Chapter 39: Q. 41 (page 1118)

What is the smallest one-dimensional box in which you can confine an electron if you want to know for certain that the electron's speed is no more than $10 m / s$ ?

Short Answer

Expert verified

The smallest one-dimensional box is $18.2 \times 10^{- 6} m .$

Step by step solution

part (a) step 1: Given information

Electron's velocity limit: $10 m / s$

part (a) step 2: formulas for solutions

$Δ x Δ p = \frac{h}{2}$

Whereas, $Δ x$ is the change in position.

$Δ p$ is change in momentum.

h is Plank's constant $(h = 6.63 \times 10^{- 34} Js)$

part (b) step 3: calculation

The range of electron's velocity is $10 m / s$ so it can be varied from $- 10 m / s$ too $+ 10 m / s .$

So the change in velocity is $20 m / s .$

$Δ p$ Change in momentum can be given as $Δ p = m Δ v$ where; $m$ is the mass of electron

$Δ p = m Δ v Δ p = 9.1 \times 10^{- 31} kg \times 20 m / s Δ p = 1.82 \times 10^{- 29}$

Here, $Δ x$ is the uncertainty in the position of an electron which is equal to the dimension of the box L

So by using the formula $Δ x Δ p = \frac{h}{2}$ , we can determine the dimension of the box.

$Δ x Δ p = \frac{h}{2} Δ x = \frac{h}{2 Δ p} = \frac{6.63 \times 10^{- 3 Js}}{2 \times 1.82 \times 10^{- 29}} = 18.2 \times 10^{- 6} m$

The dimension of the smallest box should be

18.2 \times 10^{- 6} m .

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What is the smallest one-dimensional box in which you can confine an electron if you want to know for certain that the electron's speed is no more than $10 m / s$ ?

Short Answer

Step by step solution

part (a) step 1: Given information

part (a) step 2: formulas for solutions

part (b) step 3: calculation

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What is the smallest one-dimensional box in which you can confine an electron if you want to know for certain that the electron's speed is no more than 10m/s ?

Short Answer

Step by step solution

part (a) step 1: Given information

part (a) step 2: formulas for solutions

part (b) step 3: calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physics of Motion

Engineering Physics

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Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.

What is the smallest one-dimensional box in which you can confine an electron if you want to know for certain that the electron's speed is no more than $10 m / s$ ?