Question

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.

Hint:

Quantum effects dominate at microscopic scales. At macroscopic scales (planets), classical physics emerges due to very large quantum numbers.

Answer 1

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.

Answer 2

Step 1: Bohr's condition and the correspondence principle.
Bohr's postulate \( L = \dfrac{nh}{2\pi} \) predicts that only certain discrete values of angular momentum (and hence discrete orbits) are allowed. Whether this discreteness is noticeable in practice is governed by the correspondence principle: quantum results must reduce to classical ones when the quantum number \( n \) is very large, and the test of "how classical" a system is is the fractional spacing between consecutive allowed states, \( \Delta E_n / E_n \), not the absolute size of \( n \) itself.
Step 2: How the fractional spacing behaves with \( n \).
For a Bohr-like system, the energy of the \( n \)-th state scales as \( E_n \propto -1/n^2 \), so consecutive levels differ by \[ \frac{\Delta E_n}{E_n} \approx \frac{2}{n} \] As \( n \) grows, this ratio falls steadily toward zero: the levels get closer together relative to their own energy, and the "steps" between them shrink into insignificance.
Step 3: Apply this to a planet.
For planetary motion, the corresponding quantum number is around \( n \sim 10^{74} \) (obtained by matching the planet's actual, classically measured angular momentum to \( nh/2\pi \)). Substituting into the ratio above: \[ \frac{\Delta E_n}{E_n} \approx \frac{2}{10^{74}} \sim 10^{-74} \] a fractional change so small that no instrument, however precise, could ever detect the jump from one "allowed" orbit to the next.
Step 4: Conclusion.
Because the relative spacing between consecutive quantized states vanishes as \( n \to \infty \), Bohr's rule for planetary orbits reproduces ordinary continuous classical mechanics. The orbits remain formally quantized, but the quantization has shrunk below any conceivable threshold of observation -- which is exactly why quantization is spoken of only for atoms and never for planets.