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Chapter 6: Problem 51

Does the Heisenberg uncertainty principle allow a particle to be at rest in a designated region in space?

Short Answer

Expert verified
The Heisenberg uncertainty principle does not allow a particle to be at rest in a designated region in space with a specific position, as it states that there is a fundamental limit to the accuracy with which we can simultaneously know the position and momentum of a particle. If a particle is at rest (\(\Delta p = 0\)), it would imply that \(\Delta x \Delta p = 0\), which violates the principle given by the formula \(\Delta x \Delta p \geq \frac{\hbar}{2}\). Therefore, a particle can never be precisely at rest in a designated region in space due to the inherent uncertainty in position and momentum as described by the principle.

Step by step solution

01

Understanding the Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle is given by the formula: \(\Delta x \Delta p \geq \frac{\hbar}{2}\) where \(\Delta x\) is the uncertainty in position, \(\Delta p\) is the uncertainty in momentum, and \(\hbar\) is the reduced Planck constant (\(\hbar = h / 2\pi\), where \(h\) is the Planck constant). The inequality means that the product of the uncertainties in position and momentum cannot be less than \(\frac{\hbar}{2}\).
02

Particle at Rest

For a particle to be at rest, its momentum must be zero. Therefore, the uncertainty in momentum, \(\Delta p\), must also be zero.
03

Applying the Heisenberg Uncertainty Principle

Now let's apply the Heisenberg Uncertainty Principle to the situation where the particle is at rest. As we have stated that \(\Delta p = 0\), if a particle is at rest in a designated region in space with a specific position, its uncertainty in position, \(\Delta x\), must also be zero. This would imply that \(\Delta x \Delta p = 0\), which would violate the Heisenberg Uncertainty Principle as it must be greater than or equal to \(\frac{\hbar}{2}\), not equal to zero.
04

Conclusion

According to the Heisenberg uncertainty principle, a particle cannot be at rest in a designated region in space with a specific position without violating the fundamental limit to simultaneously knowing its position and momentum. The particle can never be precisely at rest in a designated region in space due to the inherent uncertainty in position and momentum as described by the principle.

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