Self-adjoint extensions for confined electrons: From a particle in a spherical cavity to the hydrogen atom in a sphere and on a cone

In a recent study of the self-adjoint extensions of the Hamiltonian of a particle confined to a finite region of space, in which we generalized the Heisenberg uncertainty relation to a finite volume, we encountered bound states localized at the wall of the cavity. In this paper, we study this situation in detail both for a free particle and for a hydrogen atom centered in a spherical cavity. For appropriate values of the self-adjoint extension parameter, the bound states localized at the wall resonate with the standard hydrogen bound states. We also examine the accidental symmetry generated by the Runge–Lenz vector, which is explicitly broken in a spherical cavity with general Robin boundary conditions. However, for specific radii of the confining sphere, a remnant of the accidental symmetry persists. The same is true for an electron moving on the surface of a finite circular cone, bound to its tip by a 1/r1/r potential.

The influence of yield surface shape and damage in the depth-dependent response of bone tissue to nanoindentation using spherical and Berkovich indenters

Prevention and treatment of osteoporosis rely on understanding of the micromechanical behaviour of bone and its influence on fracture toughness and cell-mediated adaptation processes. Postyield properties may be assessed by nonlinear finite element simulations of nanoindentation using elastoplastic and damage models. This computational study aims at determining the influence of yield surface shape and damage on the depth-dependent response of bone to nanoindentation using spherical and conical tips. Yield surface shape and damage were shown to have a major impact on the indentation curves. Their influence on indentation modulus, hardness, their ratio as well as the elastic-to-total work ratio is well described by multilinear regressions for both tip shapes. For conical tips, indentation depth was not statistically significant (p<0.0001). For spherical tips, damage was not a significant parameter (p<0.0001). The gained knowledge can be used for developing an inverse method for identification of postelastic properties of bone from nanoindentation.

Fate of accidental symmetries of the relativistic hydrogen atom in a spherical cavity

The non-relativistic hydrogen atom enjoys an accidental SO(4) symmetry, that enlarges the rotational SO(3) symmetry, by extending the angular momentum algebra with the Runge–Lenz vector. In the relativistic hydrogen atom the accidental symmetry is partially lifted. Due to the Johnson–Lippmann operator, which commutes with the Dirac Hamiltonian, some degeneracy remains. When the non-relativistic hydrogen atom is put in a spherical cavity of radius R with perfectly reflecting Robin boundary conditions, characterized by a self-adjoint extension parameter γ, in general the accidental SO(4) symmetry is lifted. However, for R=(l+1)(l+2)a (where a is the Bohr radius and l is the orbital angular momentum) some degeneracy remains when γ=∞ or γ = 2/R. In the relativistic case, we consider the most general spherically and parity invariant boundary condition, which is characterized by a self-adjoint extension parameter. In this case, the remnant accidental symmetry is always lifted in a finite volume. We also investigate the accidental symmetry in the context of the Pauli equation, which sheds light on the proper non-relativistic treatment including spin. In that case, again some degeneracy remains for specific values of R and γ.

Periacetabular osteotomy restores the typically excessive range of motion in dysplastic hips with a spherical head

BACKGROUND Residual acetabular dysplasia is seen in combination with femoral pathomorphologies including an aspherical femoral head and valgus neck-shaft angle with high antetorsion. It is unclear how these femoral pathomorphologies affect range of motion (ROM) and impingement zones after periacetabular osteotomy. QUESTIONS/PURPOSES (1) Does periacetabular osteotomy (PAO) restore the typically excessive ROM in dysplastic hips compared with normal hips; (2) how do impingement locations differ in dysplastic hips before and after PAO compared with normal hips; (3) does a concomitant cam-type morphology adversely affect internal rotation; and (4) does a concomitant varus-derotation intertrochanteric osteotomy (IO) affect external rotation? METHODS Between January 1999 and March 2002, we performed 200 PAOs for dysplasia; of those, 27 hips (14%) met prespecified study inclusion criteria, including availability of a pre- and postoperative CT scan that included the hip and the distal femur. In general, we obtained those scans to evaluate the pre- and postoperative acetabular and femoral morphology, the degree of acetabular reorientation, and healing of the osteotomies. Three-dimensional surface models based on CT scans of 27 hips before and after PAO and 19 normal hips were created. Normal hips were obtained from a population of CT-based computer-assisted THAs using the contralateral hip after exclusion of symptomatic hips or hips with abnormal radiographic anatomy. Using validated and computerized methods, we then determined ROM (flexion/extension, internal- [IR]/external rotation [ER], adduction/abduction) and two motion patterns including the anterior (IR in flexion) and posterior (ER in extension) impingement tests. The computed impingement locations were assigned to anatomical locations of the pelvis and the femur. ROM was calculated separately for hips with (n = 13) and without (n = 14) a cam-type morphology and PAOs with (n = 9) and without (n = 18) a concomitant IO. A post hoc power analysis based on the primary research question with an alpha of 0.05 and a beta error of 0.20 revealed a minimal detectable difference of 4.6° of flexion. RESULTS After PAO, flexion, IR, and adduction/abduction did not differ from the nondysplastic control hips with the numbers available (p ranging from 0.061 to 0.867). Extension was decreased (19° ± 15°; range, -18° to 30° versus 28° ± 3°; range, 19°-30°; p = 0.017) and ER in 0° flexion was increased (25° ± 18°; range, -10° to 41° versus 38° ± 7°; range, 17°-41°; p = 0.002). Dysplastic hips had a higher prevalence of extraarticular impingement at the anteroinferior iliac spine compared with normal hips (48% [13 of 27 hips] versus 5% [one of 19 hips], p = 0.002). A PAO increased the prevalence of impingement for the femoral head from 30% (eight of 27 hips) preoperatively to 59% (16 of 27 hips) postoperatively (p = 0.027). IR in flexion was decreased in hips with a cam-type deformity compared with those with a spherical femoral head (p values from 0.002 to 0.047 for 95°-120° of flexion). A concomitant IO led to a normalization of ER in extension (eg, 37° ± 7° [range, 21°-41°] of ER in 0° of flexion in hips with concomitant IO compared with 38° ± 7° [range, 17°-41°] in nondysplastic control hips; p = 0.777). CONCLUSIONS Using computer simulation of hip ROM, we could show that the PAO has the potential to restore the typically excessive ROM in dysplastic hips. However, a PAO can increase the prevalence of secondary intraarticular impingement of the aspherical femoral head and extraarticular impingement of the anteroinferior iliac spines in flexion and internal rotation. A cam-type morphology can result in anterior impingement with restriction of IR. Additionally, a valgus hip with high antetorsion can result in posterior impingement with decreased ER in extension, which can be normalized with a varus derotation IO of the femur. However, indication of an additional IO needs to be weighed against its inherent morbidity and possible complications. The results are based on a limited number of hips with a pre- and postoperative CT scan after PAO. Future prospective studies are needed to verify the current results based on computer simulation and to test their clinical importance.