Kinetic Energy Is Important in the Nanoscale World

Transcription

1 Kinetic Energy Is Important in the Nanoscale Worl Frank Riou Department of Chemistry College of St. Beneict & St. John's University St. Joseph, MN Most eplanations of atomic an molecular phenomena foun in tetbooks are epresse in terms of potential-energy-only (PEO) moels. Inclusion of kinetic energy in the analysis is generally consiere to be unnecessary or irrelevant. This view is of questionable valiity, an it is becoming increasing clear that ignoring kinetic energy at the nanoscopic level can lea to facile but incorrect eplanations of atomic an molecular behavior. (-4) The iea that kinetic energy shoul not be ignore is not too surprising since quantum mechanical calculations involve minimization of the total energy, which inclues both kinetic an potential energy contributions. In other wors, kinetic energy plays an important role at the computational level, an therefore shoul not be eclue at the level of analysis an interpretation. For eample, the funamental questions regaring the stability of matter, the nature of the covalent bon, an the interaction of electromagnetic raiation with matter cannot be answere without a consieration of kinetic energy in the quantum mechanical contet. (5) The following simple variational calculation on a particle in a one-imensional bo with a linear internal potential clearly illustrates the importance of kinetic energy. For a particle of unit mass in a one-imensional bo of length one bohr (a o = 52.9 pm =.529 nm) with internal potential energy V = 2 the Schröinger equation in atomic units (h = 2π) is, 2 Ψ ( ) + V ( ) Ψ ( ) = E Ψ ( ) where V ( ) := 2 Three normalize trial wave functions are consiere in this analysis, an are shown below both mathematically an graphically. The potential energy function is superimpose on the graphical representation of the trial wave functions. ( ) := 2 sin π := 5 ( ) 2 := 5 2 ( ) :=,.2.. Comparison of Trial Wavefunctions V ( ).5

2 If aske to choose the best trial wave function by inspection, one woul unoubtely be incline to select Ψ b because it is skewe to the left sie of the bo where the potential energy is lowest. Ψ a woul be net best because it is symmetric, an Ψ c woul be last because it is skewe to the right sie of the bo where the potential energy is highest. However, the quantum mechanical calculations reveal that Ψ a is the best trial function of the three because it gives the lowest total energy, the primary criterion of the variational principle. For each trial wave function the epectation values for kinetic energy (T), potential energy (V), total energy (E = T + V), an position are calculate. Atomic units are use all calculations: E h = 4.36 aj an a o = 52.9 pm) Calculations for trial wave function Ψ a 2 T a := V a := 2 T a = V a =. E a := T a + V a E a = X a := X a =.5 Calculations for trial wave function Ψ b 2 T b := V b := 2 T b = 7. V b =.75 E b := T b + V b E b = 7.75 X b := X b =.375

3 Calculations for trial wave function Ψ c 2 T c := V c := 2 T c = 7. V c =.25 E c := T c + V c E c = 8.25 X c := X c =.625 These calculations are summarize in following table. Property\Wave function Qa Qb Qc Kinetic Energy/E h Potential Energy/E h Total Energy/E h Average Position/a o Ψ a is a symmetric function which favors neither the low potential energy region nor the high potential energy region, but has the lowest total energy because it has a significantly lower kinetic energy than the other trial wave functions. The reason it has a lower kinetic energy is because it has a lower curvature than the other wave functions (curvature is the secon erivative of the function). Ψ b has a somewhat lower potential energy than Ψ a because it favors the left sie of the bo, but consequentially a much higher kinetic energy because of its greater curvature. Total energy, as note above, is what counts in a variational calculation. Ψ c is the worst trial function because it has both high kinetic energy an high potential energy. Of course, Ψ a is not the best possible wave function for this problem; it is only the best of the three consiere here. The best wave function can be foun by a more elaborate variational calculation or by numerical integration of Schröinger's equation. A Mathca (6) program for numerical integration of Schröinger's equation for a particle in a bo with linear internal potential is given in the appeni. This latter metho yiels a wave function with the following physical properties: <T> = E h ; <V> =.983 E h ; <E> = E h ; <X> =.49 a o. Note that this optimum wave function is skewe a little to the left of center, increasing kinetic energy slightly (+.7 E h ) an reucing potential energy slightly more (-.7 E h ), an overall yieling an energy reuction of -. E h. The etails of these calculations can be foun in the appeni.

4 However, it is also important to note that Ψ a, the eigenfunction for the particle in a bo problem [V()=], is a very goo trial wave function for this particlular problem. It is in error by only.7% when compare with the more accurate, an essentially eact, numerical solution. Ψ a is isplaye along with the numerical wave function in the appeni to show how little it iffers from the numerical solution. Another point that shoul be note is that Ψ b oes not become the preferre trial function until V() = 6.6. In other wors it requires a rather steeply rising internal potential energy to offset the kinetic energy avantage that Ψ a has. The energy calculations for both wave functions are given below. V a := 6.6 V a = 8.3 E a := T a + V a E a = V b := 6.6 V b = E b := T b + V b E b = In conclusion, this simple eample reveals that our intuition about the importance of potential energy in the analysis of physical phenomena at the nanoscale level shoul be tempere by a realization that the quantum mechanical nature of kinetic energy cannot be safely ignore. Literature cite:. Tokiwa, H.; Ichikawa, H. Int. J. Quantum Chem. 994, 5, Riou, F.; DeKock, R. L. J. Chem. Euc. 998, 75, Weinhol, F. Nature 2, 4, Riou, F. Chem. Eucator 23, 8, S43-47(3)65-9; DOI.333/s897365a.. 5. In the contet of quantum mechanics, confinement energy is probably a better escriptor than kinetic energy, because the latter implies classical motion. Accoring to quantum mechanical principles, confine particles, because of their wave-like charcter, are escribe by a weighte superposition of the allowe position eignestates. They are not eecuting a trajectory in the classical sense. In other wors, they are not here an later there; they are here an there, simultaneously. 6. Mathca is a prouct of Mathsoft, Cambrige, MA 242;

5 Appeni Numerical Solution for the Particle in a Slante Bo Parameters: ma := m := V := 2 V ( ) := V Solve Schröinger's equation numerically: Given 2 Ψ ( ) 2m 2 + V ( ) Ψ ( ) = E Ψ ( ) Ψ ( ) = Ψ' ( ) =. Ψ Enter energy guess: ( ) := Oesolve, ma Normalize wavefunction: Ψ ( ) := E Ψ ( ) ma Ψ ( ) 2 Ψ ( ).5 Calculate most probable position: Calculate average position: :=.5 Given X avg := Ψ ( ) Ψ ( ) = Ψ( ) Fin( ) =.485 X avg =.49 Calculate potential an kinetic energy: V avg := V X avg V avg =.983 T avg := E V avg T avg = 4.942