On Robin boundary conditions and the Morse potential in quantum mechanics

The physical origin is investigated of Robin boundary conditions for wave functions at an infinite reflecting wall. We consider both Schr\"odinger and phase-space quantum mechanics (a.k.a. deformation quantization), for this simple example of a contact interaction. A non-relativistic particle moving freely on the half-line is treated as moving on the full line in the presence of an infinite potential wall, realized as a limit of a Morse potential. We show that the wave functions for the Morse states can become those for a free particle on the half-line with Robin boundary conditions. However, Dirichlet boundary conditions (standard walls) are obtained unless a mass-dependent fine tuning (to a reflection resonance) is imposed. This phenomenon was already observed for piece-wise flat potentials, so it is not removed by smoothing. We argue that it explains why standard quantum walls are standard. Next we consider the Wigner functions (the symbols of both diagonal and off-diagonal density operator elements) of phase-space quantum mechanics. Taking the (fine-tuned) limit, we show that our Wigner functions do reduce to the expected ones on the half-line. This confirms that the Wigner transform should indeed be unmodified for this contact interaction.Comment: 19 page

SOME THROMBOCYTIC AND ERYTHROCYTIC PARAMETERS AFTER BURNING (EXPERIMENTAL INVESTIGATIONS)

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Flow of S-matrix poles for elementary quantum potentials

The poles of the quantum scattering matrix (S-matrix) in the complex momentum plane have been studied extensively. Bound states give rise to S-matrix poles, and other poles correspond to non-normalizable anti-bound, resonance and anti-resonance states. They describe important physics, but their locations can be difficult to find. In pioneering work, Nussenzveig performed the analysis for a square well/wall, and plotted the flow of the poles as the potential depth/height varied. More than fifty years later, however, little has been done in the way of direct generalization of those results. We point out that today we can find such poles easily and efficiently, using numerical techniques and widely available software. We study the poles of the scattering matrix for the simplest piece-wise flat potentials, with one and two adjacent (non-zero) pieces. For the finite well/wall the flow of the poles as a function of the depth/height recovers the results of Nussenzveig. We then analyze the flow for a potential with two independent parts that can be attractive or repulsive, the two-piece potential. These examples provide some insight into the complicated behavior of the resonance, anti-resonance and anti-bound poles.Comment: 23 pages, 15 figure

DIAGNOSTIC AND TREATMENT TACTICS IN MECHANIC JAUNDICE

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ABOUT DIAGNOSTICS AND TREATMENT OF GASTRODUODENAL HEMORRHAGE

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Exact quantization of a PT-symmetric (reversible) Li\'enard-type nonlinear oscillator

We carry out an exact quantization of a PT symmetric (reversible) Li\'{e}nard type one dimensional nonlinear oscillator both semiclassically and quantum mechanically. The associated time independent classical Hamiltonian is of non-standard type and is invariant under a combined coordinate reflection and time reversal transformation. We use von Roos symmetric ordering procedure to write down the appropriate quantum Hamiltonian. While the quantum problem cannot be tackled in coordinate space, we show how the problem can be successfully solved in momentum space by solving the underlying Schr\"{o}dinger equation therein. We obtain explicitly the eigenvalues and eigenfunctions (in momentum space) and deduce the remarkable result that the spectrum agrees exactly with that of the linear harmonic oscillator, which is also confirmed by a semiclassical modified Bohr-Sommerfeld quantization rule, while the eigenfunctions are completely different.Comment: 10 pages, 1 figure, Fast Track Communicatio

Clinico-anatomical presentation of the ankle sprain - the most common diagnosis in trauma centres

