Quantum mechanics symbols

Schwinger, J.S. and C. Clarice Schwinger Quantum Mechanics Symbolism of Atomic Measurements, Springer-Verlag, Inc., New York, NY, 2001. 

The symbols X and denote the quantum mechanical coordinates of the nuclei and electrons, respectively. The index p runs over electronic structures and y over geometries. 

The progression of sections leads the reader from the principles of quantum mechanics and several model problems which illustrate these principles and relate to chemical phenomena, through atomic and molecular orbitals, N-electron configurations, states, and term symbols, vibrational and rotational energy levels, photon-induced transitions among various levels, and eventually to computational techniques for treating chemical bonding and reactivity. 

The symbol ti, which is read h bar, means h/hr, a useful combination that is found widely in quantum mechanics. 

A set of complete orthonormal functions ipfx) of a single variable x may be regarded as the basis vectors of a linear vector space of either finite or infinite dimensions, depending on whether the complete set contains a finite or infinite number of members. The situation is analogous to three-dimensional cartesian space formed by three orthogonal unit vectors. In quantum mechanics we usually (see Section 7.2 for an exception) encounter complete sets with an infinite number of members and, therefore, are usually concerned with linear vector spaces of infinite dimensionality. Such a linear vector space is called a Hilbert space. The functions ffx) used as the basis vectors may constitute a discrete set or a continuous set. While a vector space composed of a discrete set of basis vectors is easier to visualize (even if the space is of infinite dimensionality) than one composed of a continuous set, there is no mathematical reason to exclude continuous basis vectors from the concept of Hilbert space. In Dirac notation, the basis vectors in Hilbert space are called ket vectors or just kets and are represented by the symbol tpi) or sometimes simply by /). These ket vectors determine a ket space. 

Quantum mechanically the combination of two angular momenta is more complicated since the angular momenta are operators. (They are tensor operators of rank one.) The law of combination of angular momenta can be expressed, in general, by the so-called tensor product of two operators, indicated by the symbol x,... 

An atomic unit of length used in quantum mechanical calculations of electronic wavefunctions. It is symbolized by o and is equivalent to the Bohr radius, the radius of the smallest orbit of the least energetic electron in a Bohr hydrogen atom. The bohr is equal to where a is the fine-structure constant, n is the ratio of the circumference of a circle to its diameter, and is the Rydberg constant. The parameter a includes h, as well as the electron s rest mass and elementary charge, and the permittivity of a vacuum. One bohr equals 5.29177249 x 10 meter (or, about 0.529 angstroms). 

Fig. 1.13 The Pettifor structure map for sp-valent AB2 triatomic molecules with N 16 where N is the total number of valence electrons. The full triangles and circles correspond to bent and linear molecules respectively whose shape is well established from experiment or self-consistent quantum mechanical calculations. The open symbols correspond to ambiguous evidence. The data base has been taken from Andreoni et a/. (1985).

Physicist P. A. M. Dirac suggested an inspired notation for the Hilbert space of quantum mechanics [essentially, the Euclidean space of (9.20a, b) for / — oo, which introduces some subtleties not required for the finite-dimensional thermodynamic geometry]. Dirac s notation applies equally well to matrix equations [such as (9.7)-(9.19)] and to differential equations [such as Schrodinger s equation] that relate operators (mathematical objects that change functions or vectors of the space) and wavefunctions in quantum theory. Dirac s notation shows explicitly that the disparate-looking matrix mechanical vs. wave mechanical representations of quantum theory are actually equivalent, by exhibiting them in unified symbols that are free of the extraneous details of a particular mathematical representation. Dirac s notation can also help us to recognize such commonality in alternative mathematical representations of equilibrium thermodynamics. 

Ket notation is sometimes used for functions in quantum mechanics. In this notation, the function / is denoted by the symbol j/) /—1/>. Ket notation is convenient for denoting eigenfunctions by listing their eigenvalues. Thus nlm) denotes the hydrogen-atom stationary-state wave function with quantum numbers , /, and m. 

Following quantum mechanical rules, a nucleus with total spin quantum number / may occupy (21 + 1) different energy levels when placed in a magnetic field. For nuclei with 7 = 4, e.g. 1H, 13C, 15N, 19F, 31P, two spin alignments relative to B0 arise (Fig. 1.3) these are symbolized by + and —... 

Quantum Number (Principal). A quantum number that, in the old Bohr model of the atom, determined the energy of an electron in one of the allowed orbits around the nucleus, In the theory of quantum mechanics, the principal quantum number is used most commonly to describe the atomic shell in which tlie elections are located, In a somewhat general way, it is related to the energy of the electronic states of an atom, The symbol for the principal quantum number is n. In x-ray spectral terminology, a -shell is identical to an n = 1 shell, and an L-shell to an n = 2 shell, etc. 

Often we must compute values of quantities that are not simple functions of the space coordinates, such as the y component of the momentum, py, where Equation (E.4) is not applicable. To get around this, we say that corresponding to every classical variable, there is a quantum mechanical operator. An operator is a symbol that directs us to do some mathematical operation. For example, the momentum operators are... 

The symbol h, which is read h bar, means h/ln, a useful combination that occurs widely in quantum mechanics. From inside the back cover, we see that ti = 1.054 X 10-34 J-s. Equation 6 tells us that if the uncertainty in position is very small (Ax very small), then the uncertainty in linear momentum must be large, and vice versa (Fig. 1.11). The uncertainty principle has negligible practical consequences for macroscopic objects, but it is of profound importance for electrons in atoms and for a scientific understanding of the nature of the world. 

In the following, we indicate the time derivative of a hermitian operator B with the symbol B. In the Heisenberg representation of quantum mechanics, it obeys the Heisenberg equation of motion... 

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