wavefunction and adjusting the value of until the condition is satisfied. For the lowest E energy state (ground state) the wavefunction is zero only at the extreme values for E1, x and therefore, the number of crossing is zero. The 1st excited state will have only 1 E2 crossing and the excited state will have crossings as shown in Figure 3. nth En+1 n 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 -400 eV and width 0.10 nm with = -0.10 nm, = +0.10 nm. xmin xmax [se_fdm.m] The smaller binding energy, the less accurate are the results because the of values x is not large enough to accurately define the exponential convergence to zero of the wavefunctions as approaches For the case = 4 as shown in Figure 3, = -21 x xmax. n E4 eV when = 0.1 nm and one can see that the exponential tail is not very well defined. xmax When is increased to 0.2 nm, the exponential tail is better defined and the total xmax
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 -400 eV and width 0.10 nm with xmin = -0.10 nm, xmax = +0.10 nm. [se_fdm.m] The smaller binding energy, the less accurate are the results because the of x values is not large enough to accurately define the exponential convergence to zero of the wavefunctions as x approaches xmax. For the case n = 4 as shown in Figure 3, E4 = -21 eV when xmax = 0.1 nm and one can see that the exponential tail is not very well defined. When xmax is increased to 0.2 nm, the exponential tail is better defined and the total
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