- #1
unscientific
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I don't really understand the explanation given in Binney's text about:
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, Jz and J2 share the same eigenstates ##|j, m>##.
2. <J2> = 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 Ji and ##j = m_{max}##
The rest of the argument doesn't make any sense at all..
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, Jz and J2 share the same eigenstates ##|j, m>##.
2. <J2> = 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 Ji and ##j = m_{max}##
The rest of the argument doesn't make any sense at all..
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