# I Spectra in the thermal interpretation

#### A. Neumaier

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.

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#### Mentz114

Gold Member
Typo ?

$\langle A(t)\rangle =Tr ~\rho A(t)=\sum_{j,k}\rho_{jk}A_{kj}(t)$

#### A. Neumaier

Typo ?

$\langle A(t)\rangle =Tr ~\rho A(t)=\sum_{j,k}\rho_{jk}A_{kj}(t)$
Thanks, corrected!

"Spectra in the thermal interpretation"

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