91Ó°ÊÓ

Understanding electron momentum involves exploring how electrons, like waves, are defined by wavelength. De Broglie's equation, \( \lambda = \frac{h}{p} \), shows the relationship between an electron's wavelength (\( \lambda \)) and its momentum (\( p \)). Here, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ J} \cdot \text{s}) \). By re-arranging this equation, we can find the momentum:

• \( p = \frac{h}{\lambda} \)

Plugging values for an electron with a wavelength of **5.00 nm**, we find its momentum to be approximately \( 1.325 \times 10^{-25} \text{ kg} \cdot \text{m/s} \). This calculation shows how an electron's motion can be characterized like that of a wave rather than a tiny particle, supporting wave-particle duality.

The kinetic energy of electrons is crucial to understanding their behavior when accelerated. We use the relationship between kinetic energy (KE) and momentum:

• \( KE = \frac{p^2}{2m} \)

where \( m \) is the mass of an electron \( (9.109 \times 10^{-31} \text{ kg}) \). By substituting the previously calculated momentum \( (1.325 \times 10^{-25} \text{ kg} \cdot \text{m/s}) \), we find the kinetic energy to be \( 9.61 \times 10^{-20} \text{ J} \). The kinetic energy reflects the energy that an electron possesses due to its motion. Next, this energy can be transformed into accelerating potential energy as:

• \( KE = eV \)

where \( e \) is the elementary charge \((1.602 \times 10^{-19} \text{ C}) \). This allows the calculation of voltage \( V \) for the acceleration process, giving around **0.600 volts**.

Photon energy relates closely to its wavelength. For photons, the energy \( E \) can be calculated using the equation:

• \( E = \frac{hc}{\lambda} \)

where \( c \) is the speed of light \( (3.00 \times 10^8 \text{ m/s}) \). Substituting for a given wavelength of **5.00 nm** yields \( E \approx 3.97 \times 10^{-17} \text{ J} \). This formula shows how shorter wavelengths correspond to higher energy photons. This principle is opposite to what occurs with electrons since decreasing electron wavelength increases momentum and kinetic energy, while for photons it increases energy directly.

Photons can have a specific wavelength if given a specific energy. When photons have the same energy as electrons in motion, the relation is expressed as:

• \( \lambda = \frac{hc}{E} \)

Using the electron kinetic energy \( E = 9.61 \times 10^{-20} \text{ J} \) from earlier and substituting back into the equation, we find the corresponding photon wavelength to be approximately \( 2.07 \times 10^{-6} \text{ m} \) or **2070 nm**. This method highlights the inverse relationship between energy and wavelength for photons; high-energy photons have shorter wavelengths. This distinction clarifies how we can correlate a photon's wavelength directly to the energy it carries, distinct from particle-like behavior.