I Spectra in the thermal interpretation

A. Neumaier
Science Advisor
Insights Author
Messages
8,699
Reaction score
4,771
TL;DR Summary
It is demonstrated how the thermal interpretation explains the spectra of atoms and molecules.
In any quantum system, the differences of the energy levels (the eigenvalues of the Hamiltonian ##H##) are in principle directly observable, since they represent excitable oscillation frequencies of the system and thus can be probed by coupling the system to a harmonic oscillator with adjustable frequency. Thus the observed spectral properties of quantum systems appear in the thermal interpretation as natural resonance phenomena.

To see this, we shall assume for simplicity a quantum system whose Hamiltonian has a purely discrete spectrum. We work in the Heisenberg picture in a basis of eigenstates of the Hamiltonian, such that ##H|k\rangle =E_k|k\rangle ## for certain energy levels ##E_k##. The q-expectation
$$\langle A(t)\rangle =Tr ~\rho A(t)=\sum_{j,k}\rho_{jk}A_{kj}(t)$$
is a linear combination of the matrix elements
$$A_{kj}(t)=\langle k|A(t)|j\rangle =\langle k|e^{iHt/\hbar}Ae^{-iHt/\hbar}|j\rangle=e^{iE_kt/\hbar}\langle k|A|j\rangle e^{-iE_jt/\hbar}=e^{i\omega_{kj}t}\langle k|A|j\rangle ,$$
where
$$\omega_{kj} = \frac{E_k-E_j}{\hbar}.$$
Thus every q-expectation exhibits multiply periodic oscillatory behavior whose frequencies ##\omega_{jk}## are scaled differences of energy levels. This relation is the modern general form of the Rydberg--Ritz combination principle.

Linear coupling of ##A## to a macroscopic (essentially classical, large ##m##) harmonic oscillator leads after standard approximations to a forced damped classical harmonic oscillator dynamics
$$m\ddot q + c\dot q + kq =F(t),$$ where ##F(t)=\langle A(t)\rangle## has the above form. If ##k## is adjusted according to ##k=\omega^2m,## standard analysis gives approximately amplitudes depending on ##\omega## in an approximately Lorentz-shaped way.

This is indeed what is observed in actual measured spectra.
 
Last edited:
  • Like
Likes Mentz114
Physics news on Phys.org
Typo ?

##\langle A(t)\rangle =Tr ~\rho A(t)=\sum_{j,k}\rho_{jk}A_{kj}(t)##
 
  • Like
Likes dextercioby
Mentz114 said:
Typo ?

##\langle A(t)\rangle =Tr ~\rho A(t)=\sum_{j,k}\rho_{jk}A_{kj}(t)##
Thanks, corrected!
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In her YouTube video Bell’s Theorem Experiments on Entangled Photons, Dr. Fugate shows how polarization-entangled photons violate Bell’s inequality. In this Insight, I will use quantum information theory to explain why such entangled photon-polarization qubits violate the version of Bell’s inequality due to John Clauser, Michael Horne, Abner Shimony, and Richard Holt known as the...
I understand that the world of interpretations of quantum mechanics is very complex, as experimental data hasn't completely falsified the main deterministic interpretations (such as Everett), vs non-deterministc ones, however, I read in online sources that Objective Collapse theories are being increasingly challenged. Does this mean that deterministic interpretations are more likely to be true? I always understood that the "collapse" or "measurement problem" was how we phrased the fact that...

Similar threads

Back
Top