Question

According to de Broglie, what is the relation between the wave length of the wave associated with the moving electron and momentum of the electron?

Short answer

Answer: The relationship between the wavelength of the wave associated with a moving electron (λ) and the electron's momentum (m·v) is given by the de Broglie wavelength formula: \( λ = \frac{h}{m \cdot v} \), where h is the Planck constant, m is the mass of the electron, and v is the velocity of the electron.

Step-by-step solution

Recall the de Broglie wavelength formula

De Broglie's wavelength formula shows the relationship between the momentum of a particle and its wavelength. The formula is: \[ λ = \frac{h}{p} \] where: - \(λ\) is the de Broglie wavelength - \(h\) is the Planck constant, approximately \(6.626 \times 10^{-34} Js\) - \(p\) is the momentum of the particle This formula holds for all particles, including electrons (which is what we are interested in this exercise).

Express the momentum of the electron

In order to find the relationship between the wavelength and momentum of the electron, we need to express the momentum of the electron. Momentum is the product of mass and velocity: \[ p = m \cdot v \] where: - \(p\) is the momentum - \(m\) is the mass of the electron, approximately \(9.109 \times 10^{-31} kg\) - \(v\) is the velocity of the electron We can use this equation to express the momentum of the electron in terms of velocity and mass.

Find the relationship between wavelength and momentum

Now we have the de Broglie wavelength formula, and the expression for electron momentum. We substitute the expression for momentum from step 2 into the de Broglie wavelength formula: \[ λ = \frac{h}{m \cdot v} \] This equation shows the relationship between the wavelength of the wave associated with a moving electron (\(λ\)), and the momentum of the electron (\(m \cdot v\)).

Key concepts

Wave-Particle Duality

The revolutionary concept of wave-particle duality arises from the realization that every particle exhibits both wave-like and particle-like properties. This idea was a cornerstone in the development of quantum mechanics and deeply influenced our understanding of the nature of light and matter.

Essentially, wave-particle duality tells us that particles such as electrons, which were once thought to behave strictly as tiny, solid objects, can also display wave-like characteristics under certain conditions. This is especially evident when we look at phenomena like diffraction and interference, where electrons create patterns characteristic of waves passing through slits or interacting with each other. Conversely, light, which is typically understood as a wave, can exhibit particle-like properties, as shown through the photoelectric effect where light 'particles' (photons) knock out electrons from a material.

De Broglie's hypothesis unified these concepts by suggesting that all matter has a wave associated with it, and the wavelength of this wave could be calculated using the de Broglie wavelength formula. The formula for this wave, or matter wave, is particularly significant when dealing with subatomic particles like electrons.

Electron Momentum

Momentum is a fundamental concept in both classical and quantum physics, representing the quantity of motion an object possesses. For an electron, momentum is particularly important because it directly influences the electron's de Broglie wavelength.

As stated in the exercise, momentum (\(p\)) is calculated by the product of an object's mass (\(m\)) and its velocity (\(v\)). For an electron, this calculation involves its rest mass (\(9.109 \times 10^{-31} kg\)) and its velocity, which can vary depending on the electron's energy state and external forces applied to it. However, it's vital to note that at very high velocities, approaching the speed of light, relativistic effects become significant, and the classic equation for momentum (\(p = m \times v\)) needs to be adjusted accordingly.

It's also essential to understand that the momentum of an electron isn't just a measure of its inertial characteristics but also plays a role in its wave-like behavior. According to the de Broglie hypothesis, as the momentum of an electron increases, its wavelength decreases, and vice versa. This inverse relationship has profound implications for the behavior of electrons in quantum systems like atoms and molecules.

Planck Constant

The Planck constant (\(h\)) is an incredibly fundamental quantity in quantum mechanics, as it sets the scale for the quantum of action. With the value of approximately \(6.626 \times 10^{-34} Js\), it forms a bridge between the macroscopic and quantum worlds. Named after Max Planck, who first introduced it in relation to the energy quanta of black-body radiation, the constant fundamentally describes the quantization of energy in the universe.

In the case of the de Broglie wavelength formula, the Planck constant serves as a proportionality factor linking the momentum of a particle to its wavelength: a large momentum corresponds to a small wavelength and vice versa, with the Planck constant as the conversion factor. This relationship helps in understanding phenomena at the atomic and subatomic levels, where quantum effects cannot be ignored.

The Planck constant enables us to calculate the de Broglie wavelength of moving particles like electrons, and by doing so, it reveals the inherently quantized nature of their motion. Its role is not limited to de Broglie's formula, as it also appears in several other fundamental quantum equations, such as those defining the energy of a photon (\(E = h \times u\)) and the Heisenberg uncertainty principle, which underscores the limits of precision when measuring particle properties concurrently.