An electron in a hydrogen atom is excited with electrical energy to an excited state with \(n=2 .\) The atom then emits a photon. What is the value of \(n\) for the electron following the emission?

The principal quantum number, represented by the letter 'n', is a fundamental concept in quantum mechanics particularly important in the description of an electron's position in an atom. It determines the size of the electron shell and the energy level associated with an electron within an atom. Quantum numbers only take positive integer values, with 'n=1' being the smallest possible value, which signifies the electron is in its closest orbit to the nucleus.

When discussing energy levels, larger values of 'n' correspond to electrons being further from the nucleus and at higher energy states. For example, when an electron leaps from an outer shell to an inner shell, 'n' decreases and energy is released, usually in the form of a photon. This concept is critical in understanding how electrons move between energy levels and is a stepping stone to grasp more complex atomic behaviors.

Photon emission occurs when an electron makes a transition between energy levels within an atom. This transition usually involves an electron dropping from a higher energy level (higher principal quantum number) to a lower level. According to the laws of physics, energy must be conserved, so the energy difference between these levels is carried away by a photon, a particle representing a quantum of light.

The energy of the emitted photon directly corresponds to the energy gap between the two orbits or states involved in the transition. In our hydrogen atom example, when the electron transitions from 'n=2' to a lower level, a specific energy photon is emitted. This emission can be observed as a spectral line, which is unique to the element and the energy change experienced by the electron.

An excited state of an electron in an atom is one where its energy is greater than in its most stable, lowest energy state, known as the ground state. This higher energy level is often a result of absorbing energy, such as from electrical energy or a photon. In the context of our hydrogen atom example, when the electron is in the 'n=2' state, it is in its first excited state above the ground state.

While in the excited state, electrons are unstable and tend to return to their ground state. The return to the ground state is accompanied by the release of the excess energy, usually in the form of photon emission, representing the energy difference between the excited and ground states.

The ground state of an electron within an atom is its lowest energy level where the electron is most stable. For a hydrogen atom, the ground state corresponds to the principal quantum number 'n=1'. Electrons naturally gravitate towards the ground state because of their inclination to exist at the lowest energy levels possible.

Since the ground state is the most stable configuration, any transition from a higher energy level, or excited state, to the ground state results in the release of energy. In quantum mechanics, the ground state is often a starting point for analyzing electron behavior because it provides a reference point for energy levels and electron transitions within atoms.