Electron Velocity
When we talk about electron velocity, we're referring to how fast an electron moves. Electrons, like any other particles in physics, have a property of speed, denoted as 'v'. In the context of the exercise, the electron is moving at an impressive rate: 20% the speed of light. This is a significant fraction of the fastest speed possible in the universe!
Since the speed of light (\(c\)) is approximately 299,792,458 meters per second, an electron traveling at 20% the speed of light would have a velocity of \(v = 0.2c \approx 59,958,492\text{ m/s}\). That's nearly 60 million meters every second! This high rate of travel starts to approach the relativistic realm, where we need to consider the effects of special relativity to accurately describe the electron's motion and properties.
Lorentz Factor
Understanding the Lorentz factor is crucial when dealing with velocities that are a significant fraction of the speed of light. It's a term that arises in Einstein's theory of relativity. The Lorentz factor, usually denoted as \(\gamma\), adjusts physical laws that would otherwise only apply at non-relativistic speeds, making them applicable at speeds close to the speed of light.
The formula for the Lorentz factor is \(\gamma = \frac{1}{\sqrt{1-(v/c)^2}}\). As you can see, when the velocity (\(v\) ) is small compared to the speed of light (\(c\) ), the value of \(\gamma\) is close to 1, meaning there's hardly any relativistic effect. However, as in our exercise problem, as \(v\) becomes a more substantial fraction of \(c\), \(\gamma\) increases above 1, indicating that relativistic effects become significant.\
\By substituting \(v = 0.2c\) into the equation for \(\gamma\), we find that the Lorentz factor is about 1.0206 for our electron. That means we need to take into account a 2.06% correction to classical calculations.
Relativistic Momentum
In classical physics, we learn that momentum is the product of mass and velocity. But things change when we move to high-speed scenarios, like our electron zipping along at 20% of the speed of light. In this context, we need to use the concept of relativistic momentum, accounting for changes in physics due to the principles of special relativity. It's calculated as\
\ \(p = \gamma mv\).
The presence of the Lorentz factor (\(\gamma\)) here is what adjusts our 'classical' momentum to be relativistically accurate. This reflects the fact that as particles move very quickly, they seem to gain 'effective mass'. To find the momentum for our electron problem, the Lorentz factor (obtained earlier), the electron's rest mass, and its velocity must all be considered. So relativistic effects make the electron behave as if it has more inertia—hence greater momentum—than we'd expect from just looking at its rest mass and velocity.
Planck’s Constant
At the heart of quantum mechanics lies a number with a big role: Planck's constant (\(h\)). This quantity ties together energy and frequency per the famous Einstein equation \(E = hf\), where \(E\) is the energy of a photon, \(h\) is Planck's constant, and \(f\) is the frequency. Its value is approximately \(6.626 \times 10^{-34} \text{ Js}\), quite small but with universe-scale significance.\
\Planck's constant also connects with the de Broglie wavelength, which describes the wave-nature of particles. The de Broglie equation \(\lambda = \frac{h}{p}\) tells us the wavelength (\(\lambda\)) associated with any moving particle is inversely proportional to its momentum (\(p\)). That's critical in understanding phenomena like electron diffraction. For our exercise, by using Planck's constant and the calculated electron momentum, we can determine the de Broglie wavelength, bridging the gap between the very small (quantum) world and the large-scale world we see and experience every day.
