For hydrogen-like atoms, how does the energy E_n change as the principal quantum number n increases?

For hydrogen-like atoms, how does the energy E_n change as the principal quantum number n increases?

• A. Energy becomes more negative as n increases.
• B. Energy remains constant with n.
• C. Energy oscillates with n.
• D. Energy becomes less negative as n increases.

Answer: (D)

In hydrogen-like atoms, the energy of a bound state scales as E_n ∝ -1/n^2. That negative sign means these states are bound, with the ground state being the most negative. As the principal quantum number n increases, 1/n^2 gets smaller in magnitude, so E_n becomes less negative and moves closer to zero. In other words, the electron is less tightly bound in larger, more distant orbits. A convenient way to see this is E_n = - (constant) / n^2, where the constant is positive; increasing n reduces the magnitude of the negative energy, raising the energy level toward zero.

In hydrogen-like atoms, the energy of a bound state scales as E_n ∝ -1/n^2. That negative sign means these states are bound, with the ground state being the most negative. As the principal quantum number n increases, 1/n^2 gets smaller in magnitude, so E_n becomes less negative and moves closer to zero. In other words, the electron is less tightly bound in larger, more distant orbits. A convenient way to see this is E_n = - (constant) / n^2, where the constant is positive; increasing n reduces the magnitude of the negative energy, raising the energy level toward zero.