[en] The purpose of this study is to evaluate the accompanying abnormal findings of Schmorl's nodes (SN), a normal variant of the lumbar spine. Seventy-five patients with one or more SN, as seen on lumbar spine MRI were studied. Using a 1.5T MR unit, the number and location of SN, their site on the end plate, adjacent disc changes and lesions associated and not associated with SN, and accompanying associated bony spinal stenosis were retrospectively investigated. Among the 75 patients, 230 SN were noted in 375 vertebral bodies; they were relatively frequently located on the second (65, 28.3 %) and third (65, 28.3 %) lumbar vertebrae. The most common end-plate site of SN was the posterior one-third portion (160; 69.6 %). In 450 discs of these 75 patients, 172 lesions were notes; those associated with SN (76/167, 45.5%) were more common than frequently located on intervertebral disc L2-3 or L3-4 (p<0.05). Thirty-seven SN (16.1%) were associated with bony spinal stenosis. Because it is frequently associated with disc lesions and bony spinal stenosis, SN of the lumbar spine may be a pathologic condition rather than a normal variant. (author). 17 refs., 2 tabs., 2 figs

[en] We considered a quantum system of simple pendulum whose length of string is increasing at a steady rate. Since the string length is represented as a time function, this system is described by a time-dependent Hamiltonian. The invariant operator method is very useful in solving the quantum solutions of time-dependent Hamiltonian systems like this. The invariant operator of the system is represented in terms of the lowering operator a(t) and the raising operator a^†(t). The Schroedinger solutions ψ_n(θ, t) whose spectrum is discrete are obtained by means of the invariant operator. The expectation value of the Hamiltonian in the ψ_n(θ, t) state is the same as the quantum energy. At first, we considered only θ² term in the Hamiltonian in order to evaluate the quantized energy. The numerical study for quantum energy correction is also made by considering the angle variable not only up to θ⁴ term but also up to θ⁶ term in the Hamiltonian, using the perturbation theory.