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

Circular Motion and Gravitation

Geometrical and Physical Optics

Oscillations

Dynamics

Fields in Physics

Fundamentals of 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 $10 m / s$ ?