The Harmonic Oscillator, a Review of Classical and Elementary Quantum Mechanics

The classical harmonic oscillator and an elementary discussion of the quantum mechanical solutions for the harmonic oscillator are discussed.

From ab initio quantum mechanics to molecular neurobiology: A cation–π binding site in the nicotinic receptor

The nicotinic acetylcholine receptor is the prototype ligand-gated ion channel. A number of aromatic amino acids have been identified as contributing to the agonist binding site, suggesting that cation–π interactions may be involved in binding the quaternary ammonium group of the agonist, acetylcholine. Here we show a compelling correlation between: (i) ab initio quantum mechanical predictions of cation–π binding abilities and (ii) EC50 values for acetylcholine at the receptor for a series of tryptophan derivatives that were incorporated into the receptor by using the in vivo nonsense-suppression method for unnatural amino acid incorporation. Such a correlation is seen at one, and only one, of the aromatic residues—tryptophan-149 of the α subunit. This finding indicates that, on binding, the cationic, quaternary ammonium group of acetylcholine makes van der Waals contact with the indole side chain of α tryptophan-149, providing the most precise structural information to date on this receptor. Consistent with this model, a tethered quaternary ammonium group emanating from position α149 produces a constitutively active receptor.

The importance of reactant positioning in enzyme catalysis: A hybrid quantum mechanics/molecular mechanics study of a haloalkane dehalogenase

Hybrid quantum mechanics/molecular mechanics calculations using Austin Model 1 system-specific parameters were performed to study the SN2 displacement reaction of chloride from 1,2-dichloroethane (DCE) by nucleophilic attack of the carboxylate of acetate in the gas phase and by Asp-124 in the active site of haloalkane dehalogenase from Xanthobacter autotrophicus GJ10. The activation barrier for nucleophilic attack of acetate on DCE depends greatly on the reactants having a geometry resembling that in the enzyme or an optimized gas-phase structure. It was found in the gas-phase calculations that the activation barrier is 9 kcal/mol lower when dihedral constraints are used to restrict the carboxylate nucleophile geometry to that in the enzyme relative to the geometries for the reactants without dihedral constraints. The calculated quantum mechanics/molecular mechanics activation barriers for the enzymatic reaction are 16.2 and 19.4 kcal/mol when the geometry of the reactants is in a near attack conformer from molecular dynamics and in a conformer similar to the crystal structure (DCE is gauche), respectively. This haloalkane dehalogenase lowers the activation barrier for dehalogenation of DCE by 2–4 kcal/mol relative to the single point energies of the enzyme's quantum mechanics atoms in the gas phase. SN2 displacements of this sort in water are infinitely slower than in the gas phase. The modest lowering of the activation barrier by the enzyme relative to the reaction in the gas phase is consistent with mutation experiments.

Introduction to quantum mechanics : a course at Los Alamos University /

Mode of access: Internet.

Linear operators and their spectra with applications to quantum mechanics.

From lectures given at the New York university Institute for mathematica and mechanics, by R. Cournat and others.

Complex numbers and symmetries in quantum mechanics, and a nonlinear superposition principle for Wigner functions

Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the two formulations also requires that complex superpositions of state vectors can be described in the phase space formulation, and it is shown that this leads to a nonlinear superposition principle for orthogonal, pure-state Wigner functions. It is concluded that the use of complex numbers in quantum mechanics can be regarded as a computational device to simplify calculations, as in all other applications of mathematics to physical phenomena.

Quantum Mechanics of Guiding Center Motion in Strong Magnetic Field (Coherent State Path Integral Approach)

A new approach is proposed for the quantum mechanics of guiding center motion in strong magnetic field. This is achieved by use of the coherent state path integral for the coupled systems of the cyclotron and the guiding center motion. We are specifically concerned with the effective action for the guiding center degree, which can be used to get the Bohr- Sommerfeld quantization scheme. The quantization rule is similar to the one for the vortex motion as a dynamics of point particles.

