Understanding Energy and Frequency in Rotation Spectra

[unscientific]

I don't really understand the explanation given in Binney's text about:

Hamiltonian is given by:

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

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

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

Energy is given by:

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

We are only interested in states:

$$E_{jm} = \frac{\hbar^2}{2I} j(j+1)$$

Emitted energy and frequency are:

$$\Delta E_p =\pm (E_j - E_{j-1}) = \pm j\frac{\hbar^2}{I}$$
$$v_j = j\frac{\hbar}{2\pi I}$$
Let's try to analyze the explanation here.

1. Yes, energy, J_z 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 J_i and $j = m_{max}$

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

 

[Bill_K]

He didn't say j(j+1) >> j. He said "significantly larger".

It would help to know what system this discussion is about, what the upper and lower states are, what ν_j is, and what is contained in Eq.(7.24).

 

[unscientific]

He didn't say j(j+1) >> j. He said "significantly larger".

It would help to know what system this discussion is about, what the upper and lower states are, what ν_j is, and what is contained in Eq.(7.24).

It's about the energy levels in rotation spectra of diatomic molecules - specifically CO molecule in this case.

The problem is I'm not sure what he is referring to. This is taken from Binney's book, pg 140:

 

[DrDu]

He didn't say j(j+1) >> j. He said "significantly larger".

Much more importantly, he said larger than $j^2$, not j.

 

[unscientific]

Much more importantly, he said larger than $j^2$, not j.

Ok besides that, I don't get the rest of the argument about rotation frequencies at all!

 

[DrDu]

Classically, you have two ways to determine the rotation frequency: Either from the energy of the state using: $\omega=\sqrt(2IE)$ or measuring the frequency of the emitted radiation, $\omega_\mathrm{transition}$. In QM, the energy is quantized, so ω, as determined from E will depend on J. He is saying $\omega(j-1)<\omega_\mathrm{transition}<\omega(j)$, where the transition is from j to j-1. In the limit of large j, all thre values will converge to the same classical value.