Uncertainty and Probability

Classical physics often describes objects as having definite properties at every moment.

For example, if a ball is moving through the air, we can imagine that it has:

• an exact position
• an exact velocity
• an exact momentum
• a definite trajectory

If our measuring instruments are good enough, classical physics suggests that these quantities can, in principle, be determined with extremely high precision.

Quantum physics changes this picture.

For particles such as electrons, certain pairs of physical quantities cannot simultaneously have arbitrarily precise values.

This is expressed by the:

Heisenberg uncertainty principle

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Heisenberg's Uncertainty Principle

In 1927, German physicist Werner Heisenberg developed one of the central principles of quantum mechanics.

For position and momentum:

ΔxΔp ≥ ħ/2

where:

• Δx = uncertainty in position
• Δp = uncertainty in momentum
• ħ = reduced Planck's constant

The reduced Planck constant is:

ħ = h/(2π)

approximately:

ħ = 1.05 × 10⁻³⁴ J·s

Therefore:

ΔxΔp ≥ 5.27 × 10⁻³⁵ J·s

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What Does "Uncertainty" Mean?

The word uncertainty can be misleading.

It does not simply mean:

"Scientists do not have good enough equipment."

Instead, uncertainty is built into the quantum description itself.

A quantum state generally cannot assign both an arbitrarily precise position and an arbitrarily precise momentum to a particle simultaneously.

If position becomes more precisely defined:

Δx ↓

then momentum becomes less precisely defined:

Δp ↑

Similarly:

Δp ↓ → Δx ↑

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Position and Momentum

Momentum is:

p = mv

for a non-relativistic particle.

Therefore, momentum depends on:

• mass
• velocity

The uncertainty principle tells us that an electron cannot generally be described as simultaneously having:

exact position + exact momentum

This makes the classical idea of a perfectly defined electron trajectory inappropriate.

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A Trade-Off

The uncertainty principle creates a trade-off.

Imagine reducing the uncertainty in position.

If:

Δx becomes smaller

then:

Δp must become larger

so that:

ΔxΔp ≥ ħ/2

The product cannot be made arbitrarily small.

This is a fundamental feature of quantum mechanics.

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Worked Example: Minimum Momentum Uncertainty

Suppose an electron is localized to within:

Δx = 1.0 × 10⁻¹⁰ m

Use:

ΔxΔp ≥ ħ/2

The minimum momentum uncertainty is:

Δp ≥ ħ/(2Δx)

Substitute:

Δp ≥ (1.05 × 10⁻³⁴) / [2(1.0 × 10⁻¹⁰)]

Δp ≥ 5.25 × 10⁻²⁵ kg·m/s

Therefore, tightly localizing an electron creates a significant uncertainty in its momentum.

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Velocity Uncertainty

For a non-relativistic particle:

p = mv

If mass is constant:

Δp ≈ mΔv

Therefore:

Δx mΔv ≥ ħ/2

or:

Δv ≥ ħ/(2mΔx)

This allows us to estimate the minimum uncertainty in velocity.

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Worked Example: Electron Velocity

For an electron:

m = 9.11 × 10⁻³¹ kg

Suppose:

Δx = 1.0 × 10⁻¹⁰ m

Then:

Δv ≥ ħ/(2mΔx)

Δv ≥ (1.05 × 10⁻³⁴) / [2(9.11 × 10⁻³¹)(1.0 × 10⁻¹⁰)]

Δv ≈ 5.8 × 10⁵ m/s

This is a very large velocity uncertainty.

At atomic scales, uncertainty can therefore have major physical consequences.

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Why Quantum Uncertainty Matters for Electrons

Atoms have dimensions of approximately:

10⁻¹⁰ m

This is the same general scale used in the previous example.

Therefore, attempting to describe an electron as precisely localized within an atom produces a substantial uncertainty in its momentum.

This helps explain why the classical picture of an electron travelling around a precise circular orbit is inadequate.

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Classical Orbit vs Quantum State

In the Bohr model, an electron is pictured as travelling around the nucleus along a definite orbit.

Such a picture implies that the electron has:

• a definite position
• a definite momentum
• a definite trajectory

Quantum mechanics replaces this with:

wavefunction + probability distribution

Instead of asking:

"What exact path does the electron follow?"

we usually describe:

the probabilities associated with possible measurement outcomes.

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Probability in Quantum Physics

Probability is fundamental to quantum mechanics.

The state of a particle can be represented using a:

wavefunction

written:

ψ

The probability density is related to:

|ψ|²

This allows physicists to predict where a particle is more or less likely to be detected.

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Probability Density

Suppose an electron has a particular wavefunction.

