Bohr Correspondence Principle

phy-1109 PHY-1109 Physics

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The greater the quantum number, the closer quantum physics approaches classical physics.

According to electromagnetic theory, an electron moving in a circular orbit radiates EM waves whose frequencies are equal to its frequency of revolution and to harmonics (that is, integral multiples) of that frequency.

Related PYQ (Principle Statement)

Question	Year(s) of Appearance
State correspondence principle. Define space quantization.	2024

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Topic 1: Classical Frequency of Revolution ($f$)

The classical frequency of revolution ($f$) is derived from the balance of electrostatic and centripetal forces, and the quantized orbit radii.

Main Derivations/Equations:

1. Electrostatic Force ($F_e$): $F_e=\frac{1}{4\pi {ϵ}_o}\frac{e^2}{r^2}\text{......(1)}$
2. Centripetal Force ($F_c$): $F_c=\frac{m{v}^2}{r}\text{......(2)}$
3. Orbit Stability Condition ($F_e=F_c$): This leads to the electron speed ($v$): $\frac{m{v}^2}{r}=\frac{1}{4\pi {ϵ}_o}\frac{e^2}{r^2}⟹\mathbf{v}=\frac{\mathbf{e}}{\sqrt{4\pi {ϵ}_{\mathbf{o}}\mathbf{mr}}}\text{......(4)}$ (Note: This speed $v$ is derived irrespective of the specific quantum number $n$).
4. Possible Orbit Radii ($r_n$): (Derived using quantization condition $n\lambda =2\pi r_n$ and $\lambda =h/mv$). ${\mathbf{r}}_{\mathbf{n}}=\frac{{\mathbf{n}}^2{\mathbf{h}}^2{ϵ}_{\mathbf{o}}}{\pi \mathbf{m}{\mathbf{e}}^2}\text{......(7)}$
5. Frequency of Revolution ($f$): This is the classical frequency of emission according to electromagnetic theory. $f=\frac{\text{electron speed}}{\text{orbit circumference}}=\frac{v}{2\pi r}$ Substituting $v$ and $r_n$ and simplifying results in: $\mathbf{f}=\frac{\mathbf{m}{\mathbf{e}}^4}{8{ϵ}_{\mathbf{o}}^2{\mathbf{h}}^3}\left(\frac{2}{{\mathbf{n}}^3}\right)\text{......(8)}$

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Topic 2: Quantum Frequency and the Correspondence Limit

The quantum theory describes radiation arising from a transition between two discrete energy levels.

Main Derivations/Equations:

1. Quantum Frequency ($\nu$): A hydrogen atom dropping from the initial quantum number $n_i$ to the final quantum number $n_f$ emits a photon whose frequency ($\nu$) is given by: $\nu =\frac{\mathbf{m}{\mathbf{e}}^4}{8{ϵ}_{\mathbf{o}}^2{\mathbf{h}}^3}\left(\frac{1}{{\mathbf{n}}_{\mathbf{f}}^2}-\frac{1}{{\mathbf{n}}_{\mathbf{i}}^2}\right)\text{......(9)}$
2. Substitution for Transition ($n_i$ and $n_f$): Let the initial quantum number be $n$ ($n_i=n$), and the final quantum number be $n-p$ ($n_f=n-p$), where $p=1,2,3,\dots$. $\nu =\frac{m{e}^4}{8{ϵ}_o^2{h}^3}\left(\frac{1}{(n-p{)}^2}-\frac{1}{n^2}\right)$ Simplifying the term in parentheses: $\left(\frac{1}{(n-p{)}^2}-\frac{1}{n^2}\right)=\frac{n^2-(n-p{)}^2}{n^2(n-p{)}^2}=\frac{2np-p^2}{n^2(n-p{)}^2}$
3. Applying the Correspondence Limit: This limit applies when the quantum numbers ($n_i$ and $n_f$) are very large, and $n$ is much greater than $p$. $2np-p^2\approx 2np$ $(n-p{)}^2\approx n^2$
4. Approximate Quantum Frequency ($\nu$): Substituting these large number approximations back into the frequency formula: $\nu \approx \frac{m{e}^4}{8{ϵ}_o^2{h}^3}\left(\frac{2np}{n^2(n^2)}\right)=\frac{\mathbf{m}{\mathbf{e}}^4}{8{ϵ}_{\mathbf{o}}^2{\mathbf{h}}^3}\left(\frac{2p}{{\mathbf{n}}^3}\right)\text{......(10)}$

Conclusion of Correspondence Principle:

• Comparing the classical frequency of revolution $f$ (Eq. 8, where $1/n^3$ term has a factor of 2) and the approximate quantum frequency $\nu$ (Eq. 10, where $1/n^3$ term has a factor of $2p$).
• When the transition involves $p=1$ (a jump between adjacent, highly excited orbits), the frequency of the emitted radiation ($\nu$) is exactly the same as the frequency of rotation ($f$) of the orbital electron given in Eq. (8).
• When $p=2,3,4,\dots$, the emitted radiation frequencies are multiples (harmonics) of the rotation frequency $f$.
• Result: Hence both quantum and classical pictures of the hydrogen atom make the same predictions in the limit of very large quantum numbers.

Related PYQ (Limit Proof)

Question	Year(s) of Appearance
Show that the quantum physics gives the same results as classical physics in the limit of very large quantum numbers.	2018, 2022