Timeline for How is the energy of phonon modes $\left(n+\frac{1}{2}\right)\hbar \omega_k$ when each atom in the mode has $\left(p+\frac{1}{2}\right)\hbar \omega$?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Dec 13, 2023 at 17:15 | vote | accept | Rishab Navaneet | ||
| Dec 12, 2023 at 16:13 | comment | added | Cosmas Zachos | I don't know how explicit Kittel's book is on lattice vibrations, but neat expositions of phonons (what photons are to the EM field, phonons are to a lattice) can be found in the classic text of Fetter & Walecka. | |
| Dec 11, 2023 at 15:24 | answer | added | Cosmas Zachos | timeline score: 1 | |
| Dec 11, 2023 at 14:22 | comment | added | Cosmas Zachos | This might help, but your misconception is ignoring that the hamiltonian is the sum of those of all decoupled modes, whose zero-point energies you sum, for the ground state. You may not jam all particles into the same mode, which is a linear combination of them. Do you want an illustration for N=2 ? | |
| Dec 11, 2023 at 8:54 | comment | added | Rishab Navaneet | @CosmasZachos could you please suggest some extra reading for the ground state energy u mentioned... would like to know the rationale behind summing over the normal modes... Also please clarify - is this the ground state energy of the mode? \\ \\ But still my point remains... assuming periodic boundary conditions, $\omega_n = 2\omega_0 \sin\frac{n\pi}{N}$ and the summation may not give me $\frac{N}{2}\hbar \omega_0$ but $\approx \frac{2}{\pi}N$ (for large N). even this doesnt match with the ground state energy of a mode $\frac{\hbar \omega_k}{2}$. I'm missing something | |
| Dec 10, 2023 at 16:29 | comment | added | Cosmas Zachos | " it is oscillating at a different frequency $\omega_k$. Is that why?" Indeed, that's why. The ground state energy is $\hbar ( \omega_0+ \omega_1+ \omega_2+...+\omega_{N-1})/2$, and does not collapse to your expression unless you switch off the couplings and trivialize the mode resolution. | |
| Dec 10, 2023 at 15:19 | comment | added | Jon Custer | Those N atoms are bound in a crystal, not isolated from each other. | |
| Dec 10, 2023 at 14:48 | comment | added | Cosmas Zachos | Have you fleshed out your thinking with N =2, just two coupled harmonic oscillators? What do you see in the limit of vanishing mutual coupling? How do you define a mode? | |
| Dec 10, 2023 at 5:52 | history | asked | Rishab Navaneet | CC BY-SA 4.0 |