Chapter XXI Quantum Mechan

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Chapter XXI Quantum Mechan

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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

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  1. l GENERAL PHYSICS III ia Tr m Optics F o f.c PD pd & tro ni Quantum Physics o . w itr w w N
  2. Chapter XXI l ia Quantum Mechanics Tr m F o f.c PD pd §1. The wave nature of particles tro The Heisenberg Uncertainty Principle ni §2. o . w The Schrödinger equation itr w §3. w Solutions for some quantum systems N §4.
  3.  It has been known from the previous chapter that light, and in general, electromagnetic waves have particle behavior. l ia  Some latter time than the quantum theory of light, it was discovered that particles show also wavelike behavior. Tr m F The wave-particle duality of matter is the fundamental concept of o f.c PD modern physics pd Newton’s classical physics should be replaced by the new mechanics tro ni which is able to describe the wave nature of particles o . w itr w w “QUANTUM MECHANICS” N
  4. §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, l ia 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 Tr electrons, alpha particles, protons,…, which we usually think of as m particles, may in some situations behave like waves. F o f.c PD pd A particle shoud have a wavelength  related to its momentum in exactly tro the same way as photon ni  h o= p (the de Broglie wavelength) . w itr w w N And the relation between frequency and particle’s energy is also in the same way as photon E  = h
  5. 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: l Apply the de Broglie hypothesis to two cases, the first on the ia macroscopic scale, and the second on the microscopic scale Tr  The case of macroscopic particles: A particle with m = 10 kg, v = 10 m/s  p = 100 kg.m/s m F o f.c   h/p = (6.63x10-34 / 100) m = 6.63x10-36 m . PD = pd tro With this scale of wavelengths a mcroscopic particle can not ni produce any observable effect of interference or diffraction. o . w itr w The case of microscopic particles: w An electron with m = 9.1x10-31 kg, accelerated to v = 4.4x106 m/s N  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.
  6. 1.3 Electron diffraction:  Davisson & Germer experiment (1927): • Elecrons emitted thermally from the cathode C l ia • Then they are accelated by Tr a voltage V  a parallel beam of monoenergetic electrons m F are produced. o f.c PD pd tro • A plate P & a diagraph D plays the role of ni o . w a detector which measures the number of itr w scattered electrons. w N The experimental graph shows the angular distribution of the number of scattered electrons (for V = 54 volts). There is a peak at  50o . =
  7.  Explanation: • The existence of the peak at  50o 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 l ia • With electron beam of such low intensity that the electron go through Tr the apparatus one at a time the interference pattern remains the same  the interference is between waves associated with single electron. m F o • For a quantitative consideration, we calculate the electron wavelength: f.c PD pd by using = h/p, where eV = p2/2m  p = 2meV tro Substuting V = 54 volts, one gets  1.67x10-10 m  = ni o . w by using the formula for the first order diffraction peak itr w w  d sinwhere d was detemined from X-rays diffraction  =   N experiments, d = 2.15x10-10 m. For  50o, one gets  1.65x10-10 m = = The two obtained values of agree within the accuracy with experiment !
  8. 1.4 Application of Matter Waves: Electron Microscopy  The ability to “resolve” tiny objects improves as the wavelength decreases. Consider the microscope objective: Objects d D  l to be  ia resolved diffraction Tr f disks = focal length of lens m F o  f.c • Nominal (conventional) minimumangle for resolution:   . 22 PD 1 pd c D tro • The minimum d for which we can still resolve two objects is ni  times the focal length: d min f  .22 f o . c 1 c w itr w D w A good microscope objective has f/D 2, so with  500 nm ~ N the optical microscope has a resolution of dmin 1 m. We can do much better with matter waves because, as electrons with energies of a few keV have wavelengths less than 1 nm.
  9.  Example: Observation of a virus by an electron microscopy You wish to observe a virus with a diameter of 20 nm, electron gun 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 D l ia virus with a resolution of dmin = 2 nm. The “f-number” for an electron microscope is quite large: f/D 100. Tr (Hint: First find required to achieve dmin with the given m f F o f/D. Then find E of an electron from .) f.c PD pd f d min  .22 1 tro D ni  D   D  object d min  o  2nm    0 .0164 nm  . .22 f  .22 f   1   1 w  itr w w h2 1.505 eV  2 nm E   .6 keV N 5 2 m .0164 nm  2 0 2 To accelerate an electron to an energy of 5.6 keV requires 5.6 kilovolts .
  10. §2. Heisenberg Uncertainty 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 l ia particle’s precise location. Tr  For classical waves one can produce a localized “wave packet” by superposing waves with a range of wave vectors  E.g.: k. m F o f.c  the spread in wave number k: PD  pd   the spread in coordinate (the size x: tro of the wave packet) x ni   o . w • For wave packets:   1 (see the next slides in more details) k. x itr w w Interpretation: N  To make a short wavepacket requires a broad spread in wavelengths. Conversely, a single-wavelength wave would extend forever.
  11. →0 k l ia Wave with definite k () monochromatic plane wave Tr m F o f.c  From the quantum relation between momentum and wavelength p = h/ PD pd and the relation k = 2  p = (h/2  = ħ where ħh/2 (“h- / ).k .k,  tro bar”) we have a relation between the spread in the particle’s locations ni o . x and likely momenta p w itr w ħ(  1)  (ħ k)· ħ   x· ħ   x w k· x p x N This relation is known as the Heisenberg Uncertainty Principle.  To understand the relation   1 we consider the following k· x example
