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Quantum Mechanics

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					        Quantum Mechanics
• Bohr Theory worked pretty well for one-
  electron atoms (H, He+, Li++, etc.
• Failed totally for He with two electrons
• Close examination of spectra of one
  electron atoms showed that some spectral
  lines were actually multiple lines
• Couldn’t predict intensity of lines
• Couldn’t explain chemical bonding
        Quantum Mechanics
• A new approach was needed and appeared
  in two different forms
• The first was called wave mechanics and
  was developed by Erwin Schrodinger
• The second was called matrix mechanics
  and was developed by Werner Heisenberg
• Eventually shown to be the same
        Quantum Mechanics
• Deals with the microscopic world of atoms
  and light (photons)
• Blends smoothly with classical mechanics
  as we approach the macroscopic world
• This is the correspondence principle
• Mathematics of quantum mechanics involve
  matrices and partial differential equations
• We’ll just have to look at results
           Wave Mechanics
• EM Waves have frequency, wavelength,
  energy
• Relate wave to particle properties by using
  E = hf
• Amplitude of an EM wave is the strength of
  the electric or magnetic field and is related
  to the intensity of the wave
            Wave Mechanics
•   Particles have wave properties as well
•   Wavelength is h/mv
•   We say this is a matter wave
•   What is the amplitude of a matter wave?
•   Schrodinger defined a wave function 
•    is a function of position and time
            Wave Mechanics
•  can be compared to the electric field
• For light the intensity is proportional to the
  square of the electric field strength
• Light can be considered a stream of
  particles
• Then the intensity depends on the number
  of photons
       Wave Mechanics
             So, to say how much light we
             have, we talk about the
             intensity. We can either talk
             about the square of the field
   2
I E  N     strength or the number of
             photons. The two quantities
             are proportional to each other.
       2
N E         We said that  is comparable
             to E, so 2 must be related to
             numbers of particles or
             something similar.
           Wave Mechanics
• The great leap is that we might consider the
  probability of finding photons somewhere
  in space to be proportional to E2
• Now, we make the same leap for particles
• Consider the probability of finding particles
  somewhere in space to be proportional to 2
Wave Mechanics
      Direct photons or particles at a
      pair of slits. E2 gives the
      probability of finding a photon at
      the viewing screen. 2 gives the
      probability of finding a particle
      at the viewing screen.
      Now send photons or particles
      one at a time. What happens?
      Place a piece of film at screen.
Wave Mechanics
      We will see a dot on the film
      when a photon or particle hits the
      screen. If we keep sending
      photons or particles and keep
      watching, eventually the
      interference patterns appear.
      We can’t predict what any single
      particle or photon will do, but we
      can predict what a lot of them
      will do!
Wave Mechanics
      If we cover one slit for a while
      and then the other for a while, no
      interference pattern is seen!
      So, a single photon or particle
      must somehow pass through both
      slits in order to interfere!
      Says our macroscopic view of
      waves and particles cannot be
      extended to the microscopic!
Wave Mechanics
      What we do know is that E2 and
      2 give us the probability of
      finding the photon or particle at a
      point in space and time!
      We can treat a wad of particles as
      a wave, but we treat individual
      particles by probabilities!
        Uncertainty Principle
• We assume that if we measure something
  we will have some small errors
• You know this from your lab work
• With better instruments and techniques you
  can reduce these errors
• Heisenberg showed that there is a limit to
  how small you can make the error!
         Uncertainty Principle
• There are two factors involved
• One is wave-particle duality
• The other is that to measure something you have
  to disturb it
• Place a ruler to measure length
• You must use pressure to line up the end of the
  ruler and the end of the object
• You have to apply pressure to make the alightment
• This changes the length of the object!
        Uncertainty Principle
• Example in the text about finding a ping-
  pong ball in a completely darkened room
• You grope around, moving your hand
• You touch the ball during the movement
• You know where the ball was, but you don’t
  know where it is after the touch
• You can’t predict the exact future position
• You gave the ball some momentum!
Uncertainty Principle
        Recall the diffraction limit of light.
        We can measure position to about a
        wavelength of the light we use. To
        get more accurate position, shorten
        the wavelength which ups the
        frequency. But E=hf, so the photon
        has higher energy. Whacks the
        electron harder and you don’t know
        where it goes or how fast it is
        moving. Lower the frequency and
        you get more uncertainty in position.
Uncertainty Principle

        x  
        p  h / 
        xp  h
        xp  h /2
            Uncertainty Principle
Particles have uncertainty in position of x=. We try to
detect with a photon that has speed c and takes a time t=
x/c= /c to pass through the uncertainty distance. So the time
of measurement is uncertain by t= /c.
Now the photon can transfer some or all of its energy to the
particle. The energy of the photon is E=hf=hc/. So, the
uncertainty in the particle’s energy after the photon hits it is
just this same amount. So the product of the two undertainties
in time and energy is Et= (hc/)(/c)=h. Heisenberg’s exact
calculation gives Et  h/2 
         Uncertainty Principle
• So what does all this mean?
• When dealing with the microscopic world we
  cannot simply take our macroscopic picture of
  particles and waves to a smaller level
• We can’t describe in words what a photon or an
  electron is
• These objects have both wave and particle
  properties and we have to consider both to gain
  understanding of microscopic phenomena
          Uncertainty Principle
• These issues force us to deal with probabilities
  rather than certainties when we discuss energies,
  times, positions, momenta
• Can we violate energy of momentum
  conservation?
• Sure, if we do it for a short enough time interval or
  a short enough position space since we can’t
  measure the failure!!!
• Is this real? YES, as we shall see!
                  Atoms
• So what are atoms?
• Well, Rutherford’s experiments established
  the notion of a small very dense nucleus
• We know atoms contain electrons
• But the classical orbit picture is garbage!
• We only know what we can measure!!!
                     Atoms
• What we can measure is that atoms have definite
  precise energies
• When electrons change energy levels, photons
  with precise energies emerge
• The electrons have highest probability of being in
  some well defined region of space with fuzzy
  edges
• These probability regions are not necessarily
  spherical as the classical orbits might imply
                     Atoms
• Atoms have other precise characteristics besides
  energies
• We will take up these ideas next time when we
  examine the notion of quantum numbers
• For now, we can simply say that just as a guitar
  string can generate musical tones as a series of
  harmonics, so the electrons in atoms can take on a
  precise series of characteristics
• These properties are inherent in nature

				
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posted:10/8/2012
language:English
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