Chapter 7.3, Problem 7.4AFP

(a)

Interpretation Introduction

Interpretation:

The speed of an electron that has a de Broglie wavelength of $100\text{ nm}$ is to be calculated.

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:

$\lambda =\frac{h}{mu}$

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.

Rearrange the above equation for $u$.

$u=\frac{h}{\lambda m}$ (1)

(b)

Interpretation Introduction

Interpretation:

The speed at which a $45.9\text{ g}$ golf ball needs to move to have a de Broglie wavelength of $100\text{ nm}$ 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:

$\lambda =\frac{h}{mu}$

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.

Rearrange the above equation for $u$.

$u=\frac{h}{\lambda m}$ (1)