic: exact knowledge of the present allows exact prediction of the future. Probabilistic: only the odds of different outcomes can be predicted for a single event, though averages over many events are predicted precisely. Scientists accepted it because its predictions (atomic spectra, chemical bonding, diffraction patterns, tunnelling rates, entanglement correlations) agree with experiment to extraordinary precision, and attempts to replace it with a deterministic local theory were ruled out by experiments.
Quiz
1. According to the uncertainty principle, if the uncertainty in an electron's position is made smaller, the uncertainty in its momentum
2. Why don't we notice the uncertainty principle for a thrown baseball?
3. In the probability interpretation, |ψ|² represents
4. Narrowing the slit in a single-slit experiment with electrons causes the pattern on the screen to
5. Quantum tunnelling refers to
6. An electron's position is known to within 1.0 × 10⁻¹⁰ m. The minimum uncertainty in its momentum is about
Big idea (F1.1): quantum objects do not have a definite position and momentum at the same time; Δx Δp ≥ h/(4π). Their behaviour is described by a wave function whose square gives probabilities. These ideas, debated through famous thought experiments, changed what physicists think it means to predict nature.
How to help: do the two-pencil activity together and ask your teen to explain the link between a narrow gap and a wide smear of light. Discuss Schrödinger's cat as a puzzle about interpretation, not a real experiment.
Common difficulties:
- Thinking uncertainty comes only from clumsy instruments. It is a basic property of waves.
- Forgetting the 4π (using h instead of h/(4π)); some books use ħ/2, which is the same value.
- Mixing up the direction: the momentum uncertainty must be in the same direction as the position uncertainty.
- Thinking quantum mechanics predicts nothing precisely. It predicts probabilities, and averages over many events, extremely accurately.