Fig. 2. Physically acceptable solutions are found only when the wavefunction converges to zero at the extreme values of The depth of x. the well is -400 eV and the width 0.1 nm. [se_fdm.m]. The process of finding physically acceptable values for and the corresponding E wavefunctions can be automated by counting the number of zero crossing of the wavefunction and adjusting the value of until the condition is satisfied. For the E lowest energy state (ground state) the wavefunction is zero only at the extreme E1, values for and therefore, the number of crossing is zero. The 1st excited state will x E2 have only 1 crossing and the excited state will have crossings as shown in nth En+1 n Figure 3. In the Finite Difference Method, we start with where is the maximum number of coordinates, = and = N x x(1) xmin x(N) xmax and then as is incremented, the other values of are calculated from the equation x for = 2 to c N-1 When a physically accepted solution is found for the state the wavefunction is nth normalized by numerically integrating the wavefunction using Simpson’s 1/3 rule The m-script se_fdm.m was used to find the total energies and its corresponding wavefunctions for a potential well of depth -400 eV and width 0.1 nm. The range for the was from -0.1 nm to + 0.1 nm. The value of was manually adjusted to x-coordinates E find the physically acceptable solutions as shown in Figure 3. For this potential well, there are four bound states. The total energy for the ground state, = 1 is 1 = -373.84 n E eV. Thus, the binding energy of the electron or its ionization energy (energy need to free the electron from its bound state) is = +373.84 eV. EB Fig. 3. The four states for an electron confined by a potential well of depth -400 eV and width 0.10 nm with = -0.10 nm, = +0.10 xmin xmax nm. [se_fdm.m]
Fig. 2. Physically acceptable solutions are found only when the wavefunction converges to zero at the extreme values of x. The depth of the well is -400 eV and the width 0.1 nm. [se_fdm.m]. The process of finding physically acceptable values for E and the corresponding wavefunctions can be automated by counting the number of zero crossing of the wavefunction and adjusting the value of E until the condition is satisfied. For the lowest energy state (ground state) E1, the wavefunction is zero only at the extreme values for x and therefore, the number of crossing is zero. The 1st excited state E2 will have only 1 crossing and the nth excited state En+1 will have n crossings as shown in Figure 3. In the Finite Difference Method, we start with where N is the maximum number of x coordinates, x(1) = xmin and x(N) = xmax and then as x is incremented, the other values of are calculated from the equation for c = 2 to N-1 When a physically accepted solution is found for the nth state the wavefunction is normalized by numerically integrating the wavefunction using Simpson’s 1/3 rule The m-script se_fdm.m was used to find the total energies and its corresponding wavefunctions for a potential well of depth -400 eV and width 0.1 nm. The range for the x-coordinates was from -0.1 nm to + 0.1 nm. The value of E was manually adjusted to find the physically acceptable solutions as shown in Figure 3. For this potential well, there are four bound states. The total energy for the ground state, n = 1 is E 1 = -373.84 eV. Thus, the binding energy of the electron or its ionization energy (energy need to free the electron from its bound state) is EB = +373.84 eV. Fig. 3. The four states for an electron confined by a potential well of depth -400 eV and width 0.10 nm with xmin = -0.10 nm, xmax = +0.10 nm. [se_fdm.m]
qp_se_fdm.docx