The spacing between neighboring energy levels in a hydrogen atom is determined by the principal quantum number (n) of the energy levels. The correct answer is option A, which states that the spacing is proportional to 1/n^3.

Explanation:

1. Energy levels in a hydrogen atom:
- In a hydrogen atom, the electron is bound to the nucleus by the electrostatic attraction between the negatively charged electron and the positively charged proton.
- The electron can occupy different energy levels, which are characterized by the principal quantum number (n).
- The energy of these levels increases as the value of n increases.

2. Energy formula for hydrogen atom:
- The energy of an electron in a hydrogen atom can be calculated using the formula:
E = -13.6 eV / n^2
where E is the energy, -13.6 eV is a constant, and n is the principal quantum number.

3. Spacing between neighboring energy levels:
- The spacing between neighboring energy levels can be calculated by taking the difference in energy between two consecutive levels.
- Let's consider two energy levels with principal quantum numbers n and (n+1).
- The energy difference between these levels can be calculated by subtracting the energy of the lower level from the energy of the higher level:
ΔE = E(n+1) - E(n)
= [-13.6 eV / (n+1)^2] - [-13.6 eV / n^2]
= -13.6 eV * [1 / (n+1)^2 - 1 / n^2]

4. Simplifying the energy difference formula:
- To simplify the energy difference formula, we can use the concept of fractional differences.
- We can rewrite the energy difference formula as:
ΔE = -13.6 eV * [1 / (n+1)^2 - 1 / n^2]
= -13.6 eV * [n^2 - (n+1)^2] / [n^2 * (n+1)^2]
= -13.6 eV * [-2n - 1] / [n^2 * (n+1)^2]

5. Proportional relationship with n:
- From the simplified energy difference formula, we can observe that the spacing ΔE is proportional to (-2n - 1) / (n^2 * (n+1)^2).
- As n increases, the term (-2n - 1) becomes insignificant compared to the term (n^2 * (n+1)^2).
- Therefore, we can approximate the spacing as proportional to 1 / (n^2 * (n+1)^2).

6. Final answer:
- Since the spacing is proportional to 1 / (n^2 * (n+1)^2), the correct answer is option A: 1/n^3.

In summary, the spacing between neighboring energy levels in a hydrogen atom is proportional to 1/n^3, as the value of the principal quantum number (n) increases.