Chapter XXI Quantum Mechan

GENERAL PHYSICS III GENERAL PHYSICS III Optics & Quantum Physics Nitro PDF Trial www.nitropdf.com Chapter XXI Chapter XXI Quantum Mechan Quantum Mechan ics ics §1. The wave nature of particles §2. The Heisenberg Uncertainty Principle §3. The Schrödinger equation §4. Solutions for some quantum systems Nitro PDF Trial www.nitropdf.com  It has been known from the previous chapter that light, and in general, electromagnetic waves have particle behavior.  Some latter time than the quantum theory of light, it was discovered that particles show also wavelike behavior. The wave-particle duality of matter is the fundamental concept of modern physics Newton’s classical physics should be replaced by the new mechanics which is able to describe the wave nature of particles “QUANTUM MECHANICS” Nitro PDF Trial www.nitropdf.com §1. The wave nature of particles: 1.1 De Broglie hypothesis: In 1923, de Broglie put a simple, but extremely important idea which initiated the development of the quantum theory. He proposed that, if light is dualistic (behaving in some situations like waves and in others like particles) → this duality should also hold for matter. It means that electrons, alpha particles, protons,…, which we usually think of as particles, may in some situations behave like waves. A particle shoud have a wavelength related to its momentum in exactly the same way as photon h p = (the de Broglie wavelength) And the relation between frequency and particle’s energy is also in the same way as photon = E h Nitro PDF Trial www.nitropdf.com 1.2 Macroscopic and microscopic world: Question: Why do we not observe the wave nature of particles in our experience of the macroscopic world? The answer is given by two following examples: Apply the de Broglie hypothesis to two cases, the first on the macroscopic scale, and the second on the microscopic scale The case of macroscopic particles: A particle with m = 10 kg, v = 10 m/s  p = 100 kg.m/s  = h/p = (6.63x10 -34 / 100) m = 6.63x10 -36 m . With this scale of wavelengths a mcroscopic particle can not produce any observable effect of interference or diffraction. The case of microscopic particles: An electron with m = 9.1x10 -31 kg, accelerated to v = 4.4x10 6 m/s  p = 4.x10 -24 kg.m/s  = h/p = 1.65x10 -10 m. With this scale of wavelengths one can observe interference and diffraction effects of electrons on atoms or molecules of crystal. Nitro PDF Trial www.nitropdf.com 1.3 Electron diffraction:  Davisson & Germer experiment (1927): • Elecrons emitted thermally from the cathode C • Then they are accelated by a voltage V  a parallel beam of monoenergetic electrons are produced. • A plate P & a diagraph D plays the role of a detector which measures the number of scattered electrons. The experimental graph shows the angular distribution of the number of scattered electrons (for V = 54 volts). There is a peak at  = 50 o . Nitro PDF Trial www.nitropdf.com  Explanation: • The existence of the peak at = 50 o proves qualitatively & quantitatively the de Broglie hypothesis ! Such a peak can only explained as a constructive interference of waves scattered by the periodically placed atoms • With electron beam of such low intensity that the electron go through the apparatus one at a time the interference pattern remains the same  the interference is between waves associated with single electron. • For a quantitative consideration, we calculate the electron wavelength: by using = h/p, where eV = p 2 /2m  p = 2meV Substuting V = 54 volts, one gets = 1.67x10 -10 m by using the formula for the first order diffraction peak = d sinwhere d was detemined from X-rays diffraction experiments, d = 2.15x10 -10 m. For = 50 o , one gets = 1.65x10 -10 m The two obtained values of agree within the accuracy with experiment ! Nitro PDF Trial www.nitropdf.com 1.4 Application 1.4 Application of Matter Waves: of Matter Waves: Electron Electron Microscopy Microscopy  The ability to “resolve” tiny objects improves as the wavelength decreases. Consider the microscope objective: A good microscope objective has f/D 2, so with ~ 500 nm the optical microscope has a resolution of d min 1 m. D f fd c  22.1 min  • Nominal (conventional) minimumangle for resolution: • The minimum d for which we can still resolve two objects is  c times the focal length: D c   22.1 Objects to be resolved diffraction disks D   d f = focal length of lens We can do much better with matter waves because, as electrons with energies of a few keV have wavelengths less than 1 nm. Nitro PDF Trial www.nitropdf.com  Example: Observation of a virus by an electron microscopyExample: Observation of a virus by an electron microscopy   eVk. nm. nmeV. m h E 65 01640 5051 2 2 2 2 2     D f .d min  221 nm f D nm f D d 0164.0 22.1 2 22.1 min                     To accelerate an electron to an energy of 5.6 keV requires 5.6 kilovolts . You wish to observe a virus with a diameter of 20 nm, which is much too small to observe with an optical microscope. Calculate the voltage required to produce an electron DeBroglie wavelength suitable for studying this virus with a resolution of d min = 2 nm. The “f-number” for an electron microscope is quite large: f/D 100. (Hint: First find required to achieve d min with the given f/D. Then find E of an electron from .) object f electron gun D Nitro PDF Trial www.nitropdf.com § § 2. Heisenberg 2. Heisenberg Uncertainty Uncertainty Principle: Principle: 2.1 Wave packet and uncertainty: Wave-like properties of particles (electrons, photons, etc.) reflect a fundamental uncertainty in the “knowability” (existence?) of the particle’s precise location.  