Chapter 5: Q57E (page 191)

The uncertainty in a particle's momentum in an infinite well in the general case of arbitrary $n$ is given by $\frac{n π h}{L}$ .

Short Answer

Expert verified

For $n = 0$ the uncertainty vanishes but is perfectly finite for all other values. Thus, so long as $Δ x^{2}$ is large enough (it is), the uncertainty principle is perfectly satisfied for all $n > 0$ .

Step by step solution

The concept and the formula used.

Heisenberg's uncertainty principle states that it is impossible to measure or calculate exactly, both the position and the momentum of an object.

Consider, energy $E = 0$ . Then, the momentum of the state must satisfy $E = \frac{p^{2}}{2 m}$ . Now, there are two solutions for $P$ corresponding to positive and negative momentum. The uncertainty can thus be calculated as follows:

${ΔP}^{2} = ⟨ P^{2} ⟩ - {⟨ P ⟩}^{2}$

$= 2 m E$

$= \frac{n^{2} π^{2} ℏ^{2}}{L^{2}}$

$ΔP = \frac{nπ ℏ}{L}$

Conclusion

Clearly, for $n = 0$ the uncertainty vanishes but is perfectly finite for all other values. Thus, so long as $Δ x^{2}$ is large enough (it is), the uncertainty principle is perfectly satisfied for all $n > 0$ .

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The uncertainty in a particle's momentum in an infinite well in the general case of arbitrary $n$ is given by $\frac{n π h}{L}$ .

Short Answer

Step by step solution

The concept and the formula used.

Conclusion

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The uncertainty in a particle's momentum in an infinite well in the general case of arbitrary nis given bynπhL .

Short Answer

Step by step solution

The concept and the formula used.

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Further Mechanics and Thermal Physics

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Study anywhere. Anytime. Across all devices.

The uncertainty in a particle's momentum in an infinite well in the general case of arbitrary $n$ is given by $\frac{n π h}{L}$ .