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Chapter 34: Problem 11

Energy-time uncertainty limits the precision with which we can know the mass of unstable particles (those that decay after a finite time). Why?

Short Answer

Expert verified
The energy-time uncertainty principle in quantum mechanics, represented by \(\Delta E \Delta t \geq \frac{h}{4\pi}\), establishes a fundamental limit to the precision whereby we can know both the energy and time of a subatomic event. Given an unstable particle decays after a finite time, it means there's an inherent uncertainty in energy derived from their decay time (which is proportional to their mass). Hence, precision in determining mass of such particles is fundamentally restricted by this principle.

Step by step solution

01

Understanding the Energy-Time Uncertainty Principle

The energy-time uncertainty principle is a critical concept in quantum mechanics. It is represented by the inequality \(\Delta E \Delta t \geq \frac{h}{4\pi}\), where \(\Delta E\) is the uncertainty in measurement of energy, \(\Delta t\) is the uncertainty in measurement of time, and \(h\) is Planck's constant.
02

Relating Unstable Particles to the Principle

Unstable particles decay after a finite time, meaning that there is an inherent uncertainty in their decay time. The decay of these particles can be considered a subatomic event, subject to the energy-time uncertainty principle. The energy involved in this event is proportional to the mass (according to Einstein's relativity equation, \(E=mc^2\)), hence, determining the mass of these particles would be uncertain.
03

Deriving the Correlation

A smaller decay time translates to a larger uncertainty in energy according to the energy-time uncertainty principle. Hence, a smaller lifetime would result in a larger uncertainty in the determination of the mass of these particles. The correlation therefore lies in the ability to determine both the decay time and energy simultaneously, which is fundamentally limited.

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