Given that a0=ħ/(αme c) show that the product a0 k of the...

Given that $a_{0}=\hbar /\left(\alpha m_{\mathrm{e}} c\right)$ show that the product $a_{0} k$ of the Bohr radius and the wavenumber of a photon of energy $E$ satisfies $$ a_{0} k=\frac{E}{\alpha m_{\mathrm{e}} c^{2}} $$ Hence show that the wavenumber $k_{\alpha}$ of an $\mathrm{H} \alpha$ photon satisfies $a_{0} k_{\alpha}=\frac{5}{72} \alpha$ and determine $\lambda_{\alpha} / a_{0}$. What is the connection between this result and our estimate that $\sim 10^{7}$ oscillations are required to complete a radiative decay. Does it imply anything about the way the widths of spectral lines from allowed atomic transitions vary with frequency?

Given that $a_{0}=\hbar /\left(\alpha m_{\mathrm{e}} c\right)$ show that the product $a_{0} k$ of the Bohr radius and the wavenumber of a photon of energy $E$ satisfies
$$
a_{0} k=\frac{E}{\alpha m_{\mathrm{e}} c^{2}}
$$
Hence show that the wavenumber $k_{\alpha}$ of an $\mathrm{H} \alpha$ photon satisfies $a_{0} k_{\alpha}=\frac{5}{72} \alpha$ and determine $\lambda_{\alpha} / a_{0}$. What is the connection between this result and our estimate that $\sim 10^{7}$ oscillations are required to complete a radiative decay. Does it imply anything about the way the widths of spectral lines from allowed atomic transitions vary with frequency?

Key Concepts

Spectral Line Width Variation with Frequency

The width of spectral lines is related to the lifetime of the excited state—shorter lifetimes yield broader lines. The analysis connecting the product of the Bohr radius and the photon wavenumber to the fine structure constant, and the resulting estimates for oscillation counts, suggests that the natural linewidths do not vary significantly with frequency for allowed atomic transitions. This highlights how the decay dynamics and the inherent quantum mechanical properties govern line broadening in atomic spectra.

Radiative Decay and Oscillation Count

In radiative decay processes, an excited state of an atom decays to a lower energy state by emitting a photon. The statement that around 10^7 oscillations occur before decay implies that the electron or the electromagnetic field oscillates many times during the lifetime of the excited state, providing insight into the lifetime broadening and natural linewidth of spectral lines.

Hydrogen Spectral Lines and Transition Energies

Hydrogen spectral lines, including the H? line, arise from electronic transitions between energy levels in the hydrogen atom. Analyzing these transitions involves matching energy differences with photon energies, linking atomic dimensions through the Bohr radius to the wavelengths or wavenumbers of emitted light.

Fine Structure Constant

The fine structure constant, often denoted by ?, is a dimensionless constant that characterizes the strength of the electromagnetic interaction. It appears in many areas of quantum electrodynamics and atomic physics, influencing the size of the Bohr radius and the scale of energy level separations.

Bohr Radius

The Bohr radius is a fundamental physical constant that represents the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It sets the scale for atomic dimensions in quantum mechanics and is key to understanding atomic structure and electron behaviour in the Bohr model.

Photon Energy and Wavenumber Relation

The relationship between a photon's energy and its wavenumber is fundamental in quantum mechanics. By equating the energy to the product of Planck’s constant and frequency (or using the relation E = ?ck for wavenumber), one can connect the scales of atomic physics (like the Bohr radius) with the properties of emitted or absorbed radiation.

Transcript

00:01 Hi there, so for this problem, we need to show that the equation for the radius of the first board orbit can be written as the following.

00:12 That is the wavelength c over 2 pi alpha.

00:22 And the other one that we need to show is that the energy and the lowest energy for the hydrogen atom, can be written as 1 over 2 alpha to the squared and the mass times the speed of light square.

00:49 Now we are given the expressions for alpha and alpha is equal to columns constant times the charge of an electron squared divided by the bar h.

01:10 Times the speed of light.

01:13 And while the wavelength c is equal to plam constant divided by the mass times the speed of light.

01:28 Now, we first start with the first expression and we need to begin with the following.

01:35 We have the following equation.

01:38 So that is barred, hbord, squared, and this divided by the mass times columns constant times the charge of an electron square.

01:51 Now we can write this in the following form.

01:55 We can separate this as this.

01:59 It's very trivial.

02:03 And what we can do is to multiply and divide this expression by the speed of light.

02:11 So we have this.

02:12 Now we can separate this conveniently, so i'm going to separate each part times the speed of light divided by n times the speed of light is squared, and this times the remaining terms.

02:32 So i'm going to write this as follows.

02:40 Now, what i are going to do, the next step, is that we know that h bar, is related with plumst constant by the following.

02:54 So we know this, so i'm going to substitute that in here.

03:00 So we are going to have the following.

03:03 We are going to have 1 over 2 pi times plums constant divided by the mass times the speed of light and this times 1 over the...

03:17 Overt this one right here.

03:21 As you can see, you can identify this one, this in here, as this one given in here, alpha.

03:33 So that this in here corresponds to 1 over alpha.

03:39 So we will have the following.

03:41 We will have, well, in this one right here corresponds to the wavelength c.

03:50 So we will obtain then that this is the wavelength c over 2 pi alpha.

03:57 So we have shown the expression that we wanted to show this one right here.

04:05 Now we do a similar procedure in order to show the expression for the energy...

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