Some regions may have:

high |ψ|²

These are regions where the electron is more likely to be detected.

Other regions may have:

low |ψ|²

These are regions where detection is less likely.

Some regions may have:

|ψ|² = 0

The electron will not be detected there for that state.

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Probability Does Not Mean Part of the Electron

Suppose an electron cloud is denser near the nucleus.

This does not mean:

"Most of the electron is near the nucleus."

An electron is not being divided into pieces.

Instead:

greater cloud density → greater probability of detecting the electron in that region

A measurement detects the electron as a localized event.

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Electron Clouds

The modern atomic model commonly represents electrons using:

probability clouds

A dense region represents:

high probability

A sparse region represents:

low probability

The cloud replaces the classical picture of a thin orbital track.

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Certainty vs Probability

Suppose quantum mechanics predicts:

70% probability

of detecting an electron in a particular region.

This does not mean that the electron is:

70% present

in that region.

It means that if the same quantum state is prepared and measured many times, approximately 70% of the measurements are expected to detect the electron in that region.

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Repeated Measurements

Imagine preparing:

10,000 identical atoms

in the same quantum state.

If the electron position is measured for each atom, different locations may be obtained.

After many measurements, the pattern approaches the probability distribution predicted by:

|ψ|²

Therefore:

individual measurement → uncertain outcome

but:

many measurements → predictable statistical distribution

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Probability Can Be Highly Predictable

Quantum mechanics is probabilistic, but this does not mean it cannot make precise predictions.

For example, a theory might predict:

• 25% probability for outcome A
• 75% probability for outcome B

We cannot necessarily predict the outcome of one individual measurement.

But after a very large number of measurements, the proportions can be predicted extremely accurately.

Therefore:

uncertainty about individual outcomes does not mean absence of scientific predictability.

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Example: Electron Diffraction

Electrons provide an excellent example.

An individual electron arriving at a detector appears at a particular:

location

But we cannot generally predict the exact location where each electron will arrive.

After many electrons:

a predictable diffraction or interference pattern emerges

This pattern corresponds to the probability distribution predicted by quantum mechanics.

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Single Events vs Statistical Patterns

Consider an electron double-slit experiment.

One electron

A single localized detection occurs.

Many electrons

A structured interference pattern appears.

The exact location of each individual detection is probabilistic.

The overall distribution is highly predictable.

This combination is characteristic of quantum mechanics.

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Wave Packets

A quantum particle can be represented by a combination of waves called a:

wave packet

A tightly localized wave packet gives relatively precise position information.

But creating that localized packet requires combining a wide range of wavelengths.

Because wavelength is related to momentum:

p = h/λ

a wide range of wavelengths corresponds to a wide range of momenta.

Therefore:

precise position → uncertain momentum

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A Spread-Out Wave

If a wave has a very well-defined wavelength, then its momentum is well defined because:

p = h/λ

However, a wave with a single precise wavelength extends over a large region of space.

Its position is therefore poorly defined.

Thus:

precise momentum → uncertain position

This provides a wave-based way of understanding the uncertainty principle.

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Uncertainty Is Not Just Measurement Disturbance

A common explanation says:

"Measuring an electron disturbs it."

Measurements can indeed disturb quantum systems.

However, that is not the fundamental reason for the uncertainty principle.

Even before considering experimental imperfections, quantum states themselves obey:

ΔxΔp ≥ ħ/2

The uncertainty principle is a property of quantum states.

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The Microscope Thought Experiment

Heisenberg discussed ideas involving trying to locate a particle using light.

To locate a very small particle accurately, we would want a very short wavelength.

But shorter-wavelength photons have greater momentum because:

p = h/λ

Interaction with the particle can therefore significantly change its momentum.

This thought experiment provides useful intuition.

However, modern quantum mechanics goes deeper:

uncertainty exists fundamentally in the quantum state, not merely because our measurement equipment disturbs the particle.

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Exact Position?

An electron can be detected at a specific position during a measurement.

But if the quantum state becomes strongly localized in position, the corresponding momentum distribution becomes broad.

Therefore, it is misleading to imagine that the electron always has an exact classical position and momentum that we simply have not discovered.

Quantum mechanics does not use that classical description.

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Atomic Orbitals and Probability

An atomic orbital describes a quantum state associated with an electron probability distribution.

For example:

1s orbital

Spherically symmetric probability distribution.

2p orbital

Two-lobed probability distribution.

Higher orbitals

More complicated probability structures.

Orbitals are therefore not paths.

They are connected to:

probabilities for measurement outcomes.

