Chapter 39: Q.22 (page 1137)

A thin solid barrier in the $x y -$ plane has a $10 μ m$ diameter circular hole. An electron traveling in the $z$ direction with $v_{x} = 0 m / s$ passes through the hole. Afterward, is it certain that $v_{x}$ is still zero? If not, within what range is $v_{x}$ likely to be?

Short Answer

Expert verified

The average velocity is zero. So, the range of velocity is from $- 18 m / s$ to $18 m / s$

Step by step solution

Heisenberg relation:

Heisenberg relation for position and momentum of a particle is,

$∆ x ∆ p_{x} \geq \frac{h}{2}$

Rewrite the equation for $∆ x$ ,

$∆ p_{x} = \frac{h}{2 ∆ x}$ .....(1)

The relationship between $∆ v_{x}$ and $∆ p_{x}$

$∆ p_{x} = m ∆ v_{x}$ .......(2)

Conversion diameter:

Convert diameter of circular hole of thin solid barrier is

$∆ x = 10 μ m = 10 μ m (\frac{10^{- 6} m}{1 μ m}) = 10 \times 10^{- 6 m}$

From equations (1) and (2),

$m ∆ v_{x} = \frac{h}{2 ∆ x} ∆ v_{x} = \frac{h}{2 m ∆ x}$

Substitute $6.63 \times 10^{- 23} J . s$ for $h, 9.11 \times 10^{- 31} k g$ for $m$ and $10 \times 10^{- 6} m$ for $∆ x$ .

$∆ v_{x} = \frac{6.63 \times 10^{- 23} J . s}{2 (9.11 \times 10^{- 31} k g) (10 \times 10^{- 6} m)} = 36 m / s$

Therefore, the average velocity is zero. So, the range of velocity is from $- 18 m / s$ to $18 m / s$

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A thin solid barrier in the $x y -$ plane has a $10 μ m$ diameter circular hole. An electron traveling in the $z$ direction with $v_{x} = 0 m / s$ passes through the hole. Afterward, is it certain that $v_{x}$ is still zero? If not, within what range is $v_{x}$ likely to be?

Short Answer

Step by step solution

Heisenberg relation:

Conversion diameter:

One App. One Place for Learning.

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A thin solid barrier in the xy-plane has a 10μmdiameter circular hole. An electron traveling in the zdirection with vx=0m/spasses through the hole. Afterward, is it certain that vxis still zero? If not, within what range is vxlikely to be?

Short Answer

Step by step solution

Heisenberg relation:

Conversion diameter:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism

Nuclear Physics

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Electricity

Study anywhere. Anytime. Across all devices.

A thin solid barrier in the $x y -$ plane has a $10 μ m$ diameter circular hole. An electron traveling in the $z$ direction with $v_{x} = 0 m / s$ passes through the hole. Afterward, is it certain that $v_{x}$ is still zero? If not, within what range is $v_{x}$ likely to be?