Interference and Wave-Particle Duality in Quantum Mechanics, Lecture notes of Quantum Mechanics

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INTERFERENCE  &  DIFFRACTION  OF  WAVES  +  3  BASIC  TENETS  OF  QUANTUM  MECHANICS   •  What  does  interference  mean?  (Sec3ons  17.1  &  17.5)   -­‐  Construc3ve  and  Destruc3ve  interference   -­‐  Both  Transverse  and  Longitudinal  Waves  interfere       •  Standing  Waves  &  the  1st  basic  tenet  of  QM:  Energy  Quan3za3on   -­‐  Standing  Waves  created  by  2  traveling  waves  in  opposite  direc3on;  Nodes  and  An3nodes      (Sec3ons  17.2,  17.3,  Figs.  17.5,  17.6,  17.9)   -­‐  The  energy-­‐levels  of  an  atom  are  quan3zed  (Bohr  model)  because  the  electron  is  a  wave!  (classnotes)       •  Beats  &  the  2nd  basic  tenet  of  QM:  Wave-­‐Par3cle  Duality   -­‐    Waves  can  behave  like  Par3cles   o  Consider  Beats  (Demo  with  tuning  forks)  (Sec3on  17.8).   o  Many  sinusoidal  waves  added  together  yield  a  wavepacket.   o  Time-­‐dura3on  of  wavepacket  and  frequency  bandwidth  of  the  source  are  related.   o  Spa3al  extent  of  wavepacket  –  deBroglie  wavelength  (Sec3on  38.4  and  39.5)   o  Can  the  deBroglie  wavelength  of  an  electron,  or  of  an  atom,  equal  the  op3cal  wavelength?   -­‐    Par3cles  can  behave  like  Waves   o  Consider  the  two-­‐slit  interference  experiment  (Remember  the  “Fabric  of  Cosmos”  video?)   o  Electrons  interfere!  Demo:  Davisson-­‐Germer  expt   o  Atoms  interfere!  New  state  of  Maber  –  Bose-­‐Einstein  condensate  (1997,  2001  Physics  Nobels)     •  Diffrac3on  and  the  3rd  basic  tenet  of  QM:  Heisenberg  Uncertainty  Principle  (Sec.  39.6)   -­‐  Waves  diffract,  i.e.,  bend  around  obstacles   Defining  signature  of  waves!   INTERFERENCE  &  DIFFRACTION  OF  WAVES  +  3  BASIC  TENETS  OF  QUANTUM  MECHANICS   •  What  does  interference  mean?  (Sec3ons  17.1  &  17.5)   -­‐  Construc3ve  and  Destruc3ve  interference   -­‐  Both  Transverse  and  Longitudinal  Waves  interfere       •  Standing  Waves  &  the  1st  basic  tenet  of  QM:  Energy  Quan3za3on   -­‐  Standing  Waves  created  by  2  traveling  waves  in  opposite  direc3on;  Nodes  and  An3nodes      (Sec3ons  17.2,  17.3,  Figs.  17.5,  17.6,  17.9)   -­‐  The  energy-­‐levels  of  an  atom  are  quan3zed  (Bohr  model)  because  the  electron  is  a  wave!  (classnotes)       •  Beats  &  the  2nd  basic  tenet  of  QM:  Wave-­‐Par3cle  Duality   -­‐    Waves  can  behave  like  Par3cles   o  Consider  Beats  (Demo  with  tuning  forks)  (Sec3on  17.8).   o  Many  sinusoidal  waves  added  together  yield  a  wavepacket.   o  Time-­‐dura3on  of  wavepacket  and  frequency  bandwidth  of  the  source  are  related.   o  Spa3al  extent  of  wavepacket  –  deBroglie  wavelength  (Sec3on  38.4  and  39.5)   o  Can  the  deBroglie  wavelength  of  an  electron,  or  of  an  atom,  equal  the  op3cal  wavelength?   -­‐    Par3cles  can  behave  like  Waves   o  Consider  the  two-­‐slit  interference  experiment  (Remember  the  “Fabric  of  Cosmos”  video?)   o  Electrons  interfere!  Demo:  Davisson-­‐Germer  expt   o  Atoms  interfere!  New  state  of  Maber  –  Bose-­‐Einstein  condensate  (1997,  2001  Physics  Nobels)     •  Diffrac3on  and  the  3rd  basic  tenet  of  QM:  Heisenberg  Uncertainty  Principle  (Sec.  39.6)   -­‐  Waves  diffract,  i.e.,  bend  around  obstacles   Defining  signature  of  waves!   