The Bisection Method

This is a method used to iterate the trial energy inputted by the user until it reaches the energy eigenvalue(eigenstate).

Initially, a trial energy E is chosen which lies below the ground state energy. U[0] is assigned a value of 0 and u[1] is assigned a non-zero value (in the case of the programs written, it is assigned a value of the step size) u[i max] is assigned the value of test0. Next, the trial energy increased by an amount and the process is repeated. Thus, the function u(i max) has a different value test1.

test1*test0=sign_1

If sign_1 > 0, the trial energy is changed by an amount , and the process is repeated.

If test0 and test1 has the opposite signs, then the wavefunction at has changed sign and the trial energy has passed over the ground state energy. The energy is reversed and halved, , , and the process is repeated to the desired accuracy. To determine the energy of the first excited case, one starts with a trial energy just above the ground state energy. The trial energy is stepped up or down in such manner until the energy converges. The next higher energy is found in a similar manner. A tolerance is set for an absolute value of where is the new iterated value while is the previous value and T is the tolerance set by the user of the program. To get the energy of the first excited state, one starts with a trial energy just above the ground state energy. The trial energy is stepped up in such a manner until the energy converges. The next higher energy is found in a similar manner

Here are some of the constants used in the Numerov algorithm above:-

, being squared of an electron charge and where is the where h is the Planck number and is the electron mass.