* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Lecture 27: Quantum Mechanics (Continued)
Quantum entanglement wikipedia , lookup
History of quantum field theory wikipedia , lookup
Quantum state wikipedia , lookup
De Broglie–Bohm theory wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Probability amplitude wikipedia , lookup
Quantum teleportation wikipedia , lookup
Wave function wikipedia , lookup
Coherent states wikipedia , lookup
EPR paradox wikipedia , lookup
Schrödinger equation wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Renormalization wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Elementary particle wikipedia , lookup
Atomic theory wikipedia , lookup
Path integral formulation wikipedia , lookup
Hydrogen atom wikipedia , lookup
Franck–Condon principle wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Wheeler's delayed choice experiment wikipedia , lookup
Hidden variable theory wikipedia , lookup
Double-slit experiment wikipedia , lookup
Identical particles wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Electron scattering wikipedia , lookup
Canonical quantization wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Wave–particle duality wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Lecture 27: Quantum Mechanics (Continued) Review o Particle in Box o Poor Man’s Particle in Box o Application to beta Carotene Today o Tunneling o STM o Imaging Wavefunctions o Harmonic Oscillator o Hydrogen Atom o Wavefunctions o Eigen-values (energies) Poor Man’s Particle in Box We can use a very simple approach to calculate the energies of a particle confined in a box using Bohr’s approach. In the region of box i.e., 0<x<a, let us try to describe a particle with only kinetic energy. Using the classical Newtonian approach we can immediately write. 1 2 p2 E mv 2 2m However, as we make the particle smaller and length of the box submicroscopic, we know that the particle can act as a wave with a wavelength given by de Broglie: h h mv p p2 h2 E 2m 2m2 Then the question becomes what wavelengths are acceptable? If we assume that the nodes of the wave must be present at the box boundaries then it immediately implies not all wavelengths can be accepted, i.e. wavelength is quantized just as in the case of a standing wave problem. The acceptable wavelengths depend on the size of the box. Mathematically, we must have n=2a. So E becomes: h2 h2n2 n2h2 E 2 2 2m 2m(2a) 8ma 2 This is the same result obtained from the solution of Schrodinger’s equation! However, note we cannot determine the nature of wavefunction using this approach. Tunneling Particle in box is a highly idealized case, developed to illustrate the basic ideas of quantum mechanics. More realistic cases involve potentials that are not infinite but finite. The walls in the particle in box problem denote an extremely hard or impenetrable barrier. If we reduce the magnitude of potential, i.e. make the walls bit softer, the strict requirement of wavefunction be zero at the wall boundaries is lifted. In other words, the wavefunction can penetrate the potential wall/barrier. In interesting case arises when the spatial potential is not only finite but also it is localized, i.e., the wall has certain thickness. To take a concrete example: We can set up this problem kinetically as shown in figure to the left. In classical kinetics the reactants must overcome the activation barrier to realize chemical reaction. However, in quantum mechanics, the wavefunction of electron/proton can penetrate the barrier. Therefore the reaction may also occur as shown in the figure to right. Thus measured kinetic rate constants can be enhanced. This is commonly observed in reactions involving electrons or protons (i.e., the oxidation/reduction reactions) Scanning Tunneling Microscopy (STM) Since electronic wavefunctions have nodes and anti-nodes, is it possible to actually image the electronic wavefunction? The answer is definitely yes, but it took almost 60 years to experimentally demonstrate it. 1986 Nobel prize went to two scientists at IBM to precisely demonstrate it. The experiment involves measurement of current. More recently this method was exploited to measure wavefunction on a carbon nanotube surface. Note, how clearly we can see the atomic structure of Benzene rings. Recently physicists at Delft have imaged the wavefunctions near HOMO. Remarkable similarity of the wavefunctions to simple particle in box type behavior should be obvious. Electronic wavefunctions on Carbon Nanotubes Harmonic Oscillator Consider simple harmonic oscillator again. Here we consider two masses m1 and m2 attached to an elastic spring. The potential energy of the system depends on the degree of stretching or compression: We can then immediately write down the corresponding Schrodinger’s equation. where is the reduced mass of the system given by: mm 1 2 m1 m2 The corresponding wavefunctions are sketched below The energy levels are given by Harmonic Oscillator Although the above wavefunctions may appear sinusoidal they are bit more complex They have more of a bell-shaped or gaussian appearance. Also note that for quantum number =0 we have a finite energy known as zero point energy. The fundamental frequency of the oscillator,0 is still given by classical value: 0 1 2 k m However as before the energy values are quantized.