# Diatomic molecule

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alphysicist
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Hi learning_phys,

Is this saying that the frequency of vibrations in a diatomic molecule is quantized?
In that picture of the diatomic molecule there is only one frequency for a specific diatomic molecule.

Once you have specified the effective spring constant and the masses, that gives the frequency, which is the same for all vibrations of the molecule.

isn't the energy of this diatomic molecule quantized since it is based off of the quantum harmonic oscillator?

also, didnt Debye come up with a theory of specific heat that allows for a spectrum of frequencies?

could someone reference me to some formulas/articles on debye's theory?

alphysicist
Homework Helper
Yes, the energy is quantized; but there is only one frequency in the particular model from the link in your original post.

(Just like in a classical model of a regular mass on a spring, there are an infinite number of energies possible, but only one frequency $\omega=\sqrt{k/m}$.)

how can energy increase if the frequency of the spring is constant?

alphysicist
Homework Helper
how can energy increase if the frequency of the spring is constant?
Think back to a classical spring (align it horizontally so you can ignore gravity). If the spring constant is k and the mass is m, the frequency is constant for all oscillations. But the energy depends on the amplitude $E_{\rm total}=\frac{1}{2}kA^2$ (or in terms of the frequency $E_{\rm total}=\frac{1}{2}m\omega^2 A^2$). So if you start the spring out by pulling it a greater distance, the energy is greater, but it still has the same frequency.

And in the quantum spring only discrete energy values are allowed, but there is still only one frequency in the model you were looking at.

Is this saying that the frequency of vibrations in a diatomic molecule is quantized?

http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/hosc.html#c1
The article you point to is confusing. The quantum harmonic oscillator has pure energy states which are indeed quantized by frequency. But you can also regroup these pure states into special "coherent" states that oscillate with one and the same frequency. The graph pictured in your article seems to coflate these two representations: the pure energy states are separated by hw, but the "average intermolecular distance" is really more descriptive of the coherent states.

It's a long story which isn't really dealt with in the short article shown here.

Think back to a classical spring (align it horizontally so you can ignore gravity). If the spring constant is k and the mass is m, the frequency is constant for all oscillations. But the energy depends on the amplitude $E_{\rm total}=\frac{1}{2}kA^2$ (or in terms of the frequency $E_{\rm total}=\frac{1}{2}m\omega^2 A^2$). So if you start the spring out by pulling it a greater distance, the energy is greater, but it still has the same frequency.

And in the quantum spring only discrete energy values are allowed, but there is still only one frequency in the model you were looking at.
are you saying that in the quantum spring, the amplitude is discrete?

also, the total energy does depend on the frequency according to your equation. why do you say the frequency doesnt change?

alphysicist
Homework Helper
are you saying that in the quantum spring, the amplitude is discrete?
Well, you might want to be careful. How are you defining amplitude? The quantum spring is different from the classical spring, and one of those differences is that the idea of a classical trajectory does not apply.

There are plenty of plots of harmonic oscillator wave functions on the web, and from those you can easily compare the classical amplitudes to the shape of the wavefunctions.

comparing classical amplitudes and the wavefunctions:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html

and for some comparisons between the classical and quantum probabilities:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc6.html#c2

also, the total energy does depend on the frequency according to your equation. why do you say the frequency doesnt change?
In your link, the diatomic moelcule is modelled by a spring with force constant k connecting two masses. If you have a different spring or different masses, which means a different molecule, then you'll have different frequencies. But for a specified molecule, you'll have a range of discrete energies, all based on a single frequency.

So if you compare two different molecules, their sets of energy levels will be different, because the energy depends on the frequency. But for a specific molecule, the frequency is a constant.

(All of this applies to the simple model given in your link, of course.)

alphysicist
Homework Helper
ah, i thought k was the wave number...

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc6.html#c2

what exactly is 'n' for the classical model? for the quantum oscillator, n can be the energy levels that the oscillator is in, but what is n for the classical model?
I guess they don't explicitly say it, but I believe what they would do to get a reasonble comparison is to choose the quantum n value, find the energy for the quantum state, and then use that energy to find the amplitude of a classical spring with that same energy.

then the diatomic molecule does have quantized amplitudes?

alphysicist
Homework Helper
then the diatomic molecule does have quantized amplitudes?
How are you defining amplitude? In the classical case, the position for the harmonic oscillator might be

x(t) = A cos(w t)

so x is restricted to be between A and -A, so the amplitude is A.

For the quantum harmonic oscillator, the first several wavefunctions are given in one of those links I posted. What is your definition for amplitude? What would be the amplitude of $$\Psi_0$$, for example, and how would you find it?

does a diamolecule still vibrate at T=0? Just like the quantum harmonic oscillator, at n=0, there is still energy?

i'm not sure why the quantum oscillator is used to describe the diamolecule since at T=0, the molecule should not be vibrating at all, and yet, if we use the quantum oscillator, it gives a minimum energy.

why is this?