I. Jake Campbell PHYS*410L September 2008 Prof. Tricia Shepherd Lab THREE: Varying Potential Fields Why is the bond strength between two fluorine atoms greater than the bond strength between two chlorine atoms? How does varying the bond distance between two atoms affect the associated energies of that molecule? Questions such as these can be modeled and studied using the Particle on a Line Model, or, Particle in a Box. A box defined by a length L, with sides that stretch up to infinity, particles--as wave functions--and their respective behavior through different potentials, can be studied. By varying the potential inside the box, whether by adding a step, adding a barrier, or by changing the parameters of the box, one can study wave curvature and characteristics as a it approaches, goes over, or goes through different potentials. For this, three different boxes, or well types are used.
Single Well: A single well can be used to model the bond behavior of two atoms. In the well each wave has its own respective energy and frequency. As the size of the well is increased the energy at each quantum level goes down. Respectively, increasing the bond length of two atoms will decrease its energy. This also increases the area where the particle could be found. Similarly, as the size of the well is decreased the energy at each level gets bigger. When the well was decreased to , the energy got 4 times bigger. Likewise, when the well was increased by 4, total energy got 4 times smaller. By this it can be seen that the total energy of a well, or bond, is inversely proportional to the size of the box, or the length of the bond. The number of nodes on a wave corresponded to the quantum number, n, minus one.
Step Well: Placing a step--increasing potential over some portion of the length--gave the waves different characteristics than in a single well. As they encountered the step (waves with less potential than the step) seemed to be squished, curving so that the nods, peaks, and valleys all occur before the step. The frequency of the wave was greater than with no step. Unlike the single well--waves that hit a potential of infinite
energy (end of well) stopped--waves hitting the step went into it, their energy exponentially decaying to their initial energy level. Waves above the step potential seemed to receive an increase in energy (amplitude got bigger). The first wave passing over the step exhibited the same curvature behavior as that of a quantum state n=1. The second behaved similarly to that of n=2, and so on. Having increased amplitude over the new potential--the step--gives rise to whether or not the probability of finding that particle is higher over the step potential. For those waves of energy much greater--n=20 or 30--than the step potential, their behavior, or curvature, appeared less effected. The number of nodes per wave at each quantum level remained the same; however, these nodes were squished closer together as they approached the step. The nodes of waves above the step appeared to be stretched father apart. Changing the step potential height, width, and location in the well also changed the behavior of the particle waves. The relation of each energy state and the potential height can be seen in the graph below.
F i g . 1
Increasing the step height, particles exhibit similar behaviors as before, those behaviors reaching further up into higher energy states. Changing the location the of step, so as to reach further or shorter into the box, waves exhibited their abnormal behavior sooner or later, respectively. As the step size approaches zero--dimensions of well approaching that of the single well--the energy states also approach those of a single well. By numerical analysis of the different energies from varying step parameters, it was found that as the step increases in width or height, making the overall size of well smaller, the energy of each wave was increased; the energy of a step well is inversely proportional to the size of the step.
Double Well: In a box of two symmetrically shaped wells-- divided by a barrier of some width and height--the particle waves displayed similar behaviors as in a single and step well. Before reaching the barrier the waves exhibit normal behavior. Upon reaching it the waves act like those of a step well-- tunneling into the barrier. Unlike the tunneling waves of a step potential--decaying to their initial energy level--tunneling waves of a double well began to decay to their original energy level while traveling through the first half of the barrier. Through the second half they appeared to tunnel back to where they left off. Wave curvature, for waves corresponding to an even quantum number, seemed little effected by the barrier. If a wave entered while it was increasing its amplitude, it remained at that same rate of increase until exiting the barrier. Waves of odd quantum numbers exhibited a constant value behavior; they tunneled to some value before tunneling out. It seems that, while maintaining two symmetrically equal wells, the curvature of particle waves with odd quantum numbers level out as they pass through the barrier, and those corresponding to even quantum numbers have the evenrate trait as they pass through the barrier. This strange behavior was no longer observed when the barrier was shifted right or left leaving two asymmetrical wells. Shifting the barrier right or left by a small value decreased the energy of each wave, until a shift equal to half a well of a p...