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.
Concept:
Bohr’s quantization condition:
\[
L = \frac{nh}{2\pi}
\]
is a quantum mechanical effect that becomes significant only at atomic scales.
Step 1: Compare scales.
In atomic systems:
Masses are extremely small (electron mass)
Angular momentum is comparable to Planck’s constant \( h \)
Quantization becomes observable
In planetary motion:
Masses are enormous (planet mass)
Angular momentum is extremely large
Step 2: Quantum number becomes huge.
If we apply Bohr’s condition to a planet:
\[
n = \frac{2\pi L}{h}
\]
Since \( L \gg h \), the quantum number \( n \) becomes extremely large (of order \( 10^{70} \) or more).
Step 3: Effect of very large \( n \).
For very large quantum numbers:
Energy levels are extremely closely spaced
Orbits appear continuous rather than discrete
This corresponds to the classical limit (correspondence principle).
Step 4: Observability.
The spacing between successive quantized planetary orbits is so tiny that:
Impossible to detect experimentally
Motion appears continuous and classical
Conclusion:
Bohr’s quantization is valid in principle for planetary motion, but the quantum effects are negligible because:
Planck’s constant is extremely small
Planetary angular momentum is extremely large
Hence, planetary orbits appear continuous and not quantized.
