1. What is the Eigenvalue of Energy Formula?

The Eigenvalue of Energy formula calculates the quantized energy levels for a quantum mechanical rigid rotor system. It's derived from solving the time-independent Schrödinger equation for rotational motion and provides the discrete energy values that such systems can possess.

2. How Does the Calculator Work?

The calculator uses the Eigenvalue of Energy formula:

Where:

• \( E \) — Eigenvalue of Energy (Joules)
• \( l \) — Angular Momentum Quantum Number
• \( \hbar \) — Reduced Planck's constant (1.0545718 × 10⁻³⁴ J·s)
• \( I \) — Moment of Inertia (kg·m²)

Explanation: The formula shows that rotational energy is quantized and depends on the square of the angular momentum quantum number and inversely on the moment of inertia.

3. Importance of Eigenvalue Calculation

Details: Calculating energy eigenvalues is fundamental in quantum mechanics for understanding rotational spectra of molecules, determining allowed energy transitions, and analyzing molecular structure and behavior in rotational states.

4. Using the Calculator

Tips: Enter the angular momentum quantum number (must be ≥ 0) and moment of inertia (must be > 0). The calculator uses Planck's constant value of 6.626070040 × 10⁻³⁴ J·s.

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of the angular momentum quantum number?
A: The angular momentum quantum number (l) determines the magnitude of the angular momentum and the number of possible orientations in space for a quantum system.

Q2: Why is energy quantized in quantum mechanical systems?
A: Energy quantization arises from the wave nature of particles and the boundary conditions imposed by the Schrödinger equation, allowing only discrete energy states.

Q3: What types of systems does this formula apply to?
A: This formula applies to quantum mechanical rigid rotors, such as diatomic molecules rotating in space without vibration or electronic excitation.

Q4: How does moment of inertia affect the energy eigenvalues?
A: Larger moments of inertia result in smaller energy level spacings, while smaller moments of inertia produce larger energy gaps between rotational states.

Q5: Are there selection rules for rotational transitions?
A: Yes, for pure rotational spectra, the selection rule is Δl = ±1, meaning transitions can only occur between adjacent rotational energy levels.