  12.  Numerical example of a wavepacket: • Consider the superposition of 7 sinusoidal waves with the frequencies & amplitudes l as below. The component ia waves have a “spead” in k, Tr denoted by  k. m The result has such form F o f.c PD pd tro ni o . w itr w w N k
  13. • The superposition of an infinite set of waves with the same “spread”  has the form which is called k “wave packet” Note that in the case of an infinite l ia set of component waves which cover a continum of k values, the Tr superposition is a single wave packet. m The Fourier mathematical analytics can F o f.c provide rigorously the foundation PD pd for this result. The wave packet y(x) tro is represented by the following integral: ni o Wave packet . w itr w w N From this integral one can show that  (the “spread” in k) and  (the k x “spread” in x) are related through the equation   1. k. x
  14. The meaning of the 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”. l ia Note that this uncertainty is from wave nature of particle, but not from Tr errors of experimental measurements. m F o f.c PD pd tro 2.2 Uncertainty in macro- and microscopic worlds: ni o . w Example 1: A person of mass 60 kg who is moving along the x-axis with itr w w a velocity of 1.5 m/s. The uncertainty principle gives N This uncertainty is clearly negligible in the macroscopic world.
  15. Example 2: An electron which is moving with a velocity 2.2x106 m/s (a typical value of electron velocity in atom). We have This distance is comparable to the size of the atom, so that this l ia uncertainty must be important for electrons. Tr Conclusion: The Heisenberg uncertainty becomes important only for microscopic objects. In the macroscopic world, this uncertainty is m F negligible. o f.c PD pd 2.3 Uncertainty for energy and time: tro ni The periodity of sinusoidal waves is expressed by the function o . t w cos(ω – kx). By analog to the relation   1 for the pair (k, x), we k. x itr w w can derive the relation  .  1 for the pair (ω t). ω t , N Then from E = h       h( /  ħ     ħ .  ħ, and we have  ( ω t)   ħ E. t The Heisenberg uncertainty for energy and time. The longer lifetime of a state, the smaller is its spread in energy.
  16. §3.The Schrödinger equation: Having established that matter acts qualitatively like a wave, we want to be able to make precise quantitative predictions, under given conditions. Usually the “conditions” are specified by giving a potential energy U(x,y,z) in which the particle is located. l ia E.g., * electron in the coulomb potential of the nucleus Tr * electron in a molecule m * electron in a solid crystal F o f.c * electron in a semiconductor ‘quantum well’ PD pd tro U(x) ni o . w Classically, a particle itr w For simplicity, in the lowest energy w here is a 1- N state would sit right at dimensional the bottom of the well. potential energy In QM this is not function: possible. (Why?) x
  17. 3.1 Wave function:  We will see that we can get good predictions (actually, so far they have never been wrong!!) by assuming that the state of a particle is described by a “wave function” l (x,y,z,t) ia (or “probability amplitude”):  Tr What do we measure? Often it’s: (x,y,z,t) m    F o f.c PD |(x,y,z,t)|2 = the probability density (per unit volume) for detecting pd tro a particle near some place (x,y,z), and at some time t. ni o . w itr w We need a “wave equation” describing how  (x,y,z,t) behaves. It should w  N  be as simple as possible  make correct predictions  reduce to the usual classical laws of physics when applied to “classical” objects (e.g., baseballs)
  18. 3.2 The Schrödinger Equation (time-dependent):  In 1926, Erwin Schrödinger proposed an equation that described the time- and space-dependence of the wave function for slow matter waves (i.e., electrons, protons,... NOT photons) The (1-D) Schrödinger Equation (SEQ) l ia  d 2 ( x, t ) 2 d ( x , t )   ( x )( x, t )   Tr 2 U i 2m dx dt m F o This equation describes the full time- and space dependence of a f.c PD quantum particle in a potential U(x), replacing the classical particle pd tro dynamics law, a=F/m ni o . Important feature: Superposition Principle  w itr w w The time-dependent SEQ is linear in  (a constant times  is also a N  solution), and so the Superposition Principle applies: If  and  are solutions to the time-dependent SEQ, then so is 1 2 any linear combination of  and 2 1 (example:   1 + 0.8 i  ) 0.6 2
  19. 3.3 The (time-independent) Schrödinger Equation: Before we consider the full time-dependence of states, we will look at a special set of states, called “stationary”, which do nothing interesting in time  the probability density ||2 does not change with time.  l • A state with a definite value of E is stationary: ia Since E = h.   is definite  this corresponds to the case of ħ    Tr monochromatic waves. For monochromatic waves the “t”-dependence is cos( or sin( m t), t), F o or, more conveniently, exp(-i = exp(-iEt/ħ t) ). f.c PD pd tro • In 1 dimension we can write (x,t)= (x).exp(-iEt/ħ Substituting this ). ni form in 1-D SEQ (see the previous slide), we can separate the “t” variable o . and obtain w itr w w  d 2 (x) 2 N   (x) x)   x) U ( E ( Note: It’s a time- independent eq.! 2 2m dx This is one-dimensional Schrödinger Equation for stationary states (states with a definte energy, or eigenstates).
  20.  What does the time-independent SEQ represent? It’s actually not so puzzling…it’s just an expression of a familiar result: Kinetic Energy (KE) + Potential Energy (PE) = Total Energy (E)  d 2 ( x ) 2  U ( x ) ( x ) E ( x ) l ia 2 2 m dx Tr KE term PE term Total E term m F o f.c KE  d 2 x ) PD 2 ( Why it represents the kinetic energy of pd  term: 2m dx 2 the particle? tro ni    k C o n s id e r : o c o s(k x ), p . w itr w d 2 w 2  k 2 c o s ( k x ) N dx 2 d 2 2 k 2 p 2  2     2m dx 2m 2m p2 This wave function Ψ describes a free particle: U(x) = 0   E  2m
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