Schrodinger equation angular momentum values

The three orbital quantum numbers n, /, and m appear naturally when the Schrodinger equation is solved, There is also a spin quantum number, s, the value of which can be visualized as related to the spin of the electron on its own axis. This quantum number does not appear when the wave equation is solved, but certain quantum-mechanical treatments lead to the conclusion that an elementary particle such as an electron can have two spin angular momentum values, which are specified by the spin quantum numbers -j-4 and —4. We can conveniently think of these two quantum... 

The last equation is formally identical to the radial Schrodinger equation with a non-integer value of the angular momentum quantum number. Its spectrum is bounded from below and the discrete eigenvalues are given by... 

Clough et al. [1982] found the coefficients Amn by numerically solving the Schrodinger equation. As V3- 0, all these coefficients except Ann vanish. Each eigenstate may be characterized by its eigenvalue En and the expectation value of angular momentum is defined as... 

The spinor that describes the spherical rotation satisfies Schrodinger s equation and specifies two orientations of the spin, colloquially known as up and down (j) and ( [), distinguished by the allowed values of the magnetic spin quantum number, ms = . The two-way splitting of a beam of silver ions in a Stern-Gerlach experiment is explained by the interaction of spin angular momentum with the magnetic field. 

The theory for a particle having a wavelength is represented by the Schrodinger equation, which, for the particle confined to a small region of space (such as an electron in an atom or molecule) can be solved only for certain energies, ie the energy of such particles is quantized or confined to discrete values. Moreover, some other properties, eg spin or orbital angular momentum, are also quantized. 

Result (3.1.17) or (3.1.18) is the Bohr energy for the hydrogen atom, except that Bohr had written the equation using a quantized orbital angular momentum 1 (it was discovered later by Schrodinger that for the H atom the lowest value for 1 is 0, while the principal quantum n = 1 is the correct one to use). 

The canonical real d-orbital wave functions are linear combinations of the complex eigen functions (solutions) of the hydrogenic Schrodinger equation and the orbital angular momentum operator. Thus, instead of a set of five degenerate orbitals that may be indexed by values of orbital... 

In the case of diatomic molecules, we find that the Schrodinger equation requires that the component of the angular momentum along the molecular axis be quantized. The quantum number. A, describing this component is the basis for the term symbols for diatomic molecules. The quantum number A may have the values, A = 0, 1, 2,. For diatomic molecules, we use a Greek letter code for A. 

The results presented so far, derived from solutions to the simplest form of the Schrodinger equation, do not explain the observed properties of atoms exactly. In order to account for the discrepancy the electron is allocated a fourth quantum number called the spin quantum number, s. The spin quantum number has a value of Like the orbital angular momentum quantum number, the spin of an electron in an atom can adopt one of two different directions, represented by a quantum num-... 

The wavefunctions that are eigenfunctions of the Schrodinger equation are also eigenfunctions of the angular momentum operator. Consider the eigenvalues themselves a product of h, a constant, and the quantum number m. The angular momentum of the particle is quantized. It can have only certain values, and those values are dictated by the quantum number m. 

When the Schrodinger equation is solved, it yields many solutions— many possible wave functions. The wave functions themselves are fairly comphcated mathematical functions, and we do not examine them in detail in this book. Instead, we will introduce graphical representations (or plots) of the orbitals that correspond to the wave functions. Each orbital is specified by three interrelated quantum numbers n, the principal quantum number I, the angular momentum quantum number (sometimes called the azimuthal quantum number) and mi, the magnetic quantum number. These quantum numbers all have integer values, as had been hinted at by both the Rydberg equation and Bohr s model. A fourth quantum number, nis, the spin quantum number, specifies the orientation of the spin of the electron. We examine each of these quantum numbers individually. 

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