Show that in the Bohr model, the frequency of a photon...

Show that in the Bohr model, the frequency of a photon emitted in a transition between levels $n+1$ and $n$, in the limit of large $n$, is equal to the electron's orbital frequency. (This is an example of Bohr's correspondence principle.)

Key Concepts

Orbital Frequency

The orbital frequency is the frequency with which an electron orbits the nucleus in its circular path. In the classical treatment of the Bohr model, this frequency can be derived from the centripetal force condition and Coulomb’s law, relating the electron’s velocity and the radius of its orbit.

Photon Emission

Photon emission occurs when an electron transitions from a higher energy level to a lower one, releasing a photon whose energy equals the difference between the initial and final energy levels. The frequency of the emitted photon is given by the relation E = hf, where h is Planck's constant.

Correspondence Principle

The correspondence principle states that the predictions of quantum mechanics must agree with classical physics when applied to systems with large quantum numbers. In this context, as the principal quantum number becomes large, the quantum result for the photon frequency emitted during a transition approaches the classical orbital frequency of the electron, thus reconciling quantum predictions with classical behavior.

Bohr Model

The Bohr model posits that electrons move in fixed circular orbits around the nucleus with quantized angular momentum, and that they occupy discrete energy levels. Transitions between these levels result in the emission or absorption of photons, with the energy difference between levels corresponding to the photon energy.

Quantized Energy Levels

In the Bohr model, each electron orbit corresponds to a specific energy level determined by a principal quantum number. The energy of these levels is inversely proportional to the square of the quantum number, and transitions between levels lead to quantized changes in energy, which manifest as photons with specific frequencies.

Transcript

00:01 To show that in the bore model, the frequency of a photon emitted in transition between the n plus 1 and the n, and the limit of large n is equal to the electron's orbital frequency.

00:09 So the electron orbital frequency, f sub -n, is equal to 2 e -0 over h times 1 over end of the third, where e -sub -n is equal to negative e -0 over n squared, and therefore following that logic, e -7 plus 1 is equal to negative e -0 over n plus 1 squared, where e -0 here is the ground state energy.

00:27 Okay, so delta e, which is equal to e sub n minus e sub n plus 1, because this is the transition we're wanting to consider.

00:43 We just plug those values above in.

00:47 So we can pull the negative e not out because that's common in both.

00:57 So what we're left with is 1 over n squared minus 1 over n plus 1 squared.

01:07 So using our algebra on what's in the brackets to simplify that, we have this delta e value being equal to negative e not multiplied by n plus 1 squared.

01:36 Plus signs not wanting to work with me.

01:39 There we go.

01:41 So n plus 1 squared minus n squared, all divided by n squared times n plus 1 squared.

02:10 Okay.

02:12 But delta e is also equal to.

02:14 By definition, planck's constant times the frequency.

02:24 Therefore, we can conclude from that that the frequency, f, is equal to, we have negative...

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