The Uncertainty Principle from Wave Packets
A localised wave packet combines a range of wave numbers. Position and momentum standard deviations obey ΔxΔp ≥ ħ/2, so narrowing position raises the momentum lower bound. This state-spread limit is separate from instrument error.
What a learner can do afterwards
- Explains how superposing waves with different wave numbers can make a localised packet
- Estimates a momentum-spread lower bound from a stated position standard deviation
- Distinguishes the prepared state's spread relation from instrument error and measurement disturbance
1 · Read
You have used |ψ|² to predict position probabilities. A plane-wave component with signed wave number k has momentum p = ħk. Its magnitude is constant across space, so it is not a normalised, localised state on the whole line. A packet combines different k values with chosen amplitudes and phases. They reinforce near its centre and mostly cancel farther away.
Δx and Δp are standard deviations of position and momentum along the same axis, across identically prepared copies of a state. Each measures spread about its mean. For example, Δx = √(⟨x²⟩ - ⟨x⟩²), where brackets mean averages. For finite spreads, ΔxΔp ≥ ħ/2, with ħ = h/(2π). A smaller Δx raises the lower bound on Δp; the product need not equal that bound.
Rearrange: Δp_min = ħ/(2Δx). Use ħ = 1.056e-34 J s. If Δx = 6.00e-10 m, the bound is 8.80e-26 kg m/s. For a nonrelativistic electron of mass 9.11e-31 kg, Δp = mΔv along this axis. Dividing gives Δv ≥ 9.66e4 m/s, to three significant figures. These bounds do not specify the actual spreads.
The relation constrains the state before measurement. Instruments may add error or disturb a particle, but those effects are separate. Better equipment cannot prepare a state below ħ/2. A Gaussian packet without a quadratic position-dependent phase attains equality. Other states can have larger products.
Halving Δx doubles Δp_min = ħ/(2Δx). A hypothetical nucleus-sized Δx therefore gives a larger bound than an atom-sized Δx. Actual spreads can lie above their bounds, so they need not change inversely between two arbitrary states. To check a proposed state, multiply its spreads and compare with 0.50ħ. A product of 0.75ħ passes this check without attaining equality.
Position and momentum standard deviations obey ΔxΔp ≥ ħ/2. Narrower position raises the momentum lower bound. Instrument error and disturbance are separate.
Lesson, questions and flashcards written for this dataset using standard wave-packet and uncertainty-relation methods.
Readings and flashcards written for this dataset. Video references: Chad Orzel's TED-Ed lesson and Khan Academy's Heisenberg uncertainty principle video.
More to read
One state, two distributions
A position description gives ψ(x). Its squared magnitude is the position probability density. The same state also has a momentum distribution. Position density alone does not generally determine that distribution, because the wavefunction's phase matters too.
A plane-wave component has momentum p = ħk, where k is the signed wave number along the chosen axis. Combining a range of these components with suitable amplitudes and phases can produce a localised packet. Interference concentrates the amplitude in some regions and reduces it elsewhere. It need not remove every tail outside one spot.
An exact plane wave on the whole line has constant magnitude. A nonzero constant density has an infinite integral there, so this idealised component is not a normalised localised particle state. A normalisable packet combines components instead.
What the uncertainty spreads describe
Prepare many copies of one state. Measure position on one set of copies and momentum on another. Each set gives a distribution of outcomes. Δx and Δp are the standard deviations of those distributions along the same axis.
The inequality ΔxΔp ≥ ħ/2 constrains these state distributions. It does not require simultaneous measurements on one copy. It also does not say that each product equals ħ/2.
A measuring instrument may disturb the state or have limited resolution. Those are separate experimental effects. Removing instrument error would not remove the inequality. For a given position spread, the state must still have at least the momentum spread allowed by the bound.
A bound does not fix the actual spread
Rearranging the relation gives Δp_min = ħ/(2Δx). If Δx decreases by a factor of three, this lower bound increases by a factor of three. A state's actual Δp may be greater than its lower bound.
To test a proposed pair of spreads, multiply them and compare the result with ħ/2. A pair can satisfy this test without having the smallest possible product. Starting well above the bound also leaves room for both actual spreads to decrease between two states.
For a nonrelativistic particle of fixed mass, momentum and velocity components obey p = mv. Their standard deviations therefore obey Δp = mΔv. Divide the momentum lower bound by m to obtain a velocity-component lower bound. This is not generally a spread relation for speed magnitude.
A Gaussian packet without a quadratic position-dependent phase reaches equality. A Gaussian envelope alone does not guarantee the minimum product.
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.