Introduction: Ankle sprain injuries are the most common injury sustained during sport activities and in every day live. Three-quarters or more of ankle injuries involve the lateral ligamentous complex, with an equal incidence between males and females. The lateral collateral ligament complex (LCL) consists of the anterior talofibular, calcaneofibular, and posterior talofibular ligaments. The mechanism of injury to the ankle ligaments is inversion and plantar flexion. The ATFL is the weakest of the lateral ankle ligaments. Acute tears are visualized as ligament swelling, discontinuity and hypo echogenicity. Objectives: To present clinic-anatomical aspects of the ankle sprain.Methods: Writing of this report, we studied 7 ankle cadavers and 21 patients with acute ankle instability due to injury to the lateral ligament complex we operated for one year from 09.2015 to 08.2016.Results: The anterior talofibular ligament is the most frequently injured ligament of the ankle and is the most frequently observed injury in the emergency room. This ligament is closely related to the ankle joint capsule. From its origin, it runs anteromedially to the insertion on the talar body immediately anterior to the joint surface occupied by the lateral malleolus. The ligament is virtually horizontal to the ankle in the neutral position but inclines upward in dorsiflexion and downward in plantar flexion.The calcaneofibular ligament originates from the anterior part of the lateral malleolus. In the neutral ankle position, the ligament runs obliquely downwards and backwards to attach to the posterior region of the lateral calcaneal surface. The calcaneofibular ligament becomes horizontal during plantarflexion and vertical in dorsiflexion, remaining tense throughout its entire arc of motion. The posterior talofibular ligament originates from the malleolar fossa, located on the medial surface of the lateral malleolus, coursing almost horizontally to insert in the posterolateral talus.Conclusions: Ankle sprain is the most common injury of the musculo-skeletal system. It is almost 40% of all athletic traumas and more than 12% of all human injuries. Anatomical variations are not of high frequency. The most commonly seen is ATFL consisting of two bands

Peroneus brevis tendon tear after acute ankle sprain

Background: Longitudinal peroneus brevis tendon (PBT) tears are very uncommon. Bassett and Speer hypothesized that the cause of a longitudinal peroneus brevis tear is likely an extrinsic phenomenon, with the tendon injured by a portion of the distal fibula after inversion trauma of the ankle. The typical patient describes the pain localized posterior to the lateral malleolus and palpable swelling behind the lateral malleolus can raise the suspicion of a tear. There is no specific diagnostic and treatment algorithm. Operative treatments include repair of the tendon, resection of the tear, debridement of the tendon, or tenodesis of the peroneus brevis to the peroneus longus.Case Presentation: Case AA 27 year old man practicing amateur football complained of persistent swelling and pain on the posterolateral aspect of his right ankle. He reported for ankle sprain before 6 weeks. Retrofibular tenderness and palpable popping with dorsiflexion eversion stress was found on physical examination, but without peroneal luxation or subluxation. Peroneal compression test was positive. An MRI showed longitudinal tear and thickening of the right peroneus brevis tendon with increased fluid in the peroneal sheath. Conservative treatments such as phisiotherapy and steroid injections were not effective. Case BA 22 year old waiter in night club complained of persistent pain in the lateral aspect of the right foot despite any recent history of trauma. He had a history of ankle sprains before 6 months of his right ankle. The patient underwent physical therapy including manipulations and steroid injections.Outcomes: A longitudinal incision is centered over the course of the peroneal tendons, beginning above the ankle 1 cm posterior and proximal to the tip of the fibula and then extending distally to the base of the fifth metatarsal. We identified the superior peroneal retinaculum and it was noted to be uninjured in both cases. After that we incised the retinaculum and peroneal tendons are carefully examined. We found longitudinal tear of peroneus brevis tendon about 5 cm long in the first case and 4 cm long in the second case, but with more expansion of the synovium. In the first case we did suture with 3-0 polypropylene and in the second, an effusion was noted in the synovial sheath, and after debridment we made side-to-side suture of the tendon with 4-0 polypropylene. Postoperatively we applied a plaster cast for 3 weeks for both patients. After two months the patients have no pain, and they evaluated surgery as very good. They returned to full activity about three months after the operation.Discussion: PBT longitudinal tears are conjunction in almost cases with lateral ankle sprains. Symptoms similar to those demonstrated with peroneal tendon tenosynovitis. The injury may be combined with ruptures of the lateral ankle ligaments. MRI and sonography are the best examinations for visualising the peroneal tendon. In our two cases the pain was localized to the posterolateral aspect of the lateral malleolus and the patients had stable ankles. MRI and sonography showed longitudinal tear of the PBT. Operative treatment was successful and recovery was in normal time. Patients with persistent posterolateral foot or ankle pain with a history of an ankle sprain or injury should be considered susceptible to a peroneus brevis tendon injury. However, it should be used to evaluate other associated disorders, which can then be dealt at the time of the peroneal tendon repair