Calculate the frequency (Hz) and wavelength (nm) of the emitted photon when an electron drops from the \(n=4\) to the \(n=2\) level in a hydrogen atom.

When an electron in a hydrogen atom transitions between energy levels, it either absorbs or emits a photon. The frequency of this photon () is directly related to the energy difference between the two levels. To understand this frequency, we use the Rydberg formula which relates the frequency, wavelength, and the quantum numbers of the involved energy levels.

The frequency can be obtained through the relationship between the speed of light (), wavelength (), and frequency: is given by (the speed of light) divided by (the wavelength). The speed of light is a constant (), approximately . Once the wavelength is known, the frequency can simply be calculated using the formula . This is crucial in spectroscopy and quantum mechanics, where the frequency of emitted or absorbed light reveals important information about the atomic structure.

The calculation of wavelength () of the emitted photon during an electron transition is an important aspect of understanding atomic spectra. Following the Rydberg formula, the inverse of the wavelength () is equal to the Rydberg constant () multiplied by the absolute difference between the reciprocals of the squares of the principal quantum numbers ().

Here's how you do it step by step: First, you apply the Rydberg formula to find the wave number (). Next, you take its reciprocal to find the wavelength in meters. It's essential to remember to convert this value into nanometers () since this unit is more commonly used in spectroscopy. The Rydberg formula is an incredible tool because it allows us to predict the position of the spectral lines solely based on the quantum numbers of the electron transitions.

Electron transitions in a hydrogen atom refer to the movement of an electron between different energy levels or shells around the nucleus. According to Niels Bohr's model of the atom, these levels are quantized which means electrons can only exist in certain allowed levels. When an electron drops from a higher-level () to a lower-level (), energy is released in the form of a photon, which leads to the emission of light that can be observed in a spectrum.

These transitions are governed by certain rules, one of which is the selection rule that dictates the change in quantum number between the two levels involved in the transition. Hydrogen atom electron transitions are quantitatively described by the Rydberg formula, making it a cornerstone for understanding atomic structure and the underlying principles of quantum mechanics.