Heisenberg Uncertainty Principle: Videos & Practice Problems

Werner Heisenberg's uncertainty principle states that the position and velocity of an electron cannot be known simultaneously due to its dual wave-particle nature. This principle is mathematically expressed as ${\Delta }_x{\Delta }_p\ge \frac{h}{4}$, where $h$ is Planck's constant. This relationship highlights the limitations in measuring quantum systems, emphasizing the concept of complementarity in quantum mechanics.

Heisenberg Uncertainty Principle

Werner Heisenberg's uncertainty principle highlights a fundamental limitation in quantum mechanics: the simultaneous measurement of an electron's velocity and position is impossible. This principle asserts that if you accurately determine the speed of an electron, its exact location becomes uncertain, and vice versa. This phenomenon arises from the dual nature of electrons, which exhibit both wave-like and particle-like characteristics.

The wave nature of electrons relates to their velocity, similar to how light energy propagates as a wave. Conversely, the particle nature of electrons pertains to their position, akin to light being composed of discrete packets known as photons. This duality is encapsulated in the concept of complementarity, which posits that electrons can be observed as either particles or waves, but not both at the same time. This intrinsic property of matter is a cornerstone of quantum mechanics, illustrating the complex behavior of subatomic particles.

Uncertainty Principle Formula

The Heisenberg uncertainty principle highlights the fundamental limits in measuring both the position and momentum of particles, particularly electrons. This principle can be articulated through the relationship between uncertainty in position and momentum, encapsulated in the uncertainty principle formula.

Momentum, defined as mass in motion, is represented by the equation:

$$\Delta p = m \cdot \Delta v$$

where Δp is the uncertainty in momentum (measured in kilograms times meters per second), m is the mass of the electron in kilograms, and Δv is the uncertainty in velocity.

The uncertainty principle formula itself is expressed as:

$$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$$

In this equation, Δx represents the uncertainty in position (in meters), and h is Planck's constant, valued at approximately 6.626 × 10^{-34} joules times seconds. This formula indicates that the product of the uncertainties in position and momentum has a lower bound determined by Planck's constant.

To further understand the implications of this principle, it is essential to recognize that the units of Planck's constant can be expressed in two ways: either as joules times seconds or as kilograms times meters squared per second. This equivalence is derived from the relationship:

$$1 \text{ joule} = 1 \text{ kg} \cdot \text{m}^2/\text{s}^2$$

Thus, when considering the units of Planck's constant, it can also be interpreted as:

$$\text{kg} \cdot \text{m}^2/\text{s}$$

This understanding of the Heisenberg uncertainty principle is crucial for grasping the limitations of quantum measurements and the inherent nature of particles at the quantum level.

Uncertainty Principle Formula Example

To calculate the uncertainty in the velocity of a neutron given the uncertainty in its position, we apply the Heisenberg uncertainty principle, which states that the product of the uncertainty in position (Δx) and the uncertainty in momentum (Δp) is greater than or equal to a constant. The formula can be expressed as:

Δx × m × Δv ≥ \(\frac{h}{4\pi}\)

Where:

• Δx is the uncertainty in position
• m is the mass of the neutron
• Δv is the uncertainty in velocity
• h is Planck's constant, approximately \(6.626 \times 10^{-34}\) Joules·seconds

In this scenario, the uncertainty in position is given as 712 picometers. To convert this to meters, we use the conversion factor where 1 picometer equals \(10^{-12}\) meters:

Δx = 712 pm = \(712 \times 10^{-12}\) m = \(7.12 \times 10^{-10}\) m

The mass of the neutron is provided as \(1.675 \times 10^{-27}\) kg. Substituting these values into the uncertainty principle equation, we have:

\(7.12 \times 10^{-10} \, \text{m} \times (1.675 \times 10^{-27} \, \text{kg}) \times Δv ≥ \frac{6.626 \times 10^{-34}}{4\pi}\)

Calculating the right side:

\(\frac{6.626 \times 10^{-34}}{4\pi} \approx 5.2728 \times 10^{-35} \, \text{kg} \cdot \text{m}^2/\text{s}\)

Now, we can simplify the left side:

Multiplying the values gives:

Left side: \(7.12 \times 10^{-10} \, \text{m} \times 1.675 \times 10^{-27} \, \text{kg} \approx 1.19 \times 10^{-36} \, \text{kg} \cdot \text{m}\)

Thus, we can rewrite the inequality as:

\(1.19 \times 10^{-36} \, \text{kg} \cdot \text{m} \times Δv ≥ 5.2728 \times 10^{-35} \, \text{kg} \cdot \text{m}^2/\text{s}\)

To isolate Δv, divide both sides by \(1.19 \times 10^{-36} \, \text{kg} \cdot \text{m}\):

Δv ≥ \(\frac{5.2728 \times 10^{-35}}{1.19 \times 10^{-36}}\) m/s

Calculating this gives:

Δv ≥ 444.2 m/s

Therefore, the uncertainty in the velocity of the neutron is greater than or equal to 444.2 m/s.

