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The first-order energy shift is not well defined, since there is no unique way to choose a basis of eigenstates for the unperturbed system. The various eigenstates for a given energy will perturb with different energies, or may well possess no continuous family of perturbations at all.
The first-order perturbation matrix on basis of the unperturbed rigid rotor function is non-zero and can be diagonalized. This gives shifts and splittings in the rotational spectrum. Quantitative analysis of these Stark shift yields the permanent electric dipole moment of the symmetric top molecule.
First-order phase transitions exhibit a discontinuity in the first derivative of the free energy with respect to some thermodynamic variable. [6] The various solid/liquid/gas transitions are classified as first-order transitions because they involve a discontinuous change in density, which is the (inverse of the) first derivative of the free ...
A CPT describes a cusp in the thermodynamic properties of a system. It signals a reorganization of the particles; A typical example is the freezing transition of water describing the transition between liquid and solid. The classical phase transitions are driven by a competition between the energy of a system and the entropy of its thermal ...
Here we consider the case where the system has a symmetry and the energy is invariant when the order parameter changes sign. A first-order transition will arise if the quartic term in is negative. To ensure that the free energy remains positive at large , one must carry the free-energy expansion to sixth-order, [5] [6] (,) = +,
Using perturbation theory, the first-order energy shift can be calculated as = >, which requires the knowledge of accurate many-electron wave function. Due to the 1 / M N {\displaystyle 1/M_{N}} term in the expression, the specific mass shift also decrease as 1 / M N 2 {\displaystyle 1/M_{N}^{2}} as mass of nucleus increase, same as normal mass ...
A key example of this phenomenon is the spin–orbit interaction leading to shifts in an electron's atomic energy levels, due to electromagnetic interaction between the electron's magnetic dipole, its orbital motion, and the electrostatic field of the positively charged nucleus.
This equation is known as the Breit–Rabi formula and is useful for systems with one valence electron in an (= /) level. [ 9 ] [ 10 ] Note that index F {\displaystyle F} in Δ E F = I ± 1 / 2 {\displaystyle \Delta E_{F=I\pm 1/2}} should be considered not as total angular momentum of the atom but as asymptotic total angular momentum .