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Why Electron Orbits Were Replaced

The Bohr model pictured:

electron → definite circular orbit

The quantum mechanical model describes:

electron → quantum state represented by a wavefunction

This change was necessary because experiments showed:

• electron diffraction
• electron interference
• discrete atomic spectra
• uncertainty relationships
• probability distributions

The fixed-orbit picture could not explain all these observations.

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Uncertainty and Atomic Stability

The uncertainty principle also provides useful insight into why electrons do not simply collapse into the nucleus.

Suppose we tried to confine an electron to an extremely tiny region near the nucleus.

Then:

Δx becomes extremely small

According to:

ΔxΔp ≥ ħ/2

the uncertainty in momentum becomes very large.

A large momentum corresponds to substantial kinetic energy.

Therefore, extreme localization carries an energetic cost.

This contributes to the quantum-mechanical stability of atoms.

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Worked Example: Extreme Localization

Suppose an electron were confined to:

Δx = 1.0 × 10⁻¹⁵ m

approximately a nuclear length scale.

Minimum momentum uncertainty:

Δp ≥ ħ/(2Δx)

Δp ≥ (1.05 × 10⁻³⁴)/(2 × 10⁻¹⁵)

Δp ≥ 5.25 × 10⁻²⁰ kg·m/s

This is enormously larger than the uncertainty obtained when the electron is spread across atomic dimensions.

Attempting to confine an electron to nuclear dimensions therefore produces very large momentum uncertainty.

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Why Uncertainty Is Usually Invisible

The uncertainty principle applies to:

all quantum objects

including macroscopic objects.

But Planck's constant is extremely small:

h = 6.63 × 10⁻³⁴ J·s

For large objects, quantum uncertainties are normally negligible compared with ordinary measurement scales.

That is why:

• cars
• footballs
• planets

appear to follow well-defined classical trajectories.

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Worked Example: A Baseball

Suppose a 0.15 kg ball has its position known to within:

Δx = 1.0 × 10⁻⁶ m

Minimum momentum uncertainty:

Δp ≥ ħ/(2Δx)

Δp ≥ (1.05 × 10⁻³⁴)/(2 × 10⁻⁶)

Δp ≥ 5.25 × 10⁻²⁹ kg·m/s

Velocity uncertainty:

Δv ≈ Δp/m

Δv ≈ (5.25 × 10⁻²⁹)/(0.15)

Δv ≈ 3.5 × 10⁻²⁸ m/s

This is extraordinarily small.

Classical physics therefore works extremely well for the ball.

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Quantum vs Classical Probability

There is an important distinction between classical and quantum uncertainty.

Classical example

Suppose a card is face down.

It already has a particular identity.

We simply do not know which card it is.

The uncertainty comes from:

missing information

Quantum example

Quantum theory assigns probabilities to possible measurement outcomes as part of the physical description itself.

The probability is not merely a convenient description of poor measuring equipment.

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Probability and Certainty

Probabilities range from:

0 to 1

or:

0% to 100%

Probability 0 means:

the outcome will not occur under those conditions

Probability 1 means:

the outcome is certain under those conditions

Values between 0 and 1 indicate different likelihoods.

For example:

P = 0.25 → 25% probability

P = 0.80 → 80% probability

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Normalization

If all possible outcomes are included, their total probability must equal:

1

or:

100%

For example:

P(A) = 0.20

P(B) = 0.35

P(C) = 0.45

Then:

0.20 + 0.35 + 0.45 = 1.00

Quantum probability distributions must also satisfy this basic requirement.

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Probability Density Is Different from Probability

A probability density describes how probability is distributed through space.

The probability of finding a particle within a particular region is obtained from the probability density over that region.

Conceptually:

probability in region = contribution of |ψ|² across that region

For advanced mathematics, this is expressed using integration.

The important idea is:

|ψ|² tells us how detection probability is distributed through space.

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Nodes

Some quantum states contain regions where:

|ψ|² = 0

These are called:

nodes

At a node, the probability of detecting the electron is zero.

Nodes are evidence of the wave-like structure of atomic states.

They are similar to nodes in standing waves.

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Energy-Time Uncertainty

Another uncertainty relationship often encountered is:

ΔEΔt ≳ ħ/2

where:

• ΔE = characteristic energy uncertainty
• Δt = characteristic time interval

Its interpretation requires more care than the position-momentum uncertainty relation because time is treated differently from position in standard quantum mechanics.

At this level, the important point is that quantum uncertainty relationships are not limited to position and momentum.

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Quantum Measurements

Measurement plays an important role in quantum physics.

Before measurement, the quantum state may predict several possible outcomes with different probabilities.