Solution of Space-Time Fractional Schrödinger Equation Occurring in Quantum Mechanics

Dedicated to Professor A.M. Mathai on the occasion of his 75-th birthday. Mathematics Subject Classi¯cation 2010: 26A33, 44A10, 33C60, 35J10.

Supersymmetry and Ghosts in Quantum Mechanics

2000 Mathematics Subject Classification: 81Q60, 35Q40.

A disappearing act performed by magnesium: the nucleotide exchange mechanism of Ran GTPase by Quantum mechanics/Molecular mechanics studies

The Ran GTPase protein is a guanine nucleotide-binding protein (GNBP) with an acknowledged profile in cancer onset, progression and metastases. The complex mechanism adopted by GNBPs in exchanging GDP for GTP is an intriguing process and crucial for Ran viability. The successful completion of the process is a fundamental aspect of propagating downstream signalling events. QM/MM molecular dynamics simulations were employed in this study to provide a deeper mechanistic understanding of the initiation of nucleotide exchange in Ran. Results indicate significant disruption of the metal-binding site upon interaction with RCC1 (the Ran guanine nucleotide exchange factor), overall culminating in the prominent shift of the divalent magnesium ion. The observed ion drifting is reasoned to occur as a consequence of the complex formation between Ran and RCC1 and is postulated to be a critical factor in the exchange process adopted by Ran. This is the first report to observe and detail such intricate dynamics for a protein in Ras superfamily.

Sense-making with Inscriptions in Quantum Mechanics

This document provides supporting materials for a paper submitted for review to the Physics Education Research Conference proceedings in July 2016, "Sense-making with Inscriptions in Quantum Mechanics."

Quantum key distribution in the classical authenticated key exchange framework

Key establishment is a crucial primitive for building secure channels in a multi-party setting. Without quantum mechanics, key establishment can only be done under the assumption that some computational problem is hard. Since digital communication can be easily eavesdropped and recorded, it is important to consider the secrecy of information anticipating future algorithmic and computational discoveries which could break the secrecy of past keys, violating the secrecy of the confidential channel. Quantum key distribution (QKD) can be used generate secret keys that are secure against any future algorithmic or computational improvements. QKD protocols still require authentication of classical communication, although existing security proofs of QKD typically assume idealized authentication. It is generally considered folklore that QKD when used with computationally secure authentication is still secure against an unbounded adversary, provided the adversary did not break the authentication during the run of the protocol. We describe a security model for quantum key distribution extending classical authenticated key exchange (AKE) security models. Using our model, we characterize the long-term security of the BB84 QKD protocol with computationally secure authentication against an eventually unbounded adversary. By basing our model on traditional AKE models, we can more readily compare the relative merits of various forms of QKD and existing classical AKE protocols. This comparison illustrates in which types of adversarial environments different quantum and classical key agreement protocols can be secure.

Explanation of relevance judgement discrepancy with quantum interference

A key concept in many Information Retrieval (IR) tasks, e.g. document indexing, query language modelling, aspect and diversity retrieval, is the relevance measurement of topics, i.e. to what extent an information object (e.g. a document or a query) is about the topics. This paper investigates the interference of relevance measurement of a topic caused by another topic. For example, consider that two user groups are required to judge whether a topic q is relevant to a document d, and q is presented together with another topic (referred to as a companion topic). If different companion topics are used for different groups, interestingly different relevance probabilities of q given d can be reached. In this paper, we present empirical results showing that the relevance of a topic to a document is greatly affected by the companion topic’s relevance to the same document, and the extent of the impact differs with respect to different companion topics. We further analyse the phenomenon from classical and quantum-like interference perspectives, and connect the phenomenon to nonreality and contextuality in quantum mechanics. We demonstrate that quantum like model fits in the empirical data, could be potentially used for predicting the relevance when interference exists.