The statement that the wavelength of electromagnetic waves should be less than the dimensions of electrons in order to determine the position of an electron is related to the Heisenberg Uncertainty Principle in quantum mechanics.
The Heisenberg Uncertainty Principle states that there is a fundamental limit to how precisely we can simultaneously know certain pairs of physical properties of a particle, such as its position (x) and momentum (p). Mathematically, the principle is expressed as:
Δx * Δp ≥ h / (4π)
where Δx is the uncertainty in the position of the particle, Δp is the uncertainty in its momentum, and h is the reduced Planck's constant (approximately 6.626 x 10^-34 Js).
Now, when we want to measure the position of an electron, we often use electromagnetic waves like light (photons) to probe its location. The basic idea is to bounce the electromagnetic wave off the electron and detect its position based on the interaction. However, there's a limitation due to the Uncertainty Principle.
If the wavelength of the electromagnetic wave used to probe the electron is too large (compared to the dimensions of the electron), it means the wave's momentum becomes relatively small, and thus Δp becomes small. According to the Uncertainty Principle, if Δp becomes small, Δx (uncertainty in position) must become large. As a result, we get a fuzzy or broad idea of where the electron might be.
On the other hand, if the wavelength of the electromagnetic wave is small (i.e., comparable to or smaller than the dimensions of the electron), the wave's momentum becomes larger, which increases Δp. Consequently, Δx must become smaller, providing a more precise estimate of the electron's position.
In summary, to determine the position of an electron with higher precision, it's essential to use electromagnetic waves (like light) with wavelengths smaller than the dimensions of the electron. This is one of the underlying principles that govern the behavior of particles at the quantum level, and it highlights the inherent limitations we encounter when trying to precisely measure certain properties of subatomic particles.