When a measurement occurs:

one particular outcome is recorded

For example, an electron detector records:

electron detected here

rather than:

30% of an electron detected here

The probabilities describe possible outcomes across repeated measurements.

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Example: Probability Distribution

Suppose a quantum model predicts:

The most likely location is:

Region C

But it would be incorrect to say:

"The electron must be in Region C."

There is only a:

50% probability

of detecting it there.

Most likely does not mean certain.

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Certainty in Quantum Physics

Quantum mechanics does allow certain predictions to be definite in appropriate quantum states.

For example, a system prepared in a particular energy eigenstate can have a precisely defined energy.

However, other quantities may then have distributions of possible outcomes.

Therefore, quantum mechanics does not say:

"Everything is always uncertain."

Instead:

different observables can have different degrees of uncertainty depending on the quantum state.

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Uncertainty Does Not Mean Random Science

Another common misconception is:

"If quantum mechanics uses probability, it cannot make accurate predictions."

In reality, quantum mechanics is one of the most accurately tested theories in science.

It predicts probability distributions with extraordinary precision.

The uncertainty concerns particular measurement outcomes, not the ability of the theory to make testable predictions.

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From Probability to Technology

Quantum probability and uncertainty are not merely philosophical ideas.

Quantum mechanics underlies technologies such as:

• transistors
• semiconductor electronics
• lasers
• LEDs
• atomic clocks
• electron microscopes
• MRI technologies
• quantum sensors
• quantum computing

Modern electronics would not be understandable without quantum mechanics.

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Comparing Classical and Quantum Descriptions

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How Uncertainty Changed the Atomic Model

Earlier atomic models attempted to describe electrons using ideas borrowed from planetary motion.

For example:

planet → definite orbit around Sun

so scientists initially imagined:

electron → definite orbit around nucleus

Quantum mechanics showed that this analogy has major limitations.

Electrons must instead be described using:

• wavefunctions
• orbitals
• probability distributions
• quantized energies
• uncertainty relationships

This produced the modern:

quantum mechanical model of the atom

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From Bohr Orbits to Orbitals

Bohr:

"The electron occupies this allowed orbit."

Quantum mechanics:

"The electron occupies this quantum state, which predicts a probability distribution for possible measurements."

This is a major conceptual difference.

The word:

orbital

should therefore never be interpreted simply as a fuzzy Bohr orbit.

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Worked Example: Comparing Localization

Consider two identical electrons.

Electron A:

Δx = 1.0 × 10⁻⁹ m

Electron B:

Δx = 1.0 × 10⁻¹¹ m

Electron B is much more precisely localized.

Since:

ΔxΔp ≥ ħ/2

Electron B must have the larger minimum:

momentum uncertainty

Therefore:

greater position precision → greater momentum uncertainty

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Worked Example: Finding Position Uncertainty

Suppose an electron's momentum uncertainty is:

Δp = 2.0 × 10⁻²⁴ kg·m/s

Use:

Δx ≥ ħ/(2Δp)

Δx ≥ (1.05 × 10⁻³⁴)/(2 × 2.0 × 10⁻²⁴)

Δx ≥ 2.63 × 10⁻¹¹ m

Therefore, the minimum position uncertainty is approximately:

2.6 × 10⁻¹¹ m

This is comparable with atomic dimensions.

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Worked Example: Most Likely vs Certain

Suppose a calculation gives:

P(A) = 0.90

and:

P(B) = 0.10

Outcome A is:

much more likely

But it is not certain.

If the experiment is repeated many times, we expect approximately:

90% A

and:

10% B

Individual outcomes can still vary.

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Why the Quantum Model Is More Accurate

The uncertainty principle and quantum probability allow modern atomic theory to explain observations that fixed-orbit models cannot.

These include:

• electron diffraction
• interference
• atomic spectra
• orbital shapes
• atomic stability
• chemical bonding
• multi-electron atoms
• electron configurations

The quantum model is therefore more accurate because it agrees with a much wider range of experimental evidence.

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Common Mistakes

Mistake: Saying uncertainty comes only from bad measuring equipment.
The uncertainty principle is fundamental to quantum mechanics.

Mistake: Saying measuring instruments are the entire cause of uncertainty.
Measurement can disturb systems, but the uncertainty relation is a property of quantum states themselves.

Mistake: Saying an electron has no position.
An electron can be detected at a position. Quantum mechanics predicts probabilities for possible positions.

Mistake: Saying the electron is spread out as pieces throughout the cloud.
The cloud represents probability density.

Mistake: Saying the darkest part of an electron cloud is where the electron definitely is.
It represents greater probability, not certainty.