For classical waves one can produce a localized “wave packet” by superposing waves with a range of wave vectors k. E.g.:  k: the spread in wave number  x: the spread in coordinate (the size  of the wave packet) • For wave packets: k.x 1 (see the next slides in more details)  Interpretation: To make a short wavepacket requires a broad spread in wavelengths. Conversely, a single-wavelength wave would extend forever. x Nitro PDF Trial www.nitropdf.com [...]... What’s a quantum mechanical problem? For a given potential function U(x), you must • substitute U(x) in the Schrödinger equation • identify the boundary conditions for wave function  (x) • find the eigenvalues of energy E (the energy levels) • find the corresponding eigenstates (the specified wave functions) §4 Solutions for some quantum systems: By some examples you will see how can solve a quantum. .. x ≤ and x ≥ 0 a ni tro pd F 0 when 0 < x < a PD U(x) = Tr o A potential well is of the following form: • In classical mechanics, particle can • Inside the well: w N w w itr o move in the 1-D box, and have any energy (a continuum od energy levels)  How do particle behave in quantum mechanics? The time independent SEQ for the region 0 < x < a (U = 0):  where  The general solution of this equation... energy is U(x) = ½ kx2 ia l The wave function for a quantum harmonic oscillator obeys the following m F Tr o SEQ: f.c pd PD By investigating and solving the SEQ it is found that: ni tro The energy levels of a quantum oscillator are w N w w itr o k  1 n  , 2,3, 1 where   E n     n  m  2 1 1 8 mk  2  1  0  2 2 e • The ground quantum state: E0       2 • The next higher state:... Classically, a particle of total energy E (II) (I) in the region x < 0 (the region I) • will remain in (I) as if E < U0 • can move to the region (II) & (III) when E >U0 But the situation is very different in quantum mechanics ! N d 2 ( x ) 2 mE In the regions A & C: k 12 ( x )  0 , k1  dx 2  2 d (x) 2m(U 0  ) E 2 In the region B: k 2  ( x )  0 , k2  2  dx (We are interested in the case E < U 0 ... solving the equations for the coeficients (in the last slide) it is found f.c pd ni tro PD  2 k 2 a  e   w  E  1  U  0 k2  w w itr o E T 16 U0 2 m (U 0 E )  N that m Tr o 2 F A  In quantum mechanics, a particle with E < U0 , which is incident to a potential barrier, has a certain probability T of penetrating through the barrier and appearing on the other side This phenomenon is called... Heisenberg uncertainty principle: “we cannot know both the position and the momentum of a particle simuntaneously with complete certainity” This principle is of fundamental importance in quantum physics It means also that in quantum physics there exists not the concept of a particle’s “path” m f.c pd ni tro PD F Tr o Note that this uncertainty is from wave nature of particle, but not from errors of experimental... problems:  particle in the indefinte potential well: A 2 2 E0  2ma 2 l n  , 2 ,3 , 1 f.c pd n  , 2,3, 1 ni tro PD F k  1 where   E n     n  m  2 m Tr o  quantum simple harmonic oscillator: A ia E n n 2 E 0 , itr o Quantum tunnelling: A particle can penetrate through a potential N w w height U0 of the barrier w barrier even in the case when the particle’s energy E is less than the...ia l →0 k Wave with definite k ( ) m f.c F Tr o monochromatic plane wave pd PD  From the quantum relation between momentum and wavelength p = h/ ni tro w N w w itr o and the relation k = 2  p = (h/2  = ħ where ħh/2 (“h/ ).k k,   bar”) we have a relation between the spread in the particle’s... 1dimensional potential energy function: N U(x) PD F Tr o E.g., * electron in the coulomb potential of the nucleus * electron in a molecule * electron in a solid crystal * electron in a semiconductor quantum well’ x Classically, a particle in the lowest energy state would sit right at the bottom of the well In QM this is not possible (Why?) 3.1 Wave function:  We will see that we can get good predictions... The time-dependent SEQ is linear in  (a constant times  is also a solution), and so the Superposition Principle applies: N  itr o PD This equation describes the full time- and space dependence of a quantum particle in a potential U(x), replacing the classical particle dynamics law, a=F/m If  and  are solutions to the time-dependent SEQ, then so is 1 2 any linear combination of  and 2 1 (example: . GENERAL PHYSICS III Optics & Quantum Physics Nitro PDF Trial www.nitropdf.com Chapter XXI Chapter XXI Quantum Mechan Quantum Mechan ics ics §1. The wave. physics should be replaced by the new mechanics which is able to describe the wave nature of particles QUANTUM MECHANICS” Nitro PDF Trial www.nitropdf.com