2.3.7 · D5Modern Physics
Question bank — Heisenberg uncertainty principle — Δx Δp ≥ ℏ - 2, ΔE Δt ≥ ℏ - 2
This page assumes the picture from the parent note: a particle is a wave packet, and the bound comes from the Fourier fact plus the de Broglie bridge . Keep those two facts in your hand for every answer.
True or false — justify
True/false: A more precise ruler will eventually beat the floor.
False — is the intrinsic statistical spread of identically prepared particles, a property of the state itself; a perfect instrument still measures that spread.
True/false: The uncertainty product can equal exactly .
True, but only for a Gaussian wave packet; every other shape does strictly worse, so is a minimum that is rarely reached.
True/false: A plane wave has zero position uncertainty.
False — it has infinite ; its is flat everywhere, so the particle is equally likely to be anywhere while its momentum is perfectly sharp.
True/false: lets a system briefly violate energy conservation.
False — energy is conserved exactly; is the spread in a measured energy, fixed by how long the state lives, not an energy loan.
True/false: If you could set , the particle would sit at one exact point.
False — forces (a single-wavelength wave fills all space), so the particle would be maximally unlocalized.
True/false: The uncertainty principle only matters when you measure the particle.
False — it holds with no measurement at all, because it is a statement about what the wave packet is; measurement disturbance is a consequence, not the cause.
True/false: Because is tiny, uncertainty simply does not exist for a cricket ball.
False — it exists but is unmeasurably small; the floor is ~ m/s for macroscopic mass, drowned by every real effect.
True/false: in the energy–time relation is the error of your stopwatch.
False — is the timescale over which the state changes appreciably (its lifetime); a perfectly accurate clock does not shrink it.
True/false: A stationary energy eigenstate () must live forever.
True — a sharp energy means ; only truly stable states have exactly defined energy, which is why excited states with finite lifetimes have blurred energies.
Spot the error
Find the flaw: "We write because that's the theorem."
The factor of is dropped — the exact theorem for RMS deviations is ; alone is only an order-of-magnitude shorthand and gives wrong "minimum value" answers.
Find the flaw: "The uncertainty comes from the photon kicking the electron when we look at it."
This describes a real disturbance, but the principle is deeper: it follows from the Fourier width relation and holds even with zero interaction; disturbance is downstream of the wave nature.
Find the flaw: "A short laser pulse can be made both perfectly brief and perfectly single-colour."
Impossible — a pulse of duration must contain a frequency spread ; brief in time forces broad in colour, the same Fourier trade as .
Find the flaw: "Confine an electron and its ground-state energy can be pushed to zero by making the box just right."
No — confinement gives , so , and ; there is always a nonzero zero-point energy.
Find the flaw: "The bound is a law of physics."
It is pure mathematics of any wave/Fourier pair — sound, water, light all obey it; physics only enters when de Broglie converts into .
Find the flaw: "A Gaussian packet is worst-case because it's spread out."
Reversed — the Gaussian is the best case; it saturates the bound at exactly , the minimum possible joint uncertainty.
Why questions
Why does making the position bump narrower force a wider momentum spread?
A narrow bump needs many wavelengths superposed to cancel outside it (Fourier), and each wavelength is a momentum via , so more wavelengths means a wider .
Why is a Gaussian wave packet special among all shapes?
It is the unique shape whose own Fourier transform is also a Gaussian, minimizing the width product to exactly ; any distortion adds extra spread.
Why do short-lived particles have a broad natural linewidth?
Short lifetime means small , and , so their energy — hence emitted photon frequency — is intrinsically fuzzy (see Natural linewidth and spectral broadening).
Why can't electrons "sit still" at the bottom of an atom?
Sitting still means , which needs ; but the atom confines Å, forcing a large momentum spread and thus perpetual motion.
Why does tunnelling not contradict energy conservation despite crossing a forbidden barrier?
Energy is conserved throughout; the wave amplitude simply leaks through the barrier, and the energy–time relation only describes the spread, not a temporary energy loan (see Quantum tunnelling).
Why is the de Broglie wavelength the crucial ingredient, not the Fourier relation alone?
Fourier gives only , a statement about waves; is the physics that turns the abstract into a measurable momentum.
Edge cases
What happens to as (perfectly localized)?
— a point-like particle needs infinitely many wavelengths, so momentum becomes completely undefined; this is the extreme opposite of a plane wave.
What is the energy uncertainty of a perfectly stable ground state?
Exactly zero — it never changes, so and ; only decaying states pick up an energy width.
Does the principle apply to a free particle with no walls or forces?
Yes — any wave packet, confined or not, obeys ; a free Gaussian packet even spreads over time as its components travel at different speeds.
Can two different quantities always be measured together precisely?
No only for complementary pairs (like and , or and ) that are Fourier conjugates; unrelated quantities can be simultaneously sharp.
What is the smallest possible energy of a particle in a box of size ?
About — never zero, because zero energy would demand and unbounded , contradicting confinement (see Particle in a box).
In the limit (imagined classical world), what happens to the bound?
The floor , so position and momentum can both be sharp — this is exactly why classical mechanics works and quantum fuzziness vanishes.
Recall One-line self-test
If someone says "uncertainty is just measurement error," what single word fixes them? ::: "Intrinsic" — it is a property of the state's wave nature, present with a perfect instrument.