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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 If E=p2/2m it is a solution to a complex wave free particle equation Here we identify operators: Copyright © 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” Look for a solution harmonic in time of the form We arrive at the time-independent Schrodinger equation 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, onedimensional 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 is Copyright © 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 parts Copyright © 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 is 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. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Quantum dot: particle in 3D box CdSe quantum dots dispersed in hexane (Bawendi group, MIT) Color from photon absorption Decreasing particle size Energy level spacing increases as particle size decreases. i.e 2 2 n + 1 h n 2h 2 ( ) E n +1 − E n = − 2 8mL 8mL2 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Determined by energylevel spacing • 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 number Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The solutions are similar to waves in a finite potential well, oscillating in the classical region, exponentially damped outside. The harmonic oscillator is an important model in QM just as it is in classical physics. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. QUESTION: Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The level spacing of typical bonds corresponds to emission and absorption of infrared radiation. Heat is associated with quantum oscillations of bond lengths. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Like sound passing through a wall, a matter wave can pass through a classically impenetrable barrier. In general a particle is partly reflected and partly transmitted so in a sense in two places at once. Even a potential well reflects a matter wave. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Once the penetration distance η is calculated using Equation 41.41, the probability that a particle striking the barrier from the left will emerge on the right is found to be Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Tunneling between conductors • Make one well deeper: particle tunnels, then stays in other well. • Well made deeper by applying electric field. • This is the principle of scanning tunneling microscope. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Scanning Tunneling Microscopy Tip, sample are quantum ‘boxes’ Tip Potential difference induces tunneling Tunneling exponentially sensitive to tip-sample spacing Sample • Control position tip and sample at nm scale with peiezoelectric effect – distortion induced by electric field in a crystal • Scan and record current or scan at constant current. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Surface steps on Si Images courtesy M. Lagally, Univ. Wisconsin Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Manipulation of atoms • Take advantage of tip-atom interactions to physically move atoms around on the surface This shows the assembly of a circular ‘corral’ by moving individual Iron atoms on the surface of Copper (111). The (111) orientation supports an electron surface state which can be ‘trapped’ in the corral Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. D. Eigler (IBM)