02 Introduction to quantum mechanics

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02 Introduction to quantum mechanics

  1. 1. History of Physics• Christian Huygens on wave nature of light (~1678)• Newton’s Principia for mechanics (~1687)• Maxwell’s A Dynamic Theory of the Electromagnetic Fields (~1864)• Electron was discovered in 1897 (J.J. Thomson)• A few strange phenomena had to be resolved, but that would be a matter of time 1Department of Microelectronics & Computer Engineering
  2. 2. Key points of this chapter• Where and why did classical mechanics fail?• Photoelectric effect: photons behave as particles• Duality principle of de Broglie• Davisson-Germer experiment• Uncertainty relation of Heisenberg• Schrodinger’s Wave equation• Pauli’s exclusion principle 2Department of Microelectronics & Computer Engineering
  3. 3. h= Planck’s constant 6,625*10-34 Js Proposed that electromagnetic energy is emitted only in quantized form and is always a multiple of smallest unit E=hѵ, where ѵ is the frequency of emitted radiation. 3Department of Microelectronics & Computer Engineering
  4. 4. In 1900: Experimental setup for the study of photoelectric effect 4 Department of Microelectronics & Computer Engineering
  5. 5. Observations of photoelectric effect Photoelectric effect The maximum kinetic energy of photoelectrons as a function of incident frequency. 5Department of Microelectronics & Computer Engineering
  6. 6. Observations :•When monochromatic light is incident on a clean surface of a materialphotoelectrons are emitted from the surface.•According to classical physics, if light intensity is high enough toovercome work function, photoelectron should be emitted. But this is notalways observed.•Max kinetic energy (Tmax) Frequency of incident lightof electron Ѵ > Ѵ0•Rate of photoelectron intensity of incident lightemission 6 Department of Microelectronics & Computer Engineering
  7. 7. Einstein’s explation• Light behaves as a particles that transfers its energy to an electron• The energy of a light particle (photon) is E=hν• The energy in excess of workfunction (2-5 eV) is converted into kinetic energy of the photoelectrons. where hѵ0 is the work function of the material But if a photon can behave as a particle, is it also possible that a particle can behave as a wave? 7 Department of Microelectronics & Computer Engineering
  8. 8. Someone was thinking about it… Can we express momentum in terms of wavelength? Who & where?de Broglie was a telecomengineer who spent most ofhis time on top of the Eiffeltower thinking about waves ! 8 Department of Microelectronics & Computer Engineering
  9. 9. Louis Victor Pierre Raymond duc de BroglieBorn: 15 Aug 1892 in Dieppe, FranceDied: 19 March 1987 in Paris, France Louis de Broglie (1892 – 1987) 9 Department of Microelectronics & Computer Engineering
  10. 10. Wave-particle duality principleFrom Einstein’s special theory of relativity, momentum ofa photon is given by - p=h/λde Broglie Hypothesized that wavelength of allparticles can also be expressed as – λ=h/pThis is the duality principle of de Broglie 10 Department of Microelectronics & Computer Engineering
  11. 11. Let us look at Braggs Law first: light interfering with a singlecrystalBased on the angle of incidence of light, there is eitherconstructive interference or destructive interference. 11 Department of Microelectronics & Computer Engineering
  12. 12. Instead of light, what if weuse a particle beam such aselectrons? Experimental arrangement of Davisson- Germer experiment (1927) 12Department of Microelectronics & Computer Engineering
  13. 13. Davisson Germer Experiment Result similar to Bragg reflection (interference)Scattered electron flux as a function ofscattering 13 Department of Microelectronics & Computer Engineering
  14. 14. Werner Heisenberg 1901-1976Friend of Bohr, who wasimportant for nuclearprogram of alliesHead of German nuclearwar program.Did he do intentionallymiscalculations? 14 Department of Microelectronics & Computer Engineering
  15. 15. Heisenberg’s uncertainty principle (1927):the Heisenberg uncertainty principle states that locating aparticle in a small region of space makes the momentum of theparticle uncertain; and conversely, that measuring themomentum of a particle precisely makes the position uncertain. ∆ p∆ x ≥ In formula: ∆ E∆ t ≥ So we can only express things in terms of probabilities!!! 15 Department of Microelectronics & Computer Engineering
