Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Chapter 41. One-Dimensional Quantum Mechanics Quantum effects are important in nanostructures such as this tiny sign built by scientists at IBM’s research laboratory by moving xenon atoms around on a metal surface. Chapter Goal: To understand and apply the essential ideas of quantum mechanics.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Consider a particle with mass m and mechanical energy E in an environment characterized by a potential energy function U(x). The Schrödinger equation for the particle’s wave function is Conditions the wave function must obey are 1. ψ(x) is a continuous normalizable function. 2. ψ(x) = 0 if U(x) is infinite. 3. ψ(x) → 0 as x → +∞ and x → −∞.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Consider a complex wave with wave vector k associated with momentum by de Broglie and energy associated with frequency by Einstein’s expression E=hf for photons Here we identify operators: If E=p2/2m it is a solution to a complex wave free particle equationCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. To describe a wave interacting with a source of potential, add a potential energy term to the “energy” We arrive at the time-independent Schrodinger equation Look for a solution harmonic in time of the form We must find E and the spatial wave function simultaneously.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The complex wave function can be written in terms of real and imaginary parts linked through two equations Like E and B, the two parts go hand in hand.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Consider a particle of mass m confined in a rigid, one-dimensional box. The boundaries of the box are at x = 0 and x = L. 1. The particle can move freely between 0 and L at constant speed and thus with constant kinetic energy. 2. No matter how much kinetic energy the particle has, its turning points are at x = 0 and x = L. 3. The regions x < 0 and x > L are forbidden. The particle cannot leave the box. A potential-energy function that describes the particle in this situation isCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The solutions to the Schrödinger equation for a particle in a rigid box are standing waves What “waves” is an oscillation between real and imaginary partsCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The normalization condition, which we found in Chapter 40, is This condition determines the constants A: The normalized wave function for the particle in quantum state n isCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Quantum dot: particle in 3D box Energy level spacing increases as particle size decreases. i.e CdSe quantum dots dispersed in hexane (Bawendi group, MIT) Color from photon absorption Determined by energy-level spacing Decreasing particle size En +1− En=n + 1( )2h28mL2−n2h28mL2Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. • Niels Bohr put forward the idea that the average behavior of a quantum system should begin to look like the classical solution in the limit that the quantum number becomes very large—that is, as n → ∞. • Because the radius of the Bohr hydrogen atom is r = n2aB, the atom becomes a macroscopic object as n becomes very large. • Bohr’s idea, that the quantum world should blend smoothly into the classical world for high quantum numbers, is today known as the correspondence principle.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. As n gets even bigger and the number of oscillations increases, the probability of finding the particle in an interval δx will be the same for both the quantum and the classical particles as long as δx is large enough to include several oscillations of the wave function. This is in agreement with Bohr’s correspondence principle.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The quantum-mechanical solution for a particle in a finite potential well has some important properties: • The particle’s energy is quantized. • There are only a finite number of bound states. There are no stationary states with E > U0 because such a particle would not remain in the well. • The wave functions are qualitatively similar to those of a particle in a rigid box, but the energies are somewhat lower. • The wave functions extend into the classically forbidden regions.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The wave function in the classically forbidden region of a finite potential well is The wave function oscillates until it reaches the classical turning point at x = L, then it decays exponentially within the classically forbidden region. A similar analysis can be done for x ≤ 0. We can define a parameter η defined as the distance into the classically forbidden region at which the wave function has decreased to e–1 or 0.37 times its value at the edge:Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The wave functions of the first three states are Where ω = (k/m)–½ is the classical angular frequency, and n is the quantum numberCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The solutions are similar to waves in a
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