Discuss the degeneracy of energy of energy states. Solve the Schrodinger’s equation for a free particle in three dimensional boxes and find Eigen values and Eigen function of free particle.
Discuss the degeneracy of energy of energy states. Solve the Schrodinger's equation for a free particle in three dimensional boxes and find Eigen values and Eigen function of free particle.
Write down Schrodinger's equation for a particle trapped in an infinitely deeped cubical potential well of de 'a'. Write expression for its energy Eigen values. What is degeneracy if the well is a rectangular parallelepiped shape with sides a =b #c?
A free particle is confined in a cubical box of side a. Write the Eigen value and Eigen functions for an energy state represented by nX + ny + nz =4. What is the order of degeneracy in this case?
Particle in a three dimensional box
Consider a single particle confined within a rectangular box with edges parallel to X, Y, and Z axes as shown in fig. Let the sides of rectangular box be a, b and c respectively. The particles can move freely within the region 0 i.e. the potential outside the box is infinite. Schrodinger wave equation inside the box is given by which is partial differential equation in three independent variables and may be solved by the method of separation of variables. The solution of Eq. (1) we have the form (x, y, z) = X(x) Y(y) Z (z) = XYZ where X(x) is a function of x alone. Y(y) is a function of y alone and Z (z) is a function of z alone. Putting the value of from equation (2) in equation (1) and dividing by X(x) Y(y) Z (z) we have. The left hand side of equation (3) is a function of x alone, while the right hand side is a function of y and z and is independent of x. Since both sides are each side should separately be equal to a constant equally, I . e... The general solution of equation (6) will be a sine function of arbitrary amplitude frequency and phase, i.e. . Where A B and C are constants which are determined by boundary condition. Also represents the probability of finding the particles at any point within the box. So [X(x)] which is a function of z coordinates only represents the probability of finding the particles at any point along f the X-axis. As the potential is very high at the walls of the box, the probability of finding the particles at the wall will be zero, i.e., where nx, ny,nz denote any set of three positive numbers. When the box is cub, i.e., a = b = c, the energy expression may be written as. Shows the energy levels of few states of a particle enclosed on cubical box. Now when there is only one wave function corresponding to a particular Eigen value, the level is known as non-degenerate but when there are a number of wave functions corresponding to a single Eigen value, the level corresponding to the energy is non-degenerate. Similar level with nx =ny = nz =2, nx =ny =nz =3 and so on will be non degenerate. In case of energy (h2/8 ma*6), nx ,ny and nz may have calues (2, 1, 1); (1,2,1);()1, 1,2)i.e., energy is degenerate level with degeneracy 3. Similarly we can consider the degeneracy of all other levels.