Chapter 7, Problem 7.35P

Interpretation Introduction

Interpretation:

The analogy between the motion of a guitar string with the motion of an electron is to be explained.

Concept introduction:

According to de-Broglie, just like light, matter also has a dual character as a wave and as a particle. The term wave 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.

When Bohr's theory is connected with de Broglie equation, the wavelength of the electrons can be determined, in Bohr's orbit and relate it with circumference of orbit and multiply with a whole number as follows:

  $2nr=n\lambda$        (2)

Substitute $\frac{h}{mu}$ for $\lambda$ in the equation (2).

  $2nr=n\left(\frac{h}{mu}\right)$        (3)

Rearrange equation (3) for $\lambda$ as follows:

  $2\pi r=n\left(\frac{h}{mu}\right)$