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In statistical physics, the rotation of molecules in a gas can be excited if the temperature is high enough. This rotation is excited in a quantum mechanical way. But is the axis of rotation also quantized? Could a molecule of $N_2$ rotate around any axis when excited?

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That depends on the molecule and its symmetries. For example, a molecule of an ideal monoatomic gas cannot rotate because it is spherically symmetric. A typical diatomic gas molecule, like $N_2$ could rotate about two axes because it is axisymmetric so it cannot rotate about its axis of symmetry. A more complicated gas molecule, like water vapor, can rotate about all three axes since it is not symmetric at all.

Note that rotation about three axes does not imply anything about quantization of the axes since any direction is a vector sum of three axes. However, the amount of energy in a rotational degree of freedom is quantized and thus corresponds to a specific wavelength, just as with energy in any other degree of freedom.

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