Show that if we add a constant to all of the energies in hydrogen, the energies of the various transitions are unaffected.

Energy transitions are a key concept in atomic physics, particularly for hydrogen atoms. An energy transition occurs when an electron moves between two energy levels. These levels are quantized, meaning electrons can only exist in certain discrete energy states. An electron emits or absorbs a photon when it transitions from a higher energy level to a lower one or vice versa. The energy of the photon corresponds to the difference between these energy levels, which is calculated as: \( \Delta E = E_{n_2} - E_{n_1} \). This means that the energy change (\( \Delta E \)) depends solely on the electronic levels involved. Adding a constant to all energy levels does not change this difference, hence the transition energy remains the same.

Quantum mechanics is the framework that explains the behavior of particles at microscopic scales. It particularly describes how electrons inhabit discrete energy levels in atoms. Unlike classical mechanics, where particles can have any energy, quantum mechanics restricts energies to specific values. The energy of an electron in a hydrogen atom is dictated by formulas derived from quantum mechanics. One such formula for hydrogen is: \( E_n = - \frac{13.6 \text{ eV}}{n^2} \). The negative sign indicates that the electron is bound to the nucleus. Any changes in the energy levels, such as adding a constant, do not alter the fundamental quantum mechanical principles or the transition energies.

The principal quantum number, symbolized as \( n \), is a fundamental concept in quantum mechanics. It indicates the main energy level an electron occupies and can be any positive integer (1, 2, 3, ...). Larger values of \( n \) correspond to higher energy levels further from the nucleus. The energy level formula for hydrogen is: \( E_n = - \frac{13.6 \text{ eV}}{n^2} \), where \( E_n \) gets less negative (higher energy) as \( n \) increases. Understanding \( n \) is crucial for calculating energy transitions. The difference in energy levels between two states \( n_1 \) and \( n_2 \) determines the transition energy, not the absolute energy values themselves.

Adding a constant to all energy levels means each energy level is increased by the same amount. In the case of hydrogen, the energy levels are initially given by \( E_n = - \frac{13.6 \text{ eV}}{n^2} \). If we add a constant \( C \): \( E_n' = E_n + C \). For any two energy levels \( n_1 \) and \( n_2 \), the energy transition is: \( \Delta E = E_{n_2} - E_{n_1} \). With the added constant: \( \Delta E' = (E_{n_2} + C) - (E_{n_1} + C) = E_{n_2} - E_{n_1} \). Thus, the constant cancels out, showing that the transition energies remain unaffected by the constant addition. This demonstrates the relative nature of energy measurements in quantum systems.