𝐅𝐫𝐨𝐦 𝐭𝐡𝐞 𝐀𝐫𝐜𝐡𝐢𝐯𝐞𝐬 Erwin Schrödinger’s Wave Equation 1926 De Broglie’s matter waves weighing heavy upon his mind, Schrödinger wanted time to ponder, time to consider all the implications. Schrödinger took flight to a villa in the Swiss Alps in 1925, leaving his wife behind and gathering a former Viennese girlfriend. What would come of this (presumably) quiet period of reflection and thought would forever change the landscape of physics. Indeed, it would change the way we as a species reckons the universe we live in. A sort of microscopic solar system, with electrons orbiting about the nucleus like planets to stars -- the Bohr atomic model was proving to be of limited utility. For hydrogen atoms, the agreement between predicted and observed behavior was sterling. However, for atoms with more than one electron -- even helium with only two electrons -- predicted and observed behaviour radically diverged. Schrödinger desired to develop a model that agreed with the experimental evidence. What came of that illicit vacation to the Swiss Alps was a model that was not derived from any other, a model that can be called an intuitive guess, a leap of imagination, a model that is astonishingly accurate. In these pages, we will, for the sake of both brevity and simplicity, only consider the time-independent Schrödinger wave equation in one dimension. We will not consider the full equation in all of its gruesome splendor. The time-independent Schrödinger Wave Equation, which could validly be called Schrodingers law, is given by the differential equation d²Ψ/dx² = −2m/ħ² [E – U(x)] Ψ(x) where Ψ(x) is the is the wave function, m is mass, ħ is Plancks constant divided by 2π, E is the total energy of the particle, and U(x) is the potential energy function of the particle. As when one ingests something disagreeable and the natural reaction is nausea, so too is the natural reaction to this equation. However, comfort may be taken if we consider that acceleration is the second derivative of the position function and, therefore, could be written a = d²x/dt² As surely as acceleration simplifies to something more palatable, The Schrödinger wave equation must simplify (a little, at least). To find general solutions to this equation, boundary conditions must be established. The principle conditions that it must adhere to are • Ψ(x) → 0 as x → ±∞ • Ψ(x) = 0 if x is in some place it is physically impossible to be • Ψ(x) is a continuous function • Ψ(x) is a normalised function We will, again for brevity and simplicity, consider the case of a particle in a one-dimensional box of ideal rigidity, such that its walls are impenetrable. Let the box have length L. As may be seen in the illustration, the potential (See the Diagram) energy function has two states: U(x) = 0 for 0≤ x ≤ L U(x) = ∞ for x ‹ 0 or x › L Since it is physically impossible for the particle to be outside of the box, it is the first state that is of interest. Indeed, this simplifies the wave equation considerably, with the term U(x) dropping out. Therefore, the wave equation corresponding to the particle in the box is given by d²Ψ/dx² = −2m/ħ² EΨ(x) Before assailing this equation with a display of mathematical acumen, let us ask ourselves what functions second derivative is merely some negative constant -- all of the terms on the right-hand side save Ψ(x) -- multiple of itself? To simplify, let B² = −2mE/ħ² Therefore, the wave equation becomes d²Ψ/dx² = − B² Ψ(x) It becomes clear a trigonometric function like sine or cosine would be a good candidate for Ψ(x). Therefore, our guess for the solutions to the wave equations is Ψ(x) = sin Bx By the first above boundary condition, it is known that Ψ(x = L) = sin BL = 0 Therefore, BL = nπ ⇒ B = nπ/L where n = 1, 2, 3, ... When the smoke clears, we have that Ψ(x) = A sin nπx/L where A is the functions amplitude. To determine the amplitude, recall the fourth boundary condition, Ψ(x) is a normalised function. Mathematically, this means ∫ │Ψ(x)│² = 1 In words, this states that the probability of finding the particle somewhere on the x-axis is one or 100%. Waving hands a bit to omit the gory details, this gives A = √2/L Gasping for breath, we at last have unearthed the solution to the wave equation for the particle of the nth quantum state in the rigid box. Ψn(x) = √2/L sin nπx/L → for 0 ≤ x ≤ L (‘n’ in subscript) Ψn(x) = 0 → x ‹ 0 or x › L The utility of this solution lies primarily in that the probability of finding the particle at some position x is given by the square of Ψ(x). Pn(x) = │Ψ(x)│² = 2/L sin² nπx/L (‘n’ in subscript) The importance of this relationship is best illustrated graphically. Consider a particle in the third quantum state. It can be seen that there are regions where the probability of finding the particle is zero -- so-called nodes. This is not unique to simply the particle in a rigid box model. It is observed in more sophisticated ones such as the model of an electron orbiting a nucleus ... . From Schrödinger wave equation, probability functions for electrons orbits about nuclei can be developed. These are electron shells called probability densities or orbitals, clouds of probability as to where the electron might be or where it is forbidden to be. This is demonstrated in the above illustration, which shows the 2Px (‘x’ in subscript) orbital and a probability wave corresponding to electrons likelihood of being found at some x-position. However, these are not hard shells, but rather soft shells of where an electron is eighty or ninety percent likely to be. Among the thought-provoking implications this brings is the possibility, however unlikely, that some given electron orbiting some given nucleus -- say one in the eye you are using to read this -- is some great distance away: the other side of the Planet, the outer edges of our solar system, some distant spiral arm of the Milky Way Galaxy ... . No longer do electrons emulate the heavens, with electrons playing the planet to the nucleus star. The Bohr model gave way to a new order in the microscopic realm. A new order not of deterministic electron orbits that can by predicted as surely as the hour and minute hands of a clock. A new order of probabilities. A new order were the position of an electron with respect to the nucleus is given by troubling, blurry clouds of the probability, devoid of deterministic certainty, called probability densities. A new order called quantum physics. The hand of Schrödinger and other countless luminaries has reached across time to touch virtually every field of science: physics, chemistry, biology ... . Indeed, it has quietly and subtly crept into popular culture. The word quantum inspires an almost mystical reverence from some. Certainly no high-brow literature is complete without some cryptic reference to it, some attempt to connect the puzzling dichotomy of the subatomic world to our everyday existence. But like Einstein puzzling over light quanta for many years to no avail, we are all left only to gratefully ponder, for the quantum world is inherently counterintuitive. [• In writing this paper, both Dr. David Mills and David Arnold were invaluable. Dr. Mills guided me to appropriate resources concerning the Schrödinger wave equation and patiently answered my questions about quantum physics. And Dave Arnold allowed me the liberty to stray a little bit from the criteria of this assignment for the differential equations class. The result, I would like to think, is well worth it.] References Feynman, Richard P. “Six Easy Peices: The Feynman Lectures on Physics”. New York, New York: Addison-Wesley, 1994. Knight, Randall D. “Physics: A Contemporary Perspective”. Preliminary Edition. New York, New York: Addison-Wesley, 1997. Kotz, John C. and Vininng, William J. “Saunders Interactive Chemistry” CD-ROM. CD-ROM. Harcourt Brace & Co., 1996. Kotz, John C. and Treichel, Paul. “Chemistry and Chemical Reactivity”. Third Edition. Harcourt Brace & Co., 1996. Tipler, Paul. “Physics for Scientists and Engineers Volume II”. Third Edition. New York, New York: Worth Publishing, 1991.
Posted on: Mon, 10 Nov 2014 16:26:09 +0000
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