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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

Expert verified
The claim violates the Heisenberg Uncertainty Principle.

Step by step solution

01

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).
02

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: \( \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \).
03

Calculate Left Side of Inequality

Multiply \( \Delta x \) and \( \Delta p \) to find the left side of the inequality: \( 0.12 \times 10^{-9} \times 3.0 \times 10^{-25} = 3.6 \times 10^{-35} \).
04

Calculate Right Side of Inequality

Calculate the right side of the inequality using \( \frac{\hbar}{2} = \frac{1.0545718 \times 10^{-34}}{2} \approx 5.272859 \times 10^{-35} \).
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

(a) A particle with mass \(m\) has kinetic energy equal to three times its rest energy. What is the de Broglie wavelength of this particle? (\(Hint\): You must use the relativistic expressions for momentum and kinetic energy: \(E^2 = (pc^2) + (mc^2)^2\) and \(K = E - mc^2\).) (b) Determine the numerical value of the kinetic energy (in MeV) and the wavelength (in meters) if the particle in part (a) is (i) an electron and (ii) a proton.

(a) What accelerating potential is needed to produce electrons of wavelength 5.00 nm? (b) What would be the energy of photons having the same wavelength as these electrons? (c) What would be the wavelength of photons having the same energy as the electrons in part (a)?

A certain atom has an energy state 3.50 eV above the ground state. When excited to this state, the atom remains for 2.0 \(\mu\)s, on average, before it emits a photon and returns to the ground state. (a) What are the energy and wavelength of the photon? (b) What is the smallest possible uncertainty in energy of the photon?

A beam of electrons is accelerated from rest through a potential difference of 0.100 kV and then passes through a thin slit. When viewed far from the slit, the diffracted beam shows its first diffraction minima at \(\pm\)14.6\(^\circ\) from the original direction of the beam. (a) Do we need to use relativity formulas? How do you know? (b) How wide is the slit?

What is the de Broglie wavelength of a red blood cell, with mass 1.00 \(\times\) 10\(^{-11}\) g, that is moving with a speed of 0.400 cm/s? Do we need to be concerned with the wave nature of the blood cells when we describe the flow of blood in the body?

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