(adsbygoogle = window.adsbygoogle || []).push({}); Consider the molecule CN, which may be described by a dumbbell consisting of two masses [itex]M_1[/itex] and [itex]M_2[/itex] attached by a rigid rod of length [itex]a[/itex]. The dumbbell rotatesin a planeabout an axis going through the center of mass and perpendicular to it.

Now, am I right in saying that the effective mass is:

Write down the Hamiltonian that describes the motion.What is the energy spectrum?Write down an expression for the difference in energy between the ground state and the first excited state in terms of the masses and[itex]a[/itex].

[tex]\mu = \frac{M_1 + M_2}{M_1M_2}[/tex]

and treating the problem as that of a point mass, [itex]\mu[/itex], travelling in a planar circular orbit or radius [itex]a[/itex]? Given this, I believe I would have:

[tex]H = \frac{p^2}{2\mu} + \frac{1}{2}(\mu a^2)\omega ^2[/tex]

Now, my book isn't clear on what the "energy spectrum" specifically is, but does it have to do with the spectral decomposition of a (linear) operator? What exactly am I to do for (b)?

My book has some stuff on harmonic oscillator, where the energy eigenvalues are given:

[tex]E_n = \left (n + \frac{1}{2}\right )\hbar \omega[/tex]

If I were to find the energy difference between the ground state and first eigenstate, would that simply be:

[tex]E_1 - E_0 = \hbar \omega[/tex]

If I can find the energy eigenvalues for my Hamiltonian (since this isn't a harmonic oscillator in this problem) is that all I need to do for (c): express [itex]E_1 - E_0[/itex]?

Thanks.

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# Homework Help: Quantum Mechanics Problem - [Rotating Molecule]

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