Dirac

1
Diracs Quantum Condition

• Classical mechanics relates two conjugated
  variables by using the Poisson bracket. Diracs
  quantum condition extends this relation to
  quantum mechanical operators
• The commutator between two operators must relate
  to the classical Poisson bracket between the two
  corresponding functions through the following
  relationship

HW 2
Operators which follow Diracs quantum condition
form an internally consistent set, though there
may be more than one set (different
representations)
2
Poisson bracket and commutator

• For example, for f(x,p) x and g(x,p)p the
  Poisson bracket is

3
Variance

• Finally, we can connect everything we know about
  commutators and the Diracs quantum condition and
  obtain the most fundamental property of the
  Quantum World

4
Uncertainty Principle

• For the product of the standard deviations of two
  properties of a quantum mechanical system whose
  state wavefunction is y, it can be shown (youll
  do it in your HW2)

5
The Born Interpretation

• The explicit function representing a state of a
  system in a particular coordinate system and in a
  particular representation is called
  WAVEFUNCTION ?

Remember? Probability of an event is given by
f2 where f is a complex number (probability
amplitude)
6
cont

• Normalization The probability of finding the
  particle SOMEWHERE in space and time MUST be 1

Wavefunctions which obey the eigenvalue equation
are called eigenfunctions