What will be the wavelength of an electron moving with \(\frac{1}{10}\) th of velocity of light? (a) \(2.43 \times 10^{-11} \mathrm{~m}\) (b) \(243 \times 10^{-11} \mathrm{~m}\) (c) \(0.243 \mathrm{~m}\) (d) \(2.43 \times 10^{-4} \mathrm{~m}\)

When we talk about the momentum of an electron, we're referring to the movement and mass of the electron as it behaves both as a particle and a wave, an idea central to quantum mechanics. In classical physics, momentum is simply the product of an object's mass and its velocity, described by the equation

\( p = m \times v \).

For electrons, which have a very small mass of approximately \( 9.109 \times 10^{-31} \) kilograms, their momentum becomes significant when moving at high velocities. The higher an electron's velocity, the greater its momentum. Remarkably, this concept allows us to connect an electron's physical movement to its wave-like properties through the de Broglie wavelength equation.

By understanding the momentum, we can dive deeper into the nature of electrons and their behavior on a subatomic level, where they exhibit properties that challenge our classical understanding of physics.

The velocity of an electron can greatly vary depending on the forces acting upon it. In our study of elementary particles like electrons, velocity not only denotes speed but also helps in understanding kinetic energy and other wave-like behaviors. For example, the velocity of an electron orbiting an atom can significantly differ from one that is being accelerated in a vacuum.

Within the context of the de Broglie wavelength calculation, electron velocity is especially crucial. As the exercise illustrates, an electron moving at \( \frac{1}{10} \) the speed of light—an exceptional speed in human terms, yet completely plausible on the quantum scale—greatly influences the wavelength associated with that electron. In essence, knowing the electron's velocity allows us to calculate its momentum, which in turn determines the de Broglie wavelength, bridging the gap between particle physics and wave phenomena.

The Planck constant is a fundamental quantity in quantum mechanics, symbolized by 'h', with a value of \( 6.62607015 \times 10^{-34} \) joule seconds. It is a pivotal factor in quantifying the relationship between energy and frequency of a photon, and it plays a starring role in the renowned equation

\( E = h \times u \),

where \(E\) is energy, and \(u\) is the frequency. This constant also is essential when examining the de Broglie hypothesis, which posits that all matter exhibits wave-like properties. The hypothesis utilizes the Planck constant as a proportionality factor in the de Broglie wavelength formula \( \lambda = \frac{h}{p} \), where \(\lambda\) is the wavelength and \(p\) is the momentum of the particle. The Planck constant's small magnitude underscores the minute scale of quantum phenomena, bridging our macroscopic and quantum understanding of the physical world.