The collective excitations from the spin density wave pinned by some impurities are investigated with the self-consistent harmonic approximation applied for the phase Hamiltonian. It is found that the pinning-depinning transition due to the quantum fluctuation in the ground state cannot occur in two-dimensional nor three-dimensional isotropic system. In contrast, it has already been revealed that the quantum depinning (melting) can occur in one dimension. It suggests that the critical dimension of the quantum localization-delocalization phase transition is one, within the present approximation. The present result is also available for the charge density wave pinned by some impurities.
Using the functional-integral method applied to the BCS Hamiltonian we have calculated the partition function, off-diagonal correlation function, specific heat, and conductivity of a super-conductor in the transition region. The partition function is obtained in the form $\ensuremath{\int}D\ensuremath{\psi}{e}^{\ensuremath{-}\ensuremath{\beta}F[\ensuremath{\psi}]}$, where
F is of the Ginzburg-Landau form but with a time-dependent order parameter. An examination of the coefficient
b of the fourth-order term in
F, however, shows that
b is sharply peaked at zero frequency, thus justifying the use of a time-independent order parameter. We then apply a self-consistent mean-field approximation similar to that of Mar\ifmmode \check{c}\else \v{c}\fi{}elja to the fourth-order term. The self-consistency condition determines the mean-square fluctuation in the order parameter and a renormalized temperature shift in terms of which the nature of the phase transition (or lack of it) can be understood. We find that only in three dimensions does a phase transition strictly occur, at a slightly lowered transition temperature with critical indices $\ensuremath{\gamma}=2$, $\ensuremath{\nu}=1$. In two, one, and zero dimensions the mean-square fluctuation in the order parameter is bounded for
T>0 and the mean value of the order parameter vanishes, indicating that the transition is completely suppressed in samples of reduced dimensionality. This behavior is reflected in the off-diagonal correlation function which manifests off-diagonal long-range order only in three dimensions. The specific heat in three dimensions is found to be finite at
Tc, with critical index $\ensuremath{\alpha}=\ensuremath{-}1$. In two, one, and zero dimensions the specific-heat transition is rounded relative to the BCS result. We find an anomalous factor of 2 in the specific-heat calculation deriving from our treatment of the fourth-order term in
F. As a result we conjecture that the Hartree approximation of Mar\ifmmode \check{c}\else \v{c}\fi{}elja may be better than the Hartree-Fock approximation of Tucker and Halperin. The equations for the conductivity are identical to those of Mar\ifmmode \check{c}\else \v{c}\fi{}elja.
A method is presented for solving the one-dimensional Schroedinger equation. The method gives the wave function in the form of an infinite series. The first term of the series is the WBK approximation. Each of the higher terms in turn represents the reflections generated by the preceding term. The expansion is obtained by approximating the potential by a "staircase" potential, which is constant over short regions and changes its value discontinuously between these regions. The method is applied to periodic potentials. By taking the WBK approximation and the first reflection term into account, the zones of the periodic potential can be found in terms of a coefficient which gives the probability that a particle will be reflected in passing through one cell of the periodic potential.