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

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.

Answer 2

This question asks why we never observe quantized orbits for planets even though Bohr's angular momentum condition, in principle, applies to any system moving under a central force. The way to see this clearly is to compare the size of a planet's orbital "wave" to the size of its orbit, using the de Broglie picture of quantization rather than working with the raw angular momentum number.

Step 1: Bohr's condition as a standing-wave condition.
Bohr's postulate \( L = \dfrac{nh}{2\pi} \) is equivalent, through de Broglie's hypothesis \( \lambda = \dfrac{h}{p} \), to requiring that the circumference of the orbit contain a whole number of de Broglie wavelengths: \[ 2\pi r = n\lambda \] Quantization is only a meaningful, observable restriction when \( \lambda \) is comparable to the size of the orbit \( r \) -- only then does "fitting a whole number of waves" pick out noticeably different orbits.

Step 2: de Broglie wavelength for an electron in an atom.
For an electron in the first Bohr orbit of hydrogen, momentum \( p \sim 10^{-24}\,\text{kg m/s} \), giving \[ \lambda = \frac{h}{p} \sim \frac{6.6 \times 10^{-34}}{10^{-24}} \sim 10^{-10}\,\text{m} \] which is of the same order as the orbit radius itself (\( \sim 10^{-10}\,\text{m} \)). The wave nature is not a small correction here -- it controls the geometry of the orbit completely, so only certain discrete radii allow a standing wave to close on itself.

Step 3: de Broglie wavelength for a planet.
For a planet such as Earth, momentum is enormous: \( p = mv \sim (6 \times 10^{24}\,\text{kg})(3 \times 10^{4}\,\text{m/s}) \sim 10^{29}\,\text{kg m/s} \), so \[ \lambda = \frac{h}{p} \sim \frac{6.6\times 10^{-34}}{10^{29}} \sim 10^{-63}\,\text{m} \] This is vanishingly small compared to the orbital radius (\( \sim 10^{11}\,\text{m} \)). The ratio \( \lambda / r \) is about \( 10^{-74} \), meaning an astronomically large number of wavelengths fit into the orbit.

Step 4: Why the quantization becomes unobservable.
When \( \lambda \ll r \), neighbouring allowed orbits (differing by one wavelength in circumference) are separated by a change in radius that is a correspondingly minuscule fraction of \( r \). No measurement can resolve such a difference, so the allowed radii form what is effectively a continuum. The restriction is still technically present, but it has no observable consequence.

This is why quantization is a defining feature of atomic-scale motion, where \( \lambda \) and \( r \) are comparable, but is never mentioned for planetary motion, where \( \lambda \) is many tens of orders of magnitude smaller than the orbit -- the discreteness is real but far too fine to detect.