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Chapter 7: Problem 9

Can we measure both the position and momentum of a particle with complete precision?

Short Answer

Expert verified
No, we cannot measure both the position and momentum of a particle with complete precision due to the Heisenberg Uncertainty Principle in quantum mechanics. This principle states that the product of the uncertainties in position (\(\Delta x\)) and momentum (\(\Delta p\)) is always greater than or equal to a specific constant value: \[ \Delta x \Delta p \geq \dfrac{\hbar}{2} \] Here, \(\hbar\) represents the reduced Planck constant. Consequently, as the precision of one property increases, the precision of the other property decreases. This is a fundamental property of nature and not a limitation of our measuring techniques.

Step by step solution

01

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle is a fundamental concept in quantum mechanics that places a limit on the precision with which certain pairs of physical properties (in this case, position and momentum) can be simultaneously measured. Mathematically, the principle states that the product of the uncertainties in position (\(\Delta x\)) and momentum (\(\Delta p\)) is always greater than or equal to a particular constant value, as per the following inequality: \[ \Delta x \Delta p \geq \dfrac{\hbar}{2} \] Here, \(\hbar\) is the reduced Planck constant, which is approximately \(1.0545718 脳 10^{-34} \, \text{J} \cdot \text{s}\).
02

Can We Measure Both Position and Momentum with Complete Precision?

According to the Heisenberg Uncertainty Principle, it is fundamentally impossible to measure both the position and momentum of a particle with complete precision. As the uncertainty in position decreases (\(\Delta x \rightarrow 0\)), the uncertainty in momentum must increase (\(\Delta p \rightarrow \infty\)), in order to satisfy the Heisenberg Uncertainty Principle inequality. Similarly, if we try to measure momentum with extreme precision, the position becomes highly uncertain. This is not a limitation of our measuring techniques, but a fundamental property of nature. In conclusion, due to the Heisenberg Uncertainty Principle in quantum mechanics, we cannot measure both the position and momentum of a particle with complete precision. The very act of measuring one of these properties with higher precision introduces greater uncertainty in the other.

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Key Concepts

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

Heisenberg Uncertainty Principle
The Heisenberg Uncertainty Principle is a cornerstone of quantum mechanics. It dictates that you cannot precisely know both the position and momentum of a particle at the same time. Imagine you're trying to catch a bouncing ball in the dark. If you shine a flashlight to see exactly where the ball is (its position), it might bounce away quickly, altering its momentum. This principle is expressed mathematically by the inequality: \[ \Delta x \Delta p \geq \dfrac{\hbar}{2} \]Here,
  • \(\Delta x\) represents the uncertainty in position,
  • \(\Delta p\) represents the uncertainty in momentum,
  • \(\hbar\) is the reduced Planck constant.
This principle is not due to limitations in our measurements but is a fundamental aspect of nature. It shapes how we understand particles at the quantum level.
Position and Momentum Measurement
In the realm of quantum mechanics, measuring both position and momentum of a particle accurately is akin to a balancing act. The more accurately you measure one, the less accurately you can know the other. This is not just a theoretical constraint but a law of nature according to the Heisenberg Uncertainty Principle.
  • If you try to pinpoint a particle's position with exceptional accuracy, its momentum becomes increasingly uncertain.
  • Conversely, if a particle's momentum is measured with precision, its position becomes blurred and uncertain.
This is not a defect in our instruments but a fundamental characteristic of how particles behave at tiny scales. Therefore, perfect simultaneous measurement remains an unattainable goal in the quantum world.
Reduced Planck Constant
The reduced Planck constant, symbolized as \(\hbar\), is a small yet significant value in quantum mechanics. It is critical in the Heisenberg Uncertainty Principle, which dictates a limit to how precisely certain pairs of quantum properties like position and momentum can be known simultaneously.Its value is approximately \(1.0545718 脳 10^{-34} \text{ J} \cdot \text{s}\). This constant provides the scale at which quantum effects become significant. In practical terms, \(\hbar\) defines the 'quantum fuzziness' that exists due to the uncertainty principle.Understanding \(\hbar\) helps in grasping why quantum mechanics doesn't just describe smaller versions of our everyday experiences, but an entirely different set of rules where certainty is banished and probability rules.
Fundamental Limits of Measurement
At the heart of the Heisenberg Uncertainty Principle lie the fundamental limits of measurement in quantum mechanics. These limits compel us to rethink what measurement means at the smallest scales of existence. Unlike classical mechanics, where things can be accurately measured and predicted, quantum mechanics accepts uncertainty as part of the game.
  • Nature imposes a limit on the precision with which pairs of properties like position and momentum can be simultaneously known.
  • This implies that the more you try to nail down a particle's position, the more unpredictable its momentum becomes.
  • It highlights a new vision where probabilities replace certainties.
These limitations reflect the inherent properties of quantum systems and not just shortcomings of our tools. As such, they guide our understanding of the universe's fundamental laws in profound ways.

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Most popular questions from this chapter

A particle with mass \(m\) moving along the \(x\) -axis and its quantum state is represented by the following wave function: $$ \Psi(x, t)=\left\\{\begin{aligned} 0, & x<0 \\ A x e^{-\alpha x} e^{-i E t / \hbar}, & x \geq 0 \end{aligned}\right. $$ where \(\alpha=2.0 \times 10^{10} \mathrm{m}^{-1} .\) (a) Find the normalization constant. (b) Find the probability that the particle can be found on the interval \(0 \leq x \leq L\). (c) Find the expectation value of position. (d) Find the expectation value of kinetic energy.

A gas of helium atoms at \(273 \mathrm{K}\) is in a cubical container with \(25.0 \mathrm{cm}\) on a side. (a) What is the minimum uncertainty in momentum components of helium atoms? (b) What is the minimum uncertainty in velocity components? (c) Find the ratio of the uncertainties in (b) to the mean speed of an atom in each direction.

For a quantum particle in a box, the first excited state \(\left(\Psi_{2}\right)\) has zero value at the midpoint position in the box, so that the probability density of finding a particle at this point is exactly zero. Explain what is wrong with the following reasoning: "If the probability of finding a quantum particle at the midpoint is zero, the particle is never at this point, right? How does it come then that the particle can cross this point on its way from the left side to the right side of the box?

An electron confined to a box of width 0.15 nm by infinite potential energy barriers emits a photon when it makes a transition from the first excited state to the ground state. Find the wavelength of the emitted photon.

Consider an infinite square well with wall boundaries \(\begin{array}{llllll}x=0 & \text { and } & x=L & \text { Show } & \text { that } & \text { the function }\end{array}\) \(\psi(x)=A \sin k x \quad\) is the solution to the stationary Schr枚dinger equation for the particle in a box only if \(k=\sqrt{2 m E} / \hbar .\) Explain why this is an acceptable wave function only if \(k\) is an integer multiple of \(\pi / L\)
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