Established physics · Quantum Mechanics 201 · Lesson 7 of 15
Uncertainty and the width of a wave packet
A state cannot make both position and momentum distributions arbitrarily narrow; this is a relation between state spreads, beyond ordinary instrument error.
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Picture it
Build a localized pulse by combining waves with different wavelengths. A long, nearly single-wavelength wave has a narrow wavelength range but is spread out in position. Bringing many wavelengths together can make a sharper pulse. In quantum mechanics those wavelengths correspond to momenta, so tightening position localization broadens the momentum distribution needed to build the state.
Ask about an ensemble spread
Prepare the same state many times. On one set of trials measure position; on another set measure momentum. Each set produces its own distribution. Their standard deviations quantify the spreads called Δx and Δp. The uncertainty relation constrains these state distributions. It is not a claim that an individual detector is unable to print a precise number.
A physical state constraint
Measurement disturbance is a real topic, but it is not the whole content of the position-momentum uncertainty relation. Even with ideal instruments, no normalized state has both distributions arbitrarily sharp. Position and momentum are represented by incompatible operators, and their wavefunction descriptions are related by a Fourier transform. The shape tradeoff exists before adding a story about clumsy apparatus.
A minimum-width example
A suitable Gaussian wave packet saturates the bound: decreasing its position standard deviation increases its momentum standard deviation in reciprocal proportion. Other states can have a larger product. The sliders below show this specific minimum-uncertainty family, not a statement that every state must sit exactly on the lower bound.
Do not turn this into a universal excuse
Uncertainty does not mean every physical quantity is equally vague, that conservation laws are optional, or that any outcome is possible. A state can have a sharply defined energy or one definite spin component while other measurements remain distributed. Identify the pair of observables and the state before deciding which quantitative constraint applies.
- Combine momentum components
- Build a position wave packet
- Compare both distribution widths
The Gaussian example reaches the lower bound; a general state may have a larger spread product.
Try the example on paper
With ℏ = 1: Δx = 0.5 gives Δp = 1; Δx = 1 gives Δp = 0.5; Δx = 2 gives Δp = 0.25. All three minimum Gaussians have Δx Δp = 0.5.
Make it concrete
In the demonstration’s units, a Gaussian with Δx = 1 has Δp = 0.5. Tightening it to Δx = 0.5 gives Δp = 1. Their products both equal 0.5. Doubling the position precision in this sense doubles the momentum spread. Neither change requires saying that the particle borrowed energy or that the detector stopped conserving it.
The precise version
For canonical position and momentum, Δx Δp ≥ ℏ/2. For the unchirped Gaussian family used here, Δp = ℏ/(2Δx). The demonstration sets ℏ = 1 and displays complementary standard deviations. More generally, ΔA ΔB ≥ |⟨[A,B]⟩|/2; the state and the operators determine the bound.
Watch the trap
Do not interpret Δx as an ordinary ruler’s calibration error or replace the inequality with an equality for every state. The displayed Gaussian is one special family that reaches the minimum.
Keep this
Uncertainty constrains the distributions a state can support. Narrower position localization requires a broader momentum distribution.
Words for this lesson: uncertainty · wave packet · momentum · wavefunction · observable · standard deviation
Check your understanding
Reason through each question before opening its answer and explanation.
1. What do Δx and Δp describe here?
- The standard deviations of position and momentum outcome distributions
- Only the manufacturer’s instrument error ratings
- The time between two clicks
Show answer and explanation
The standard deviations of position and momentum outcome distributions
They quantify the spreads predicted for repeated measurements on equivalently prepared systems.
2. For the minimum Gaussian with ℏ = 1, Δx = 0.5 gives what Δp?
- 1
- 0
- 0.25
Show answer and explanation
1
Δp = 1/(2Δx) = 1, preserving the product 0.5.
3. Does better instrument calibration remove the position-momentum state bound?
- Only if a person reads both detectors
- Yes, it is only poor calibration
- No, the bound also holds for ideal measurements of the state distributions
Show answer and explanation
No, the bound also holds for ideal measurements of the state distributions
The canonical uncertainty relation is a constraint on quantum states and incompatible observables.
4. Must every quantum state have Δx Δp exactly ℏ/2?
- No; many states have a larger product
- Yes, the inequality is always an equality
- No; a normalized state can have both zero
Show answer and explanation
No; many states have a larger product
The Gaussian family here reaches a lower bound. The general relation permits larger products.
5. Which preparation tends to require a broader momentum range?
- A position wave extending almost uniformly over a large region with nearly one wavelength
- Either one always has the same momentum distribution
- A more sharply localized position wave packet
Show answer and explanation
A more sharply localized position wave packet
Sharper localization requires more wave-number components, which correspond to a broader momentum range.
6. Does uncertainty permit a particle to ignore energy conservation?
- Only during an unread measurement
- Yes, whenever the time is short enough
- No; it does not suspend conservation laws
Show answer and explanation
No; it does not suspend conservation laws
The uncertainty relation does not grant an energy loan. Tunneling will be explained using energy-conserving dynamics.