---
title: "A hydrogen atom is initially in its second excited state (\\(n=3\\)). The energy levels of the hydrogen atom are given by the equation \\(E_n = \\dfrac{-13.6 \\text{ eV}}{n^2}\\), where \\(n\\) is the principal quantum number. The electron undergoes a transition to the first excited state (\\(n=2\\)), emitting a photon."
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url: "https://nerd-notes.com/ubq/117713/"
date_modified: "2026-08-04T07:53:36+00:00"
---

# A hydrogen atom is initially in its second excited state (\(n=3\)). The energy levels of the hydrogen atom are given by the equation \(E_n = \dfrac{-13.6 \text{ eV}}{n^2}\), where \(n\) is the principal quantum number. The electron undergoes a transition to the first excited state (\(n=2\)), emitting a photon.

A hydrogen atom is initially in its second excited state (\(n=3\)). The energy levels of the hydrogen atom are given by the equation \(E_n = \dfrac{-13.6 \text{ eV}}{n^2}\), where \(n\) is the principal quantum number. The electron undergoes a transition to the first excited state (\(n=2\)), emitting a photon.

![A vertical energy level diagram for hydrogen. Five horizontal solid lines are shown, spaced closer together as they go up. From bottom to top, the left sides of the lines are labeled with quantum numbers n=1, n=2, n=3, and n=4. The top line is dashed and labeled n=\infty. The right sides of the lines are labeled with their corresponding energies: -13.6 eV, -3.40 eV, -1.51 eV, -0.85 eV, and 0 eV. A vertical arrow points downward from the n=3 line to the n=2 line. A squiggly arrow representing a photon points outward from the middle of the vertical arrow to the right. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830016-9449aV.jpg)

**Part a)** **Calculate** the energy, in \(\text{eV}\), of the photon emitted when the electron transitions from the \(n=3\) state to the \(n=2\) state. *(2 points)*

**Part b)** **Calculate** the wavelength of this emitted photon. *(2 points)*

**Part c)** **Derive** an expression for the frequency \(f\) of a photon emitted when a hydrogen atom transitions from a higher energy state with quantum number \(n_i\) to a lower energy state with quantum number \(n_f\). Express your answer in terms of \(n_i\), \(n_f\), the ground state energy \(E_1\) (where \(E_1 = -13.6 \text{ eV}\)), and fundamental constants, as appropriate. *(3 points)*

**Part d)** A different hydrogen atom is initially in the \(n=2\) state. A photon with the exact energy calculated in Part A is incident on this atom. **Predict** the final energy state of the atom. **Justify** your prediction using physical principles. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117713/*