Which statement about the quantum mechanical solution for the hydrogen atom is correct?

• A. The energy levels depend only on the principal quantum number n.
• B. The angular momentum quantum number l determines the energy of the electron.
• C. The electron can exist at any energy within the atom.
• D. The probability of finding the electron is highest at the nucleus for all orbitals.

Answer: (A)

Verified step by step guidance

1. Recall that the quantum mechanical model of the hydrogen atom describes the electron's behavior using wavefunctions characterized by quantum numbers: the principal quantum number \(n\), the angular momentum quantum number \(l\), and the magnetic quantum number \(m_l\).
2. Understand that the energy levels of the hydrogen atom depend primarily on the principal quantum number \(n\), which determines the allowed energy states. This means all orbitals with the same \(n\) have the same energy (degenerate energy levels) in a hydrogen atom.
3. Recognize that the angular momentum quantum number \(l\) affects the shape of the orbital and the electron's angular momentum, but in the hydrogen atom, it does not affect the energy level. This is different in multi-electron atoms where \(l\) influences energy due to electron-electron interactions.
4. Note that the electron cannot have just any energy; it is restricted to specific quantized energy levels determined by \(n\). The electron's energy is not continuous within the atom.
5. Remember that the probability density of finding the electron varies with the type of orbital. For example, the probability is highest at the nucleus only for the \$1s\( orbital, but not for orbitals with higher \)n\( or different \)l$ values.