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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 states that we can't know both the exact energy of a particle and the exact time at which that energy was measured. Therefore, if we know exactly when an unstable particle decays (its time), we can't also know its exact energy, and by extension, via Einstein's relation, its mass. This inherent limitation of quantum mechanics means we can't precisely know the mass of unstable particles.

Step by step solution

01

Understand the Energy-Time Uncertainty Principle

In quantum physics, Heisenberg's uncertainty principle tells us that it's impossible to measure precisely both the position and the momentum (or similarly, energy and time) of a particle at the same time. The more accurately we know one of these values, the less accurately we know the other. It's important to remember that this isn't due to any lack in our measurement methods; it's a fundamental aspect of quantum mechanics.
02

Apply this Principle to Energy and Time

The energy-time version of the uncertainty principle states that the uncertainty in energy, \( \Delta E \), and the uncertainty in time, \( \Delta t \), must satisfy the relationship \( \Delta E \cdot \Delta t \geq \hbar/2 \). Here, \( \hbar \) is reduced Planck's constant. According to this principle, you can never simultaneously know the exact energy and the exact time at which that energy is measured.
03

Connect to Unstable Particles

Unstable particles are particles that decay into other particles after a finite time. If we know exactly when an unstable particle decays (i.e., if \( \Delta t \) is 0), the energy uncertainty (\( \Delta E \)) must be infinitely large due to the principle discussed in step 2. As mass and energy are related by Einstein's relation \( E = mc^2 \), the more uncertain we are about energy, the more uncertain we are about mass; therefore, the energy-time uncertainty principle limits the precision with which we can know the mass of unstable particles.

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