I don't really understand the explanation given in Binney's text about:(adsbygoogle = window.adsbygoogle || []).push({});

Hamiltonian is given by:

[tex]H = \frac{\hbar^2}{2} \left( \frac{J_x^2}{I_x} + \frac{J_y^2}{I_y} + \frac{J_z^2}{I_z} \right)[/tex]

Orient axes such that ##I_x = I_y = I##.

[tex]H = \frac{\hbar^2}{2} \left( \frac{J^2}{I} + J_z^2(\frac{1}{I_z} - \frac{1}{I})\right)[/tex]

Energy is given by:

[tex]E_{jm} = \frac{\hbar^2}{2} \left[ \frac{j(j+1)}{I} + m^2(\frac{1}{I_z} - \frac{1}{I}) \right][/tex]

We are only interested in states:

[tex]E_{jm} = \frac{\hbar^2}{2I} j(j+1) [/tex]

Emitted energy and frequency are:

[tex]\Delta E_p =\pm (E_j - E_{j-1}) = \pm j\frac{\hbar^2}{I}[/tex]

[tex]v_j = j\frac{\hbar}{2\pi I}[/tex]

Let's try to analyze the explanation here.

1. Yes, energy, J_{z}and J^{2}share the same eigenstates ##|j, m>##.

2. <J^{2}> = j(j+1) : Yes, since that is the eigenvalue and eigenvalue correspond to real observables.

3. Why do low lying states with ##m = 0## and ##j~O(1)## lead to: ## j(j+1) >> j ##? Firstly, doesn't low lying states correspond to a low ##j##? And what does m have to do with anything? ##m## was defined as the eigenvalue of J_{i}and ##j = m_{max}##

The rest of the argument doesn't make any sense at all..

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# Rotation Spectra; energies

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