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

Turning Points in Physics

Electric Charge Field and Potential

Fundamentals of Physics

Rotational Dynamics

Wave Optics

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?