To what uncertainty (in m) can the position of a baseball traveling at 51.0 m/s be measured if the uncertainty of its speed is 0.12%? The mass of the baseball is 150 g.

• A. 4.2 x 10^-33 m
• B. 5.7 x 10^-33 m
• C. 5.9 x 10^-33 m
• D. 4.8 x 10^-33 m

Answer: (B)

An electron with a mass of 9.11 x 10^-31 kg has an uncertainty in its position of 630 pm. What is the uncertainty in its velocity?

• A. 9.0 x 10³ m/s
• B. 5.9 x 10³ m/s
• C. 9.2 x 10⁴ m/s
• D. 7.8 x 10⁴ m/s

Answer: (C)

A proton with a mass of 1.67 x 10^-27 kg traveling at 4.7 x 10⁵ m/s has an uncertainty in its velocity of 1.77 x 10⁵ m/s. Determine its uncertainty in position.

• A. 1.78 x 10 ^-13 m
• B. 3.56 x 10^-12 m
• C. 2.50 x 10^-13 m
• D. 4.24 x 10^-12 m

Answer: (A)

Here’s what students ask on this topic:

What is the Heisenberg uncertainty principle and why is it important?

The Heisenberg uncertainty principle states that it is impossible to simultaneously determine both the exact position and velocity (momentum) of an electron or any subatomic particle. This arises due to the dual wave-particle nature of electrons, known as complementarity. The principle is mathematically expressed as:

${\mathrm{\Delta x}}_{}\cdot {\mathrm{\Delta p}}_{}\ge \frac{h}{4\pi }$

where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and h is Planck's constant (6.626 × 10^-34 Joule seconds). This principle is crucial because it sets fundamental limits on our ability to measure and predict the behavior of particles, influencing quantum mechanics and chemistry.

How does the wave-particle duality relate to the Heisenberg uncertainty principle?

The wave-particle duality of electrons is central to the Heisenberg uncertainty principle. Electrons exhibit both wave-like and particle-like properties, but these characteristics cannot be observed simultaneously. The velocity of an electron is tied to its wave nature, while its position is linked to its particle nature. This duality creates a fundamental limit: if you measure the electron's position precisely, its wave-like velocity becomes uncertain, and vice versa. This relationship is called complementarity and is mathematically expressed by the uncertainty principle:

${\mathrm{\Delta x}}_{}\cdot {\mathrm{\Delta p}}_{}\ge \frac{h}{4\pi }$

Understanding this duality helps explain why precise measurements of subatomic particles are inherently limited.

What is the formula for the Heisenberg uncertainty principle and how is it derived?

The formula for the Heisenberg uncertainty principle is:

${\mathrm{\Delta x}}_{}\cdot {\mathrm{\Delta p}}_{}\ge \frac{h}{4\pi }$

where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and h is Planck's constant (6.626 × 10^-34 Joule seconds). Momentum (p) is defined as mass (m) times velocity (v), so Δp can be expressed as m × Δv, where Δv is the uncertainty in velocity. The derivation stems from quantum mechanics and the wave-particle duality of electrons. It reflects the fundamental limit imposed by the nature of subatomic particles, where precise measurements of position and momentum cannot coexist.

What are the units of Planck's constant in the Heisenberg uncertainty principle?

Planck's constant (h) has units of Joule seconds (J·s) in the Heisenberg uncertainty principle. A Joule is defined as kilograms times meters squared per seconds squared (kg·m²/s²), so the units of h can also be expressed as:

$\mathrm{kg}\cdot m^2\cdot s$

This unit reflects the relationship between energy, time, and the limits of precision in quantum mechanics. Understanding these units is essential for solving problems involving the uncertainty principle and interpreting the behavior of subatomic particles.

How does the Heisenberg uncertainty principle impact our understanding of electrons in chemistry?

The Heisenberg uncertainty principle profoundly impacts our understanding of electrons in chemistry. It explains why electrons cannot be precisely located within an atom, leading to the concept of atomic orbitals—regions where electrons are most likely to be found. This principle also influences the development of quantum mechanical models of atoms, replacing classical deterministic models. By acknowledging the limits of precision in position and momentum, chemists can better predict electron behavior, chemical bonding, and reactivity. The principle underscores the probabilistic nature of electron distributions, which is fundamental to modern chemistry.