Mistake: Saying quantum mechanics cannot predict anything because outcomes are probabilistic.
It predicts statistical distributions extremely accurately.

Mistake: Saying everything in quantum mechanics is uncertain.
The uncertainty principle concerns specific relationships between observables and depends on the quantum state.

Mistake: Saying uncertainty is important for cars and footballs in everyday situations.
It applies to them, but the effects are normally negligibly small because Planck's constant is tiny.

Mistake: Treating an orbital as an uncertain Bohr orbit.
An orbital is a quantum state, not a poorly measured classical trajectory.

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Key Terms

Heisenberg uncertainty principle — fundamental quantum principle limiting how precisely certain pairs of observables can simultaneously be defined.

Position uncertainty (Δx) — spread associated with possible position measurements.

Momentum uncertainty (Δp) — spread associated with possible momentum measurements.

Reduced Planck constant (ħ) — constant equal to h/(2π), approximately 1.05 × 10⁻³⁴ J·s.

Probability — numerical measure of the likelihood of a possible outcome.

Probability density — distribution describing how detection probability varies through space.

Wavefunction (ψ) — mathematical representation of a quantum state.

|ψ|² — quantity related to probability density.

Electron cloud — visual representation of an electron's spatial probability distribution.

Atomic orbital — quantum state associated with a characteristic electron probability distribution.

Wave packet — localized combination of waves used to represent a quantum particle.

Measurement — physical process that produces an observable outcome.

Quantum state — mathematical description containing information used to predict measurement probabilities.

Node — region where a particular wavefunction has zero probability density.

Trajectory — path followed by an object through space; classical trajectories are generally inappropriate for atomic electrons.

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Key Takeaways

• The Heisenberg uncertainty principle is a fundamental part of quantum mechanics.
• For position and momentum: ΔxΔp ≥ ħ/2.
• More precise position means greater minimum uncertainty in momentum.
• More precise momentum means greater minimum uncertainty in position.
• Quantum uncertainty is not simply caused by poor measuring instruments.
• Measurement disturbance can occur, but it is not the fundamental origin of the uncertainty principle.
• The uncertainty principle makes fixed classical electron trajectories inappropriate.
• Quantum mechanics describes electrons using wavefunctions and probability distributions.
• |ψ|² is related to the probability density for detecting a particle.
• Electron clouds represent probability, not physical electron material.
• Individual quantum measurements can produce unpredictable outcomes.
• Large numbers of measurements produce highly predictable statistical patterns.
• "Most likely" does not mean "certain."
• Quantum probability is more fundamental than ordinary uncertainty caused by missing information.
• Electrons can be detected at particular locations even though their quantum states describe distributions of possible positions.
• Atomic orbitals are quantum states, not fixed electron paths.
• Localizing an electron more strongly increases its momentum uncertainty.
• Quantum uncertainty is significant at atomic and subatomic scales.
• The same principles technically apply to macroscopic objects, but the effects are normally extraordinarily small.
• Uncertainty and probability were essential in replacing fixed-orbit atomic models with the modern quantum mechanical model.
• Quantum mechanics is probabilistic but still makes extraordinarily precise, experimentally testable predictions.

Check Your Understanding

1. State the Heisenberg uncertainty principle for position and momentum.
2. What do Δx and Δp represent?
3. What is the approximate value of ħ?
4. What happens to the minimum momentum uncertainty when position becomes more precisely defined?
5. Why is quantum uncertainty not simply caused by poor measuring instruments?
6. Explain why the uncertainty principle creates problems for the idea of a fixed electron orbit.
7. What does |ψ|² represent?
8. What does a dense region in an electron cloud indicate?
9. Explain why an electron cloud should not be interpreted as electron material spread through space.
10. What is the difference between a probable outcome and a certain outcome?
11. If an electron has an 80% probability of being detected in Region A, does that guarantee it will be detected there? Explain.
12. How can individual electron detections be unpredictable while the overall distribution remains predictable?
13. Explain how the electron double-slit experiment demonstrates quantum probability.
14. What is a wave packet?
15. Explain why a strongly localized wave packet has a large momentum uncertainty.
16. Calculate the minimum momentum uncertainty for an electron localized to Δx = 2.0 × 10⁻¹⁰ m.
17. Why are uncertainty effects normally negligible for everyday macroscopic objects?
18. Explain how the uncertainty principle helps show why the Bohr picture of fixed electron paths is incomplete.
19. Compare classical uncertainty caused by missing information with quantum uncertainty.
20. Explain how uncertainty and probability changed scientists' understanding of atomic structure and contributed to the modern quantum mechanical model.