(a) Show that in the Bohr model, the frequency of revolution...

(a) Show that in the Bohr model, the frequency of revolution of an electron in its circular orbit around a stationary hydrogen nucleus is $f=m e^{4} / 4 \epsilon_{0}^{2} n^{3} h^{3}$ . (b) In classical physics, the frequency of revolution of the electron is equal to the frequency of the radiation that it emits. Show that when $n$ is very large, the frequency of revolution does indeed equal the radiated frequency calculated from Eq. $(39.5)$ for a transition from $n_{1}=n+1$ to $n_{2}=n .$ (This illustrates Bohr's correspondence principle, which is often used as a check on quantum calculations. When $n$ is small, quantum physics gives results that are very different from those of classical physics. When $n$ is large, the differences are not significant, and the two methods then "correspond." In fact, when Bohr first tackled the hydrogen atom problem, he sought to determine $f$ as a function of $n$ such that it would correspond to classical results for large $n . )$

Key Concepts

Bohr Model

The Bohr model is an early quantum model of the hydrogen atom where electrons orbit the nucleus in fixed, quantized circular paths rather than following the arbitrary paths of classical mechanics. In this model, only certain orbits with specific energies and angular momentum values are permitted, and electrons can transition between these orbits by absorbing or emitting discrete energy quanta.

Frequency of Revolution

Within the Bohr model, the concept of the frequency of revolution refers to the rate at which an electron moves along its circular orbit around the nucleus. This frequency is determined by balancing the centripetal force with the Coulomb force between the negatively charged electron and the positively charged nucleus, and it depends on quantum numbers that reflect the allowed orbit sizes and energies.

Energy Transition and Radiation Frequency

When an electron transitions from one quantized orbit to another, it emits or absorbs a photon whose frequency corresponds to the energy difference between the orbits, as described by Planck's relation. The radiation frequency calculated for these transitions directly links the quantized energy levels of the electron with the observable spectral lines of the atom.

Correspondence Principle

The correspondence principle is the concept that quantum mechanical predictions must converge to classical physics results in the limit of large quantum numbers. This principle is used to check that the behavior predicted by quantum models aligns with classical expectations for high energy states or large orbitals, ensuring consistency between the two regimes.

Transcript

00:01 In this question, the main idea that we want to drive across is a correspondence between the quantum picture to a classical picture.

00:15 So for a quantum picture, usually we have something that is quantized, such as the angular momentum, the energy, when we have some corresponding quantum number associated, the n value.

00:31 Now what we want to show is that when n goes to infinity, this will correspond to a classical picture.

00:42 So this is the correspondence postulate.

00:49 And the thing that you are interested in for this question is the frequency.

00:55 The frequency of radiation that is being emitted by an atom.

01:05 So the frequency in the quantum picture is basically due to the energy difference between the energy states.

01:15 This will correspond to the energy of the photon, so lambda is equal to c over delta e.

01:24 On the other hand for the classical picture, the frequency of radiation is actually the frequency of electron that is orbiting around its orbit.

01:37 So the frequency of the electron orbit.

01:41 This goes to 2 pi r, the radius of the orbit, divided by the velocity of the electron.

01:53 So this is the circumference divided by the velocity which will give us the period.

02:01 And therefore the frequency is just 1 over period.

02:05 Just vn over 2 pi rn.

02:09 So we want to show that there is some connection between the wavelength over here where we can then find the frequency, which goes to c over lambda.

02:33 Right, we want to find whether there's a correspondence between these two very different pictures of the radiation when and goes to infinity.

02:49 Now we're going to start off with the classical picture of the electron orbiting.

02:59 And to find the frequency, we just have to substitute in the velocity of the electron as well as the radius of the orbit from the boss model...

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