
Formalism of Wave Mechanics
 De Broglie Equation
 Matter Waves and Wavelength
 Wave Packet Phase & Group Velocity
 Schrodinger’s Time Independent and Time Dependent Wave Equations
 Derivation of Time Independent Schrodinger Equation in Spherical CoordinatesHydrogen Atom
 Well behaved wave function  Normalization Constant
 Particle in a One Dimensional Box
 Symmetric and Anti symmetric Eigen functions: Degenerate states
 The Pauli’s Exclusion Principle
 Exchange Forces & The Helium Atom
 The Symmetry Character of Various Particles
 Space Quantization and Atomic Dipole Moment
 Zeeman effect1
 Zeeman Effect Series Part2
 Anomalous Zeeman Effect
Particle in a One Dimensional Box
In this video Lecture you will found: The eigenfunction and eigenvalues of a free particle and relation with the momentum and de Broglie wavelength. What is free particle? a particle is free if that is not bound by any internal or external force (it means potential is zero). What is one dimensional box or potential well? This is just picturization of the concept to show any quantum mechanically system like an atom or nucleus, or related solid state or molecular systems. An electron revolve around the nucleus in an atom this can be treated as the particle in a box concept and also a neutron or proton in nucleus are simple example. A free particle is confined in a one dimensional box, the dimension of the box is L along the Xaxis. The potential is infinite at x=0 and x=L ; so there is zero probability to find the particle at these points. You can say that wave function associated with the particle is zero. we use these two extreme points as a boundary condition to find the value of some arbitrary constants. Now, the potential is zero inside the well (or Box) but infinite at x=0 and x=L or beyond that. Also, potential energy is independent of the time, so using the timeindependent Schrodinger equation and putting the value of potential that is zero. This becomes second order differential equation. Assume the solution of this equation with two constant with A and B. Find the values of these constant using the boundary condition. After finding the values of the arbitrary constants define the possible energy states of the particle in one dimensional potential well and normalize the wave function. Because without normalization of the wave function you can not find out the probability of the particle in any system, it is must. This quantum particle is associated with the matter waves so there will be a relationship between the momentum of this particle and the wavelength associated with it as per the de Broglie wave concept. For this, as we know the total energy is the sum of kinetic energy and potential energy, here potential energy is zero so the particle’s total energy will be equal to the kinetic energy. This kinetic energy can be define in terms of the momentum as p^2/2m where p is the momentum of the particle and m is the mass. Compare it with the total energy and find out the momentum of the particle in any state (p=nh/2L), this momentum depends on the dimension of the box and the energy state of the particle. Further, if you want to calculate the de Broglie wavelength of the wave associated with this particle then lembda=eh bar upon p, put the value of p in terms of lambda in the above result and you will find that lembda= 2L/n ; for any state (n=1,2,3….)you can calculate the wavelength of the wave.