What is the de Broglie wavelength of an electron traveling at \(1.35 \times 10^{5} \mathrm{~m} / \mathrm{s} ?\)

Quantum mechanics is the branch of physics that deals with the behavior of microscopic particles at the atomic and subatomic levels. It's a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. Quantum mechanics differs significantly from classical physics, as it introduces concepts such as quantization of energy, wave-particle duality, and the uncertainty principle.

Unlike classical physics, which predicts definite outcomes, quantum mechanics is probabilistic. This means that it usually provides the likelihood of where a particle might be found. The de Broglie wavelength is a part of this quantum description and shows that particles have wave-like properties.

Electrons are subatomic particles with a negative charge, found in all atoms. Their behavior in different conditions is a cornerstone of quantum mechanics. In comparison to objects in everyday life, electrons don't follow the same physical laws. Instead of moving in a predictable path, like planets around the sun, they exist in areas called 'orbitals' with varying probabilities.

Implications of Quantum Theory

Quantum theory suggests that due to their wave-like characteristics, electrons have certain quantized energy levels. This means they can 'jump' from one energy state to another but cannot exist in between these states. When electrons move between energy states, this has implications for the emission or absorption of energy, seen in phenomena such as spectral lines and the photoelectric effect.

Wave-particle duality is a core concept in quantum mechanics that proposes that every particle or quantum entity exhibits both wave and particle properties. The de Broglie wavelength, named after physicist Louis de Broglie, represents the wave nature of particles.

De Broglie hypothesized that just as light has wave-particle duality, matter also exhibits this property. The de Broglie wavelength equation demonstrates this concept mathematically by implying that any particle with momentum has an associated wavelength. Thus, not only photons (light particles) but also electrons (matter particles) have this duality. This insight was pivotal in developing quantum theory and has profound implications for understanding electron behavior in atoms and molecules.

In classical physics, the momentum of an object is calculated by the product of its mass and velocity, denoted as \( p = mv \). However, in the quantum realm when we want to determine the de Broglie wavelength of a particle like an electron, we first calculate the momentum using the same formula.

Applying the Momentum Concept

In the provided exercise, we follow the classical approach in calculating the momentum of an electron, acknowledging the quantum effects later when we use that value to find the de Broglie wavelength. By multiplying the electron's mass by its velocity, you can determine its momentum. This calculated momentum allows you to find the wavelength using de Broglie's formula, bridging the concepts of classical momentum with quantum wave-like behavior.