Chapter 28, Problem 1OQ

(a)

To determine

Whether the statement is true or not for an electron.

Answer to Problem 1OQ

True.

Explanation of Solution

In 1905 it was Einstein who suggested the concept of light having a wave particle duality. In a similar way Louise de Broglie suggested that electron also exhibits a dual nature.

Broglie derived a mathematical expression to prove the wave nature of electron along with the particle nature by the given expression,

    $\lambda =\frac{h}{mv}$        (I)

Here, $\lambda$ is the wavelength, $h$ is the Planck’s constant, $m$  is the mass, $v$ is the velocity.

Conclusion:

Therefore, the given statement is true for electron.

(b)

To determine

Whether the rest energy is zero.

Answer to Problem 1OQ

The statement is False.

Explanation of Solution

The expression for the rest mass energy is given by,

    $E_0=m_0{c}^2$        (II)

Here, $E_0$ is the rest mass energy of electron, $m_0$ is the mass of electron, $c$ is the speed of light.

For the electron,

    $\begin{array}{c}{E}_0=\left(9.1\times {10}^{-31}\text{kg}\right){\left(3\times {10}^8\text{m}/\text{s}\right)}^2\\ =0.511\text{MeV}\end{array}$

Conclusion:

Therefore, the rest mass energy of the electron will not be zero, the statement is False.

(c)

To determine

Whether it carries energy in its motion.

Answer to Problem 1OQ

The statement is true.

Explanation of Solution

Electrons in atoms can act as charge carriers. It carries energy in its motion. For a moving electron it will have a velocity along with the mass. For a moving particle it possess kinetic energy. Thus while moving electron also possess energy.

Conclusion:

Therefore, the statement is true.

(d)

To determine

Whether the electron carries momentum in its motion.

Answer to Problem 1OQ

The given statement is true.

Explanation of Solution

Write the expression for the momentum of a particle.

    $p=mv$

Here, $p$ is the momentum and $v$ is the speed.

Electron possess a mass along with velocity in a motion. Thus electron will have momentum.

Conclusion:

Therefore, electron carries momentum in its motion so the given statement is true.

(e)

To determine

Whether the motion of the electron is described by a wave function that has a wavelength and satisfies a wave equation.

Answer to Problem 1OQ

The given statement is true.

Explanation of Solution

Electron exhibits both wave and particle nature. Its motion is described by a wave function that has a wavelength and satisfies a wave function. Equation (I) shows that electron have dual nature.

Conclusion:

Therefore, the given statement motion of the electron is described by a wave function that has a wavelength and satisfies a wave equation is true.