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Chapter 7: Problem 8

Can we measure the energy of a free localized particle with complete precision?

Short Answer

Expert verified
No, we cannot measure the energy of a free localized particle with complete precision. According to the Heisenberg Uncertainty Principle, the product of the uncertainties in position and energy is always greater than or equal to \(\frac{\hbar}{2}\). If we try to make the energy uncertainty very small, it would mean there is infinite uncertainty in the position, making the particle no longer localized. Therefore, it is not possible to measure the energy of a free localized particle with complete precision without sacrificing information about its position.

Step by step solution

01

Definition of the Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle is a fundamental principle in quantum mechanics that states the following relation for position (\(x\)) and momentum (\(p\)): \[\Delta x \cdot \Delta p \geq \frac{\hbar}{2}\] Here, \(\Delta x\) represents the uncertainty in position, \(\Delta p\) represents the uncertainty in momentum and \(\hbar\) is the reduced Planck constant (approximately \(1.05 \cdot 10^{-34} \, Js\)).
02

Energy uncertainty

Since the particle under consideration is free, it is not subject to any potential energy. Therefore, its total energy is equal to its kinetic energy. The kinetic energy of a particle can be expressed in terms of its momentum: \[E = \frac{p^2}{2m}\] Here, \(E\) is the energy, \(p\) is the momentum, and \(m\) is the mass of the particle. To find the uncertainty in energy, we can differentiate the kinetic energy expression with respect to the momentum: \[\frac{dE}{dp} = \frac{p}{m}\] Now, we can express the uncertainty in energy (\(\Delta E\)) in terms of the uncertainty in momentum (\(\Delta p\)): \[\Delta E = \frac{dE}{dp} \cdot \Delta p = \frac{p}{m} \cdot \Delta p\]
03

Uncertainty in energy and position

We're trying to find out if we can measure the energy of a free localized particle with complete precision. This means that we want the value of \(\Delta E\) to be as small as possible, ideally zero. However, we know from Heisenberg's Uncertainty Principle that a decrease in the uncertainty of one variable (in this case, energy) leads to an increase in the uncertainty of its conjugate variable (position). Therefore, we can rewrite the uncertainty principle using the expression for \(\Delta E\): \[\Delta x \cdot \Delta E = \Delta x \cdot \frac{p}{m} \cdot \Delta p \geq \frac{\hbar}{2}\]
04

Analyzing the results

From the inequality derived in Step 3, we see that the product of the uncertainty in position and energy is always greater than or equal to \(\frac{\hbar}{2}\). If we try to make the uncertainty in energy (\(\Delta E\)) very small (ideally zero), it would mean there is infinite uncertainty in the position (\(\Delta x\)), making the particle no longer localized. The Heisenberg Uncertainty Principle does not allow for complete precision in measuring the energy of a localized particle without sacrificing information about its position. Therefore, it is not possible to measure the energy of a free localized particle with complete precision.

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