Abstract
In this section the same one-dimensional problems as in Sect. 5.2 will be considered, particularly the harmonic oscillator (HO) with constant frequency \(\omega = \omega _0\) and (in the limit \(\omega _0 \rightarrow 0\)) the free motion, but now including, classically, a linear velocity dependent friction force.
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- 1.
Again, non-dissipative is used here instead of conservative because, for TD frequency \(\omega = \omega (t)\), the corresponding Hamiltonian is also not a conserved quantity, though no dissipative force is present.
- 2.
The more general case is discussed in Appendix B.
- 3.
The phase velocity \(v_p\) is related to \(v_g\) via \(v_p = \frac{\omega _k}{k} = \frac{\hbar }{2 m} k = \frac{1}{2} v_g\).
- 4.
Note that \(\frac{d}{dt} |A_{\tiny \text{ NL }}| \ne |\frac{d}{dt} A_{\tiny \text{ NL }}|\).
- 5.
More precisely, magnetic induction, but the difference does not matter in this context (see, e.g., [19]).
- 6.
Because for \(t \ne t' = 0\) the functions \(f_i (t)\) can be different from 1, this also can lead to a time-dependence of the terms multiplied by \(f_i\) in the initial WP for \(t > t'\). Therefore, it is no longer the case that \(G_ {\tiny \text{ NL }} (x,x'.t,t')\) itself has also to fulfil the NLSE, like \(G_{\tiny \text{ L }}\) has to fulfil the SE in the linear case.
- 7.
The terms independent of x and \(x'\) do not necessarily all cancel due to what was mentioned in the last footnote.
- 8.
The choice \(\dot{\alpha }_{\tiny \text{ NL,0 }} = 0\) is different from the choice \(\langle [\tilde{x},\tilde{p}]_+ \rangle _{\tiny \text{ NL }} (0) = (\dot{\alpha }_{\tiny \text{ NL,0 }} - \frac{\gamma }{2} \alpha _{\tiny \text{ NL,0 }}) \alpha _{\tiny \text{ NL,0 }} = 0\) used in Sect. 5.4.2. Therefore the analytical expression for \(\alpha _{\tiny \text{ NL }}^2 (t) = \frac{\hbar }{2 m} \langle \tilde{x}^2 \rangle _{\tiny \text{ NL }}\) below (for the damped free motion) differs from the two expressions \(\langle \tilde{x}^2 \rangle _{\pm }\) in Eqs. (5.60) and (5.64), showing again the influence of the initial conditions. For \(\gamma = 0\), \(\dot{\alpha }_0 = 0\) is equivalent to \(\langle [\tilde{x},\tilde{p}]_+ \rangle _{\tiny \text{ L }} (0) = \dot{\alpha }_{\tiny \text{ L,0 }} \alpha _{\tiny \text{ L,0 }} = 0\).
- 9.
Hasse’s form of the friction term can also be regained by comparing it with \(\tilde{W}_{\tiny \text{ SCH }}\) according to \(\tilde{W}_{\tiny \text{ SCH }} = \tilde{W}_{\text{ Has }} - \langle \tilde{W}_{\text{ Has }} \rangle \). More details are below in Chap. 6.
- 10.
It might be possible to apply iterative techniques as in the Hartree–Fock method.
- 11.
This is concerning the TD quantities; in addition, for the transition from position to momentum space, \(\alpha _0\) must also be replaced by \(\epsilon _0 = \frac{1}{\alpha _0}\) (for an initial minimum-uncertainty WP) and a factor \(\mathrm{i}\) appears in the coefficient of the term depending on \(p'\).
- 12.
- 13.
On the canonical level, x and p can be expressed in terms of \(\hat{Q}\) and \(\hat{P}\).
- 14.
This also agrees with our finding in connection with the dissipative Wigner function in the previous subsection and the dissipative creation-/annihilation operators discussed in the following subsection.
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Schuch, D. (2018). Irreversible Dynamics and Dissipative Energetics of Gaussian Wave Packet Solutions. In: Quantum Theory from a Nonlinear Perspective . Fundamental Theories of Physics, vol 191. Springer, Cham. https://doi.org/10.1007/978-3-319-65594-9_5
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