A Cool Party Trick

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A Cool Party Trick
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Fundamentals of Schroedinger Notation

• Schroedinger notation is usually called position
  representation
• However, Schroedinger notation can easily be
  transformed (as we will see) into momentum
  representation
• If agt represents a state, and xgt represents the
  unit vector in the x-direction then

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• In Hamiltons formulation, the equations of
  motion were functions of r and p (or x and px)
• So we see that the momentum representation may be
  useful

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Recall

• ?kp
• Now rather than putting vector symbols
  everywhere, I am going to jump to 1-dimension so
  let ?kx px
• I will let the wavefunction in momentum space be
  represented by f(kx)

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Obviously
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Postulate 11

• The Fourier transform (FT) converts functions
  from position representation to momentum
  representation of the conjugate momenta

• This means that FT(f(x))f(kx), FT(f(y))f(ky),
  etc. and vice-versa

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Recall
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A Mystery Solved

• Lets Review
• Weve seen new formulations of classical
  mechanics, in particular, Hf(x,p)
• Weve seen evidence that energy is quantized and
  learned that we cannot necessarily measure x and
  p simultaneously

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A Mystery Solved

• We learned that massless particles only have
  momentum which is related to the wavelength
• Wavenumber is somehow related to the energy of
  the system
• Therefore

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We make a choice

• Since we dont know if velocity is a particularly
  relevant number (i.e. massless particles) but we
  do know that wavelength (related to p) is very
  relevant and so we must choose to use
  Hamiltonians formulation of mechanics!
• (But what about
  uncertainty? i.e. that x and p cannot be
  measured simultaneously?)

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Fourier Transform again

• Weve just learned that I can switch between
  position representations and momentum
  representations using the Fourier transform

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From Schaums Math Handbook
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So
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An important result
This, then, represents momentum in POSITION
representation!
This is px, the momentum in momentum
representation
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After all these years
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What about px2?
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What about the Hamiltonian?
The Hamiltonian in position representation (and
the left hand side of the Schroedinger Equation!)
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And Now for Something Completely Different

• (Not a Scotsman on a horse!)

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Maxwells Equations
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Now lets assume just a wave propagating through
space rs0
A wave equation i.e. any function that fits
this differential equation is a wave traveling
through space
Speed of propagation i.e c
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Let the wavefunction wave!
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Separating Stuff
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So we have
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We need Bohr and de Broglie

• We know from Bohr that
• mvrh/2p
• And de Broglie says that 2prl or rl/2p
• So mvl/2ph/2p
• l h/mv
• Mechanical energy is conserved under Bohr model
  so
• TV a constant E

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Just a little more here

• Thus, E-VT ½ mv2
• And from this, 2m(E-V)m2v2
• mv (2m(E-V))1/2
• Thus

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So we have
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Finally
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Recall Ehf
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And what about H?
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The Time Dependent Schroedinger Equation
TDSE
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In 3 dimensions
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Postulate 12

• The probability density, r, is defined as
  absolute square of the wavefunction

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In momentum representation
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Probability current density

• r probability in time and space
• J measure of the flow of probability from one
  place to another probability current density

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Postulate 13

• The probability current density, J, is defined as

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Huh?

• In EM, JI/A and has units of C/m2/s or
  (C/m3)(m/s) which means rv
• Where rcharge density
• vaverage velocity of charges

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Before we go on, we have to define the
expectation value of velocity
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A Faux Proof
Take the difference between these two
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Now we need to recognize that
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From EM, the Continuity Equation

• The decrease of charge in a small volume must
  correspond to the flow of charge out through the
  surface (conservation of electric charge)
• Hey, wait a minute!

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Continuity Equation for Probability

• In this case, the decrease of the probability in
  one volume must be equal to the flow of
  probability out from the surface
• Conservation of Probability!

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Another EM Analogy
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Time just keeps on slippin
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Back to TDSE
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The other side

• E is the eigenvalue of H, the energy of the
  system.
• y(r) is said to describe a stationary state
• This function c(t) is just a phase factor for
  Y(r,t) and does not contribute to the physics of
  the system. (unless you need to superimpose two
  different wavefunctions and then it is only the
  relative phase (w-w)t which matters)

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Transitions between 2 states

• In the last section, the unwritten assumption is
  that QM states are permanent that is, a system
  in a given state will always remain in that
  state.
• We know this cannot be true radioactivity, among
  other phenomena, gives lie to this argument

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Problems with our brand new toy

• The Schroedinger Equation has two major problems
• Not relativistic (see Dirac Equation)
• Only describes systems which do not change!
  QFTQuantum Field Theory covers this area

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Postulate 14

• An initial state igt can make a transition to a
  final state fgt by the emission or absorption of
  a quanta.
• The probability of this occurrence is
  proportional to
• ltfViigt2
• where Vi is the interaction potential appropriate
  to the potential.
• This is only an approximation. Further
  approximations may be necessary