  16. 16. ProbabilityProbability is the likelihood that anevent will occur. In our presentdiscussion, probability refers to thelikelihood of finding an electron at acertain positionProbability density function (pdf) is a function which gives theprobability distribution for all possible locations.Example : Imagining a one dimensional confinement ofelectron, if [z0,z4] is the set of all possible locations, then f(z)as shown in figure can be the pdf of the electron location. 02/03/08 16 16 Department of Microelectronics & Computer EngineeringDepartment of Microelectronics and Computer Engineering
  17. 17. ProbabilityWhen the set of all possiblelocations is a continuous range as inthis example, the probability offinding the electron at oneparticular location is zero.But the probability of finding the electron in a small range saybetween z1 and z1+dz is finite and is given by f(z1)dzSince the electron is confined between z0 and z4, z4 ∫ z0 f ( z )dz = 1 17 Department of Microelectronics & Computer Engineering
  18. 18. Schrodinger introduced the wave function for any particle, which was later interpreted as the square- root of probability density function of finding the particle at a particular position. 18Department of Microelectronics & Computer Engineering
  19. 19. SCHRODINGER’S WAVE EQUATIONOne-dimensional Schrodinger’s wave equation-Where Ψ(x,t) is the wave function, V(x) is the potentialfunction experienced by the particle, m is the mass of theparticle and ħ=h/2Π 19 Department of Microelectronics & Computer Engineering
  20. 20. Potential functionPotential here refers to the potential of the electron (electronicpotential).Example : electronic potential of an atom on an electron Potential function of an isolated atom 20 Department of Microelectronics & Computer Engineering
  21. 21. How to solve the wave equation?Separation of variables:Substituting this form into the Schrodinger’s equation : = CONSTANT = η Hence we have two differential equations each having one variable only 21 Department of Microelectronics & Computer Engineering
  22. 22. − j η  t   Time dependent portion of Φ (t ) = e   wave functionThis is an oscillation with radian frequency η/ ħ ⇒ ω = η/ ħ ħ ω =η, Comparing with equation E=hѵ, η is the TOTAL energy E of the particle 22 Department of Microelectronics & Computer Engineering
  23. 23. From previous slide, η = E Therefore ∴The time-independent Schrödinger wave equation (TISE) 23 Department of Microelectronics & Computer Engineering
  24. 24. Physical meaning of the wave functionThe time dependent wave function is - Hypothesis of Born (1926): |Ψ (x,t)|2dx is the probability that the particle can be found in the interval [x , x + dx] 24 Department of Microelectronics & Computer Engineering
  25. 25. 25Department of Microelectronics & Computer Engineering
  26. 26. Important potential functionsWe first apply Schrodinger’s equation to some simple situations,the results of which will be used to analyze more complicated caseslater. • (a) (b) (c) (d)a) Electron in free space, b) The step potential function, c) The infinitepotential well and d) The barrier potential function 26 Department of Microelectronics & Computer Engineering
  27. 27. Boundary Conditions 27Department of Microelectronics & Computer Engineering
  28. 28. “Probability Different probability Different probability integral” is at the same position for left to right and for right to left. infinity 28Department of Microelectronics & Computer Engineering
  29. 29. Before solving the Schrodinger’s wave equation it is helpful toconsider general solutions for- 29 Department of Microelectronics & Computer Engineering
  30. 30. Applications of Schrodinger’s wave equation Electron in free space ⇒ Potential energy V = 0 • Referring to previous 2mE c= slide,  30Department of Microelectronics & Computer Engineering
  31. 31. (  )t − j E ϕ (t ) = eTime dependent wave function is then -This is the equation for a travelling wave. 31 Department of Microelectronics & Computer Engineering
  32. 32. General expression for a travelling wave is - 2mEFrom previous solution, k =  Hence a particle with a well defined energy also has a well defined wavelength and momentum 32 Department of Microelectronics & Computer Engineering
  33. 33. The Infinite Potential Well Potential function of the infinite potential well 33Department of Microelectronics & Computer Engineering
  34. 34. TISE ⇒V(x)∞ and E finite, so Ψ(x)=0 in regions I and III In region II, where V(x)=0, TISE reduces to - 34 Department of Microelectronics & Computer Engineering
