Chapter 7, Problem 7.40P

(a)

Interpretation Introduction

Interpretation:

The de Broglie wavelength of an alpha particle of mass $6.6\times {10}^{-24}\text{ g}$ and velocity $3.4\times {10}^7\pm 0.1\times {10}^7\text{ mi/h}$ is to be determined.

Concept introduction:

According to de-Broglie, just like light, matter also has a dual character as a wave and as a particle. The term wavicle was proposed for such a particle. However, matter waves are different from electromagnetic waves. The speed of matter waves is much less compared to the speed of light. Matter waves cannot propagate through empty space. Matter waves are not emitted by the particles under consideration but are rather associated with them. Compared to the wavelengths of the electromagnetic waves, the wavelengths of the matter waves are very small.

The de Broglie equation to relate the wavelength of a moving particle with its mass and velocity is as follows:

$\lambda =\frac{h}{mu}$        (1)

Here,

$\lambda$ is the wavelength associated with the moving particle.

$h$ is the Plank’s constant.

$m$ is the mass of the particle.

$u$ is the velocity of the particle.

The conversion factor to convert the unit of mass from $\text{g}$ to $\text{kg}$ is as follows:

$1\text{ kg}=1000\text{ g}$

(b)

Interpretation Introduction

Interpretation:

The uncertainty in the position of the alpha particle is to be determined.

Concept introduction:

Heisenberg’s uncertainty principle – The Heisenberg’s uncertainty principle states that it is not possible to exactly determine both the position and the momentum for a small moving particle at the same time.

The expression for the uncertainty principle is as follows:

$\left(\Delta x\right)\left(m\Delta u\right)\ge \frac{h}{4\pi }$        (2)

Here,

$\Delta x$ is the uncertainty in the position.

$m$ is the mass.

$\Delta u$ is the uncertainty in the velocity.

$h$ is the Plank’s constant.