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Term symbols are crucial in understanding the quantum states of atoms. They describe the quantum state of electrons in an atom based on their angular momentum and spin. For helium with an electron configuration of \(1s2p\), term symbols are determined from the possible values of total electron spin \(S\) and orbital angular momentum \(L\).
\(1s\) electrons have \(L = 0\) and \(S = \pm \frac{1}{2}\). The \(2p\) electron contributes \(L = 1\).
When these electrons' spins and angular momenta combine, they give rise to possible total \(L\) values of 1 (\(^P\) terms). The spins combine into total \(S\) values of either 0 (singlet state with multiplicity 1) or 1 (triplet state with multiplicity 3).
The resulting term symbols are \(^1P\) and \(^3P\), indicating the singlet and triplet states for the \(L=1\) configuration. The multiplicity shown as \(^n\) is calculated as \(2S + 1\).

Spin-orbit coupling is a crucial effect in atomic physics. It results from the interaction between an electron's spin and its orbital motion around the nucleus. This interaction can cause splitting of term symbols, changing energy levels within an atom.
In our helium example, the triplet state \(^3P\) experiences this effect. Instead of existing as a single state, the inclusion of spin-orbit coupling causes the \(^3P\) state to divide into three separate levels with different total angular momentum \(J\) values: \(^3P_0\), \(^3P_1\), and \(^3P_2\).
The \(J\) value is calculated as the combination of \(L\) and \(S\), giving unique rotational levels, and consequently, the energy levels - a difference denoted by \(\Delta E\). This splitting does not affect the \(^1P\) state as it has a singular \(J\) value.

Degeneracy refers to the number of quantum states corresponding to a particular energy level and is pivotal for understanding atomic spectra. Calculating degeneracy involves counting how many different \(J\) states exist for a given term symbol.
For \(^1P_1\), without spin-orbit coupling, the degeneracy is computed as \(2J+1=3\). This state remains unchanged even with spin-orbit coupling because it's the only level.
For \(^3P\), the degeneracy breakdown is as follows: \(^3P_0\) has a degeneracy of \(1\), \(^3P_1\) has a degeneracy of \(3\), and \(^3P_2\) has \(5\) states.
Each of these states corresponds to different energy levels in the presence of spin-orbit coupling. Understanding these degeneracies helps predict spectral lines and understand electron transitions.