  35. 35. To satisfy boundary condition that Ψ(x) is continuous 35Department of Microelectronics & Computer Engineering
  36. 36. ½λ = 0 ⇒ A1 = 0 = 0 ⇒ A2 sin Ka = 0 ½λ This results in Ka = n π, or K = n (π/a) with n = 1, 2,……. ½λK=2π /λ = n (π/a)½λ = a/n ⇒ψ (x) = A2 sin{ n π x} ½λ awith n = 1, 2,……. 36 Department of Microelectronics & Computer Engineering
  37. 37. and 37Department of Microelectronics & Computer Engineering
  38. 38. Since particle can only exist between x=0 and x=a, a ∫ 0 Ψ ( x) Ψ * ( x) dx = 1 ψ (x) = A2 sin Kx  real function 38 Department of Microelectronics & Computer Engineering
  39. 39. 2 2 ⇒ A2 = ± ,± j a aAny of these values  same conclusions 2For simplicity we take A2 = + a 39 Department of Microelectronics & Computer Engineering
  40. 40. Particle in an infinite potential wella) Four lowest discrete energy levels, b) correspondingwave functions and c) corresponding probability functions 40 Department of Microelectronics & Computer Engineering
  41. 41. The Step Potential Function Energy of incident particle E < V0 41Department of Microelectronics & Computer Engineering
  42. 42. TISE  In region I, V=0  42Department of Microelectronics & Computer Engineering
  43. 43. η j ( k1 x − t) A given phase moves towards +x as timeIn A1e  progresses  wave travelling in +x direction η − j ( k1 x + t)Similarly B1e  wave travelling in -x direction 43 Department of Microelectronics & Computer Engineering
  44. 44. In region II, V=V0 and we assume E<V0 44Department of Microelectronics & Computer Engineering
  45. 45. 45Department of Microelectronics & Computer Engineering
  46. 46. Ψ 2 ( x) must remain finite even as x → ∞  46Department of Microelectronics & Computer Engineering
  47. 47. Reg I ⇒Reg II ⇒  47 Department of Microelectronics & Computer Engineering
  48. 48. The first derivatives of the wave functions must also be continuous at x=0 48Department of Microelectronics & Computer Engineering
  49. 49. ⇓ ⇓Two equations and three unknowns ! So we cannot solve for all three but we can express two co-efficients in terms of the remaining one. 49 Department of Microelectronics & Computer Engineering
  50. 50. What is the form of ψ and ψ ? 1 2 Potential barrier Wave function at potential barrier 50Department of Microelectronics & Computer Engineering
  51. 51. The probability density function of the reflected wave is - 51 Department of Microelectronics & Computer Engineering
  52. 52. reflected flux Reflection coefficient = R = incident flux * vr .B1.B R= 1 * vi . A1. A 1Where vi and vr are the velocities of incident and reflected waves. Now, and E=1/2mv2 when V=0 Since K1 applies to both incident and reflected waves in x ≤ 0, vr=vi 52 Department of Microelectronics & Computer Engineering
  53. 53. Total reflection at anarbitrary step function V0,if E<V0 53 Department of Microelectronics & Computer Engineering
  54. 54. Transmitted wave co-efficient A2=A1+B1, hence A2 is not equalto 0pdf |Ψ2(x)|2 of finding the particle in region II is not zeroThere is finite probability that incident particle will penetratethe potential barrier and exist in region II.But since reflection coefficient is 1, all particles in region IIwill eventually turn around and move back into region I.Quantum mechanical snooker!!! 54 Department of Microelectronics & Computer Engineering
  55. 55. The potential barrier 55Department of Microelectronics & Computer Engineering
  56. 56. Reg.I:Reg.II:Reg.III: ( since regions I and III are K3=K1) 56Department of Microelectronics & Computer Engineering
  57. 57. ψ3 can have component only in +x directionBoundary conditions: hence B3 = 0; At x = 0: ∂ ∂ ψ1 (0) = ψ2 (0); ∂ x ψ 1 (0) = 2 (0) ψ x ∂ At x = a: ∂ ∂ ψ2(a) = ψ3(a); ψ 2 a = ) 3 a) ψ ( ( ∂x x ∂ 57Department of Microelectronics & Computer Engineering
  58. 58. 58Department of Microelectronics & Computer Engineering
  59. 59. • We can see that T is not zero, a result which cannot beexplained from classical physics•T is exponentially dependent on a. Thinner the barrier,higher is the transmission co-efficient. 59 Department of Microelectronics & Computer Engineering
  60. 60. Extension of wave theory to atoms.• For simplicity potential function of one electron atom orhydrogen atom is considered.•Schrodinger’s wave equation to be solved in threedimensions.•Potential function is spherically symmetric  so usespherical co-ordinates. 60 Department of Microelectronics & Computer Engineering
  61. 61. The spherical co-ordinate system 61 Department of Microelectronics & Computer Engineering
