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The hydrogen atom is one of the simplest and most fundamental models in quantum mechanics, providing a substantial understanding of atomic physics. It consists of a single proton in the nucleus and one electron. Due to its simplicity, it serves as an excellent model for studying quantum behavior. In a hydrogen atom, electrons revolve around the nucleus in specific paths called orbitals. These orbitals are defined by quantum numbers, which are principal in determining the atom's state.
In quantum mechanics, the principal quantum number, denoted as \( n \), dictates the electron's energy and orbital size. For the hydrogen atom, the lowest energy level, or ground state, is when \( n = 1 \). Higher values of \( n \) correspond to excited states, where the electron is further from the nucleus with greater energy. Understanding the behavior of electrons in a hydrogen atom is crucial for exploring more complex atoms and molecules.

The total energy of an electron in a hydrogen atom is quantized, meaning it can only take specific values. This energy is given by a well-known formula in quantum mechanics:
\[E_n = -13.6 \frac{1}{n^2} \text{ eV}\]where \( E_n \) is the energy of the state and \( n \) is the principal quantum number.
This formula shows how the energy levels are inversely related to the square of the principal quantum number. As \( n \) increases, the energy becomes less negative, indicating that energy is required to move an electron from a lower energy level to a higher one.

• For the ground state \( n=1 \), the total energy is \(-13.6 \, \text{eV}\).
• For the first excited state \( n=2 \), the total energy becomes \(-3.4 \, \text{eV}\).
• In the second excited state \( n=3 \), which is the scenario of the exercise, the total energy is calculated as \(-1.51 \, \text{eV}\).

The quantization of energy levels explains the discrete spectral lines observed in hydrogen's emission spectrum.

Angular momentum in the context of quantum mechanics is a measure of the rotational motion of an electron in an atom. For electrons in the hydrogen atom, this is referred to as orbital angular momentum, which is quantized as well.
The orbital angular momentum \( L \) can be calculated using the formula:
\[L = \sqrt{l(l+1)} \hbar\]where \( l \) is the azimuthal quantum number and \( \hbar \) is the reduced Planck's constant.

• The quantum number \( l \) can take values from 0 to \( n-1 \), with \( n \) as the principal quantum number.
• For a given principal quantum number of 3, the maximum \( l \) value is 2.
• In this case, the maximum angular momentum is calculated as \( \sqrt{6} \hbar \).

This quantization articulates why electrons can only occupy certain paths around the nucleus, leading to stable and predictable electron configurations.

A quantum state defines the state of an electron in an atom with specific values for all quantum numbers. For the hydrogen atom, these quantum numbers are crucial:
- **Principal quantum number \( n \)**: Related to the electron's energy and size of the orbital. - **Azimuthal quantum number \( l \)**: Determines the orbital shape and has values from 0 to \( n-1 \).- **Magnetic quantum number \( m_l \)**: Relates to the orbital's orientation in space, ranging from \( -l \) to \( +l \).- **Spin quantum number \( m_s \)**: Represents the intrinsic spin of the electron, typically \( +\frac{1}{2} \) or \(-\frac{1}{2} \).
A specific quantum state is often described by the notation \(|n, l, m_l, m_s\rangle\), encapsulating all four quantum numbers.

• For the problem on hand, with a hydrogen atom at \( n=3 \), the potential configurations of \((l, m_l, m_s)\) determine the possible states and properties of the electron.

The concept of quantum states enables predicting atomic behavior accurately, providing insights into electronic transitions and the resulting emission spectra.