Question

A scientist has devised a new method of isolating individual particles. He claims that this method enables him to detect simultaneously the position of a particle along an axis with a standard deviation of 0.12 nm and its momentum component along this axis with a standard deviation of 3.0 \(\times\) 10\(^{-25}\) kg \(\bullet\) m/s. Use the Heisenberg uncertainty principle to evaluate the validity of this claim.

Short answer

The claim violates the Heisenberg Uncertainty Principle.

Step-by-step solution

Understanding the Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that it is impossible to simultaneously know the exact position and momentum of a particle. Mathematically, it is expressed as \( \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \), where \( \Delta x \) is the standard deviation of position, \( \Delta p \) is the standard deviation of momentum, and \( \hbar \) is the reduced Planck's constant (\( \hbar \approx 1.0545718 \times 10^{-34} \) Js).

Substitute Given Values

In the problem, \( \Delta x = 0.12 \) nm = \( 0.12 \times 10^{-9} \) m and \( \Delta p = 3.0 \times 10^{-25} \) kg \( \cdot \) m/s. Substitute these values into the inequality formula: \( \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \).

Calculate Left Side of Inequality

Multiply \( \Delta x \) and \( \Delta p \) to find the left side of the inequality: \( 0.12 \times 10^{-9} \times 3.0 \times 10^{-25} = 3.6 \times 10^{-35} \).

Calculate Right Side of Inequality

Calculate the right side of the inequality using \( \frac{\hbar}{2} = \frac{1.0545718 \times 10^{-34}}{2} \approx 5.272859 \times 10^{-35} \).

Compare Both Sides of Inequality

Compare the results: \( 3.6 \times 10^{-35} < 5.272859 \times 10^{-35} \). The left side is smaller than the right side, which indicates that the uncertainty product is less than \( \frac{\hbar}{2} \), violating the Heisenberg Uncertainty Principle.

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the behavior of particles at the smallest scales, such as atoms and subatomic particles. Unlike classical physics, which works well for larger objects, quantum mechanics takes into account the wave-like nature of particles. This means that particles do not have definite positions or velocities until they are measured. Instead, their properties are described by probabilities. A key feature of quantum mechanics is the concept of wave-particle duality. This idea suggests that every particle exhibits both particle-like and wave-like behavior. For example:

• Photons can exhibit wave behavior, such as interference, but also particle behavior when they hit a detector one by one.
• Electrons can form interference patterns when passed through a double-slit apparatus, highlighting their wave nature.

This fundamentally changes how we think about particles and forces us to reconsider traditional notions of measurement and observation.

Standard Deviation

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion of a set of values. In simple terms, it tells us how spread out the values are from the mean (average). In the context of quantum mechanics, it is used to express the uncertainty in the position or momentum of a particle. In the Heisenberg Uncertainty Principle, standard deviation plays a crucial role as it defines the range within which the values are likely to be found. For example:

• If the standard deviation of a particle's position is small, it means the position is known quite accurately.
• If the standard deviation is large, the position is more uncertain.

Understanding this concept is vital, as the product of the standard deviations of position and momentum must always respect the inequality prescribed by the Heisenberg Uncertainty Principle.

Particle Physics

Particle physics is a branch of physics that investigates the fundamental constituents of matter and the forces through which they interact. In this field, researchers study particles such as electrons, protons, quarks, and neutrinos. Understanding these particles is crucial for explaining how matter is constructed and interacts. In particle physics, experiments are often conducted using particle accelerators, which collide particles at very high speeds. These collisions help reveal the properties of particles and allow scientists to observe behaviors that are not evident in everyday experiences.

• The Standard Model is a theory that describes the electromagnetic, weak, and strong nuclear interactions among these particles.
• Experiments in particle physics have led to the discovery of numerous particles, including the Higgs boson, which gives other particles their mass.

Particle physics continues to search for answers to unresolved questions, such as the nature of dark matter and energy, further unraveling the mysteries of the universe.