  62. 62. 1D: CARTESIAN COORDINATES 3D: 62 Department of Microelectronics & Computer Engineering
  63. 63. 2 2TISE in spherical ∇ ψ (r , θ , ϕ ) + { E − V (r )}ψ (r , θ , ϕ ) = 0co-ordinates 2m0 1 ∂  2 ∂  1 ∂2 1 ∂  ∂ ∇ 2= 2 r + 2 2 + 2  sin θ  r ∂ r  ∂ r  r sin θ ∂ ϕ 2 r sin θ ∂ θ  ∂θ Expanding the laplacian operator the TISE in sphericalco-ordinates is - 1 ∂  2 ∂Ψ  1 ∂ 2Ψ 1 ∂  ∂ Ψ  2m0 . r + 2 2 . 2 + 2 .  sin θ . + ( E − V (r ) ) Ψ = 0 r2 ∂ r  ∂ r  r sin θ ∂ ϕ r sin θ ∂ θ  ∂ θ  2 63 Department of Microelectronics & Computer Engineering
  64. 64. Separation of variables Ψ ( r , θ , ϕ ) = R (r )Θ (θ )Φ (ϕ )Substituting this form of solution, the TISE is - sin 2 θ ∂  2 ∂ R  1 ∂ 2 Φ sin θ ∂  ∂Θ  2m + . 2+ .  sin θ .  + r sin θ 2 0 ( E − V ) = 0 2 2 . r R ∂r ∂r  Φ ∂ϕ Θ ∂θ  ∂θ  Each term is a function of only one of r, θ and φ. Groupingterms of the same variable and equating to a constant wecan get three equations. 64 Department of Microelectronics & Computer Engineering
  65. 65. The equation for φ can be written as -1 ∂ 2Φ . 2 = − m2 ⇒ Φ = e jmϕΦ ∂ϕ Wave function should be single valued  m=0,±1, ±2, ±3 65Department of Microelectronics & Computer Engineering
  66. 66. Similarly the equations in r and θ can be solved in terms ofconstants n and l. The separation of variables constants n,l and m are known as principal, azimuthal (orbital) andmagnetic quantum numbers and related by –n=1,2,3… l=n-1,n-2,n-3,…0 |m|=l,l-1,…,0Each set of quantum numbers  a quantum state whichan electron can occupy 66 Department of Microelectronics & Computer Engineering
  67. 67. Electron energy can be written as – − m0 e4 En = ( 2 ) 4π ε0 2 2 n 2 where m0 is the electron mass and n is the principal quantum number. • Energy is negative  electron is bound to nucleus • n is integer  Energy can have only discrete values. • electron being bound in a finite region of space  quantized energy 67 Department of Microelectronics & Computer Engineering
  68. 68. For the lowest energy state, n=1, l=m=0 and the wavefunction is given by -The value of r for which the probability offinding the electron is maximum is - Bohr radius 68 Department of Microelectronics & Computer Engineering
  69. 69. Radial probability density function for the one-electron atom for a) n=1,l=m=0 and b) n=2, l=m=0.• Radius of second energy shell > radius of first energy shell• finite (but small) probability that this electron can be atsmaller radius. 69 Department of Microelectronics & Computer Engineering
  70. 70. One more quantum number !Every electron has an intrinsic angular momentum or spin. Thisspin is quantized, designated by spin quantum number s and takevalues +1/2 or -1/2 70 Department of Microelectronics & Computer Engineering
  71. 71. Pauli Exclusion Principle“no two electrons in any given system (atom, moleculeor crystal) can have the same set of quantum numbers”• This principle not only determines the distribution ofelectrons in an atom but also among the energy statesin a crystal as will be seen later. 71 Department of Microelectronics & Computer Engineering
  72. 72. Periodic Table• In 1869 Mendeleev demonstrated that a periodicrelationship existed between the properties of an elementand its atomic weight.•But it was only quantum mechanics which provided asatisfactory explanation for the periodic table of elements.•Each row in the periodic table corresponds to filling up ofone quantum shell of electrons. 72 Department of Microelectronics & Computer Engineering
  73. 73. Initial portion of the periodic tableFrom the concept of quantum numbers and Pauli exclusionprinciple we can explain the periodic table. 73 Department of Microelectronics & Computer Engineering
  74. 74. • Hydrogen  single electron in lowest energy state (n=1,l=m=0). Howeverspin can be +1/2 or -1/2•Helium  two electrons in lowest energy state and this shell is full. Valenceenergy shell (which mainly determines the chemical activity) is full and hencehelium is an inert element.•Lithium  three electrons. 3rd electron must go into second energy shell(n=2)•When n=2, l=0 or 1 and when l=1, m=0,±1. In each case s=±½. So thisshell can accommodate 8 electrons.•Neon has 10 electrons and hence the n=2 shell is also full making neon alsoan inert element.•Thus the period table can be built. At higher atomic numbers, the electronsbegin to interact and periodic table deviates a little from the simpleexplanation. 74 Department of Microelectronics & Computer Engineering
  75. 75. 75Department of Microelectronics & Computer Engineering

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