Consider  two  waves  with  the  same  amplitude,    frequency,  and  wavelength    that  both  travel  in  the  +x  direc3on.   x   Observer   Wave  1   Wave  2   The  cat  observes  the  combined  wave,  and  the  way  the  waves  combine  at     his  loca3on  is  totally  determined  by  the  phase  difference.   Perfect  ConstrucDve  Interference  for:   Perfect  DestrucDve  Interference  for:   At  the  loca3on  of  the  cat:   Sec.  17.5:  1D  Wave  Interference  in  Space  w/  Sinusoidal  waves   CHAPTER17_LECTURE17.1   5   Now,  from  the  expressions  for  the  two  waves  detected  by  the  cat:   So,  the  two  waves  can  have  a  phase  difference   for  two  reasons:  a  path  difference;  and   an  iniDal  (inherent)  phase  difference.   So,  we  have  the  two  Interference  CondiDons:   Maximum  ConstrucDve  Interference:   Perfect  DestrucDve  Interference:   Note:  for  two  sources  iniDally  in  phase,  these  condi3ons  say  that  there’s  construcDve   interference  when  the  path  difference  is  an  integral  number  of  wavelengths,  and   destrucDve  interference  when  the  path  difference  is  a  half  integral  number  of  wavelengths.   Here’s  a  prac5cal  example  (noise  reducing  headphones)  using  this  principle.   Sec.  17.5:  1D  Wave  Interference  in  Space  w/  Sinusoidal  waves   CHAPTER17_LECTURE17.1   6   Whiteboard Problem 1 Two loudspeakers in a 20°C room emit 686 Hz sound waves along the x-axis. a. If the speakers are in phase, what is the smallest distance between the speakers for which the interference of the sound waves is perfectly destructive? b. Ifthe speakers are out of phase, what is the smallest distance between the speakers for which the interference of the sound waves is maximum constructive? Note: “In Phase” = Ado = 0 “Out of Phase” = Ado = 7 CHAPTER17_LECTURE17.1 7 Whiteboard Problem 2 Two loudspeakers emit sound waves along the x-axis. A lis- tener in front of both speakers hears a maximum sound intensity when speaker 2 is at the origin and speaker 1 is at x = 0.50 m. If speaker 1 is slowly moved forward, the sound intensity decreases and then increases, reaching another maximum when speaker Lis at x = 0.90 m. a. Whatis the frequency of the sound? Assume Vouna = 340 m/s. b. What is the phase e Aitference between the speakers? = ' nhs “Rn «CL when OX20.S And When OX20.9 WYN Q)AzOUmoafe ¥ a fsour AY b) 4% = 2rax +Ba O.Sm 04m par = ABs i ,5P 2@o0Sm > BEA; som OA... = UT (os) + OH, M7AM,... = 2 (0-9) + 9, >A G2-25T7 SAraE! —_—o > 4h = ae sz -9-STT Good = ow CHAPTER17_LECTURE17.1 2 o--* 10 Sec.  17.7  Interference  in  Space  in  2D  and  3D   Observer   Source  1   Source  2   As  we  did  for  1D,  consider  two  waves  of  the  same  amplitude  and  frequency  in  2D:   wavefronts   Again,  the  cat  observes  the   combined  wave  and,  the  way   the  waves  combine  depends   on  their  phase.  The  phase   difference  depends  on  the   path  difference  and  the  ini3al   phase.     ConstrucDve  Interference:    DestrucDve  Interference:   Note:  these  are  the  same  condi3ons  that  we  had  for  1D,  but  x  is  replaced  with  r.                        The  big  difference  here  is  in  the  geometry.   CHAPTER17_LECTURE17.1   11   Sec.  17.7  Whiteboard  Problem  3   wavefronts   r1   r2   r C  /  D   P Q R Δ CHAPTER17_LECTURE17.1   12