# ZPE calculations

Hi all!

I am a chemist trying to make her way through solid state computational physics problems and I have been stuck with this question for a while and couldn't find a clear answer to it:

What is a Zero Point Energy and especially, how can I calculate this for a bcc solid?

Anyone can help?

I suggest you look at how the harmonic oscillator is quantized in quantum mechanics. It is simply the ground state energy of the harmonic oscillator, and isn't zero because quantum operators don't commute in general.

Here is a pretty detailed explanation, but might be too technical: http://www.physics.thetangentbundle.net/wiki/Quantum_mechanics/harmonic_oscillator/operator_method [Broken]
Otherwise I recommend Griffith's QM book.

Now, to the harder part. A solid can be thought of as a lattice of ions with electrons wizzing about. The electrons move quickly enough that they cushion the interaction between ions, and you can very approximately model this by saying that each ion sits inside an $$(x - x_0)^2$$ potential, like it's attached to other ions by springs. (Technical: taylor expand the true potential about it's lowest energy stable configuration, and the first non-zero term is quadratic).

Using the fact that the lattice is periodic, you can fourier transform the whole shebang and write the collection of ions as decoupled harmonic oscillators (in momentum space). One harmonic oscillator for each wave-vector, essentially. Now, a quantum harmonic oscillator possesses a zero point or ground state energy. Now you have 3N such harmonic oscillators, (each with different frequencies), so the solid as a whole has quite a bit of ZPE to go around.

I hope that has helped. The quantization of solids in this way is treated in most condensed matter/solid state textbooks under the treatment of phonons. (The business about fourier transforming and whatnot is actually classical mechanics. Just a trick to decouple the ions)

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Mmm, phonons are quasi-particles, but quantum fields, which are a bit more advanced than ordinary quantum mechanics, also possess zero point energy (for pretty much the same reason). This leads to interesting consequences, e.g. the Casimir effect, and I believe a lot of work has been done to understand the Casimir effect in molecules, and its relation to the van der Waals force. But this isn't a "vibrational" issue and would probably seem a bit like black magic until you feel comfortable with the phonon business. However, I invite you to check it out.

Thank you very much, I understand a bit better what it is!