Reviews the concepts on which the solution to this problem...

Reviews the concepts on which the solution to this problem depends. The electron in a hydrogen atom is in the first excited state, when the electron acquires an additional $2.86 \mathrm{eV}$ of energy. What is the quantum number $n$ of the state into which the electron moves?

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Key Concepts

Quantum Energy Levels

In quantum mechanics, especially in the context of the hydrogen atom, electrons can only exist in certain discrete energy levels. These levels are defined by a quantized energy formula (typically E? = -13.6 eV/n² for hydrogen), meaning that an electron’s energy is assigned by its principal quantum number n. This concept is fundamental because it explains why electrons emit or absorb energy in specific packets rather than any arbitrary amount.

Principal Quantum Number

The principal quantum number, denoted as n, determines the energy and size of an electron’s orbital in an atom. It is a whole number (n = 1, 2, 3, ...) that indicates the electron’s state and its corresponding energy level. Understanding this concept is crucial for predicting how electrons behave when they transition between different energy states.

Electron Excitation

Electron excitation refers to the process by which an electron absorbs energy and moves from a lower energy state to a higher one. This additional energy must exactly match the gap between the initial and the final energy levels, making the transitions quantized. This principle underlies many phenomena in atomic physics and spectroscopy, where energy absorption leads to discrete spectral lines.

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