1. What is the Principal Quantum Number?

The principal quantum number (n) describes the energy level and size of an electron orbital in a hydrogen-like atom. It is a positive integer (1, 2, 3, ...) that determines the electron's energy and most probable distance from the nucleus.

2. How Does the Calculator Work?

The calculator uses the formula derived from the Bohr model:

Where:

• \( n \) — Principal quantum number (integer)
• \( Z \) — Atomic number (number of protons)
• \( E \) — Energy of the electron (must be negative for bound states)
• 13.6 — Rydberg energy constant for hydrogen (eV)

Explanation: The formula calculates the energy level from the electron's binding energy and the nuclear charge.

3. Importance of Quantum Numbers

Details: The principal quantum number is fundamental in quantum mechanics, determining electron configurations, spectral lines, and chemical properties of elements.

4. Using the Calculator

Tips: Enter atomic number (Z) as a positive integer and energy (E) as a negative value in eV. The calculator will return the nearest integer quantum number.

5. Frequently Asked Questions (FAQ)

Q1: Why must the energy be negative?
A: Negative energy indicates a bound electron state. Positive energy would mean the electron is unbound (ionized).

Q2: What are typical values for n?
A: For ground state hydrogen (Z=1), n=1. Excited states have higher integers (n=2,3,...).

Q3: Does this work for multi-electron atoms?
A: The formula is exact only for hydrogen-like atoms (single electron systems), but can approximate outer electrons in other atoms.

Q4: What if my calculated n isn't an integer?
A: The calculator rounds to nearest integer since n must be whole. Non-integer results suggest the electron isn't in a stationary state.

Q5: What's the physical meaning of n?
A: Higher n means higher energy, larger orbital radius, and more nodes in the electron's wavefunction.