If Bohr’s quantization postulate (angular momentum \( = \frac{nh}{2\pi} \)) is a basic law of nature, it should be equally valid for the case of planetary motion also. Why, then, do we never speak of quantization of orbits of planets around the Sun? Explain.
Show Hint
Quantum effects dominate at microscopic scales.
At macroscopic scales (planets), classical physics emerges due to very large quantum numbers.
Why Planetary Orbits Are Not Quantized
Bohr’s quantization postulate states that the angular momentum of an electron in an atom is quantized as:
\[
L = \frac{n h}{2 \pi}, \quad n = 1, 2, 3, \dots
\]
where \( h \) is Planck’s constant. If this were a universal law of nature, one might expect it to apply to planetary motion as well. However, we never talk about quantization of planetary orbits. The reason lies in the scale of Planck’s constant relative to macroscopic systems. Explanation:
1. Extremely Large Angular Momentum: The angular momentum of planets orbiting the Sun is enormous compared to \( \frac{h}{2\pi} \). For example, Earth’s orbital angular momentum around the Sun is about \( 2.66 \times 10^{40} \, \text{kg·m²/s} \), while \( h/(2\pi) \approx 1.05 \times 10^{-34} \, \text{kg·m²/s} \).
2. Quantum Number Becomes Astronomically Large: If we applied Bohr’s formula, the quantum number \( n \) would be:
\[
n = \frac{L \cdot 2\pi}{h} \sim 10^{74}
\]
Such an astronomically large quantum number makes the energy levels and orbits effectively continuous. The spacing between allowed orbits becomes so tiny that the quantization is unobservable. Conclusion:
Bohr’s quantization is only significant for microscopic systems like atoms, where angular momenta are comparable to \( h \). For macroscopic systems like planets, the quantization steps are infinitesimally small, and planetary motion appears perfectly continuous. Therefore, the concept of quantized planetary orbits is meaningless in practice.
