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Archive for the 'Quantum Physics' Category

Finding the commutator of the Hamiltonian operator, H and the position operator, x and finding the mean value of the momentum operator, p

Friday, May 24th, 2019

By Kim S. Ponce, MS Physics, MSU-IIT   Problem Given that the Hamiltonian is . Show that and the mean value of the momentum () is zero. Solution Substituting our Hamiltonian operator,     Using the commutator of and , this can be written in the form:     But , so     Therefore, […]

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The Ionization energy, radius and velocity of a Hydrogen atom

Friday, May 24th, 2019

a. Show that The Ionization energy is the energy needed to remove the electron from an atom. The formula for the Ionization energy of a hydrogen atom in ground state is given by,   First we must obtain the values of and . is the reduced mass with the formula: where, (mass of the electron) […]

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Derivation of the Energy of the Bohr model of a hydrogen atom

Friday, May 24th, 2019

 Problem: Derive   Solution We start with the three equations below,       Equation 1 is an expression of the fact that the total energy E of an electron is the sum of its kinetic energy and its potential energy . Equation 2 is the fundamental equation of Newtonian dynamics, is the Coulomb force […]

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Mathematical Tool of Quantum Mechanics

Thursday, May 23rd, 2019

  Problem:   Consider the set of all entities of the form where the entries are real numbers. Addition and scalar multiplication are defined as follows:  i.) Write down the null vector and inverse of . ii.) Show that vectors of the form do not form a vector space.   Solution:  i.) Let be the null […]

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Derivation of the ladder operators for the Orbital Angular Momentum, L

Wednesday, May 22nd, 2019

Vanessa V. Destura, MSU-IIT Let be an eigenvector of and with eigenvalues and respectively. Now, we take the square of the norm of the eigenvalue equations for and For : For :   where Note that We expand the left-hand side of the equations obtained for (equation ) and (equation ) and substitute equations and […]

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