Quantum number angular-momentum/ magnetic

Quantum numbers The four quantum numbers—principal, angular momentum, magnetic, and spin—arise from solutions to the wave equation and govern the electron configuration of atoms. 

The third quantum number is the magnetic quantum number ( /). It describes the orientation of the orbital around the nucleus. The possible values of m1 depend on the value of the angular momentum quantum number, /. The allowed values for m/ are —/ through zero to +/. For example, for /= 2 the possible values of mi would be —2, —1, 0, +1, +2. This is why, for example, if / = 1 (a p orbital), then there are three p orbitals corresponding to m/ values of—1, 0, +1. This is also shown in Figure 10.3. 

The magnetic quantum number (mi) defines the spatial orientation of the orbital with respect to a standard set of coordinate axes. For an orbital whose angular-momentum quantum number is /, the magnetic quantum number mi... 

In this table, Ml is the total magnetic quantum number of the ion. Its maximum is the total orbital angular quantum number L. Ms is the total spin quantum number along the magnetic field direction. Its maximum is the total spin quantum number 5. / = L 5, is the total angular momentum quantum number of the ion and is the sum of the orbital and spin momentum. For the first seven ions (from La + to Eu +), J =L —S, for the last eight ions (from Gd + to Lu +), J = L + S. The spectral term consists of three quantum numbers, L, S, and J and may be expressed as. The value of L is indicated by S, P, D, F, G, H, and I for L = 0,... 

The state of a spinless hydrogen atom is completely specified by the principal quantum number n, the orbital angular momentum quantum number and the magnetic (projection) quantum number m. The Schrodinger equation is... 

The atom comprises a nucleus surrounded by electrons. Each element has a specific number of electrons that are bound to the atomic nucleus in an orbital structure which is unique for each element. In the structure of the atom, each electron is characterised by the principal, angular momentum, magnetic and spin quantum numbers that define its energy level at no time do they simultaneously have the same value for two electrons of the same atom (Pauli s exclusion principle). 

It turns out that there is not one specific solution to the Schrodinger equation but many. This is good news because the electron in a hydrogen atom can indeed have a number of different energies. It turns out that each wave function can be defined by three quantum numbers (there is also a fourth quantum number but this is not needed to define the wave function). We have already met the principal quantum number, n. The other two are called the orbital angular momentum quantum number (sometimes called the azimuthal quantum number), , and the magnetic quantum number, mi. 

These results can be explained by the electron spin, which is an intrinsic angular momentum independent of the orbital behaviour of the electron. The splitting of the incident beam is interpreted on the basis of two values for a spin magnetic quantum number, m. A magnetic quantum number is related to an angular momentum quantum number, a, by taking values varying between —a and +a, in unitary jumps, that is 2a +1 values. Thus, the observation of two values for means that the spin quantum number for the electron is = 1 /2, i.e. = 1 /2. 

In quantum mechanics, three quantum numbers are required to describe the distribution of electrons in hydrogen and other atoms. These numbers are derived from the mathematical solution of the Schrodinger equation for the hydrogen atom. They are called the principal quantum number, the angular momentum quantum number, and the magnetic quantum number. These quantum numbers will be used to describe atomic orbitals and to label electrons that reside in them. A fourth quantum number—the spin quantum number—describes the behavior of a specific electron and completes the description of electrons in atoms. 

Fig. 12.10. Classical and quantum tops in space, (a) The space is isotropic and therefore the classical top ftteserves its angular momentum i.e.. its axis does not move with respect to distant stars and the top rotates about its axis with a cxxistant speed. This behavior is used in the gyroscopes that help to orient a spaceship with respect to distant stars, (b) The same tc in a homogeneous vector field. The space is no longer isotropic, and therefore the total angular momentum is no longer preserved. The projection of the total momentum on the field direction is still preserved. This is achieved by the precession of the top axis about the direction of the field, (c) A quantum top i.e., an elementary particle with spin quantum number / = in the magnetic field. The projection /- of its spin I is quantized /- =mjH with mj = —, + and, therefore, we have two energy eigenstates that correspond to two precession cones, directed up and down.

The orbital angular momentum quantum number 1 can take the values 0,1,2,3,... (also know as azimuthal quantum number) and the magnetic quantum number m must be in —/, — / + 1,..., / (also known as orientational quantum number). The eigenfunctions can be efficiently constructed through the definition of ladder operators, which is standard in nonrelativistic quantum mechanics and therefore omitted here. The general expression for the spherical harmonics reads [70]... 

Give the possible values of (a) the principal quantum number, (b) the angular momentum quantum number, (c) the magnetic quantum number, and (d) the spin quantum number. 

The nuclei of many isotopes possess an angular momentum, called spin, whose magnitude is described by the spin quantum number / (also called the nuclear spin). This quantity, which is characteristic of the nucleus, may have integral or halfvalues thus / = 0, 5, 1, f,. . . The isotopes C and 0 both have / = 0 hence, they have no magnetic properties. H, C, F, and P are important nuclei having / = 5, whereas and N have / = 1. 

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