Operators.

Operators:

Operator is a rule or set of rules which when applied to a function within a given domain produces another function within a given range.

For example, when differential operator is applied to a function f(x) it changes into another function g(x) within a given range.

In quantum mechanics, since the momentum and energy cannot be measured accurately due to uncertainty principle. The operator are important to find the average or expectation values of energy, momentum etc. The various operators are given below.

1) Energy operator.

2) Momentum operator.

3) kinetic energy operator.

4) Hamiltonian operator.

5) Angular momentum operator.

Different operators.

Comments

LS coupling and jj coupling.

Total angular momentum: The total angular momentum of an electron is the sum of the orbital angular momentum and spin angular momentum of the electron i.e Coupling Scheme Since an atom consist of large number of electrons having different orbital and spin momenta, Coupling scheme is necessary to obtain the resultant orbit and spin momenta of atom as a whole. There are two types of coupling scheme namely 1) LS Coupling 2) JJ Coupling. 1)LS Coupling: In this coupling the 'l' vectors of all electrons combine to form resultant 'L' vector and all the 's' vectors of these electrons combine to form resultant 'S' vector. Then the 'L' vector and 'S' vector undergoes vector addition to give resultant 'J' vector which represents the total angular momentum of an atom. Symbolically LS coupling is represented as This type of coupling is governed by the following principles: 1) All the three vectors (L,S and J vectors) are quantized. 2)L is an ...

Mass defect, packing fraction and binding energy.

Mass defect, packing fraction and binding energy: It was assumed that mass of the nucleus is equal to the mass of its constituents (i.e protons and neutrons). But experimentally it was found that the actual mass of the nucleus is less than the theoretical mass. Thus, the difference between the theoretical mass and experimental mass is called mass defect i.e ∆m={[Zmₚ + (A-Z)mₙ] - M} Where mₚ= mass of proton mₙ= mass of neutron M= actual mass of nucleus Z= atomic number A= mass number The ratio of mass defect and mass number (A) is called packing fraction (f) f = ∆m/A Thus packing fraction is the mass defect available per nucleon. The packing fraction explains the stability of the nucleus. The packing fraction may be positive, negative or zero. The positive value of packing fract...

Gamma decay.

Gamma decay A nucleus can exist in states whose energies are higher than that of its ground state, as an atom can. An excited nucleus is denoted by an asterisk on its usual symbol. The excited nucleus return to its ground state by emitting photon whose energy corresponds to the energy difference between initial and final states in the transition involved. The photons emitted by the nucleus have energy in the range of several MeV and is traditionally called gamma decay. Unlike, alpha decay and beta decay, the parent nucleus does not undergo any physical change in the process, daughter and parent nuclei are the same. Most of the time, gamma decay occurs after the radioactive nuclei have undergone an alpha or a beta decay. The alpha and beta decays leave the daughter nuclei in an excited state. From the excited state, the daughter nuclei can get back to the ground state by emitting one or more high energy gamma rays. The relationship between decay and energy level is shown in f...

Modern Physics

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