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Dirac’s Quantum Condition
Classical mechanics relates two conjugated variables by using the Poisson bracket.
Dirac’s 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
ˆ

i  f  g  f , gˆ 


HW #2
Operators which follow Dirac’s quantum condition form an internally consistent set,
though there may be more than one set (different representations)
for those of you who do not remember (or have not seen) the Poisson bracket...
for f  f  x, p  and g  g  x, p 
 f , g 
f g g f

x p x p
Poisson bracket and commutator
For example, for f(x,p)= x and g(x,p)=p the Poisson bracket is
 x, p 
x p p x

 1 0  1
x p x p
to construct the quantum mechanical operators related to position and momentum,
 xˆ, pˆ   i
choosing
x  xˆ
and
p  -i
x, p   xˆ, pˆ   i

x
we can see that it works by applying the commutator to a F
x F
 F
 
  
  
i
 xˆ, pˆ  F   x  i    i  x  F  i x
x  
x  
x
x
 

 F
 F 
 F
 F
 i x
i  F  x
i x
i F
  i x
x
x 
x
x

  xˆ, pˆ   i
i F

Variance
Finally, we can connect everything we know about commutators and the Dirac’s
quantum condition and obtain the most fundamental property of the Quantum World
For a state that is not an eigenstate of Aˆ , we get various possible results everytime
we measure the observable Aˆ in identical systems.
A measurement of the spread in the set of results
Ai  A is the deviation from the average.
individual
measurements
expectation
value
to avoid de cancellation of deviations (some are +, some are -), we use
A
VARIANCE   A 
2
 A2 
A A

 A2  A
2


A

2
A2  A
2
 2A A
(the average of the square of the deviations)

 A2  A
2
2 A A
2
and the STANDARD DEVIATION   A 
 A
2
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 (you’ll do it in your HW2)
1 ˆ ˆ
A  B   A, B 
2
if  Aˆ , Bˆ   0,
 A  B product can be zero  both standard deviations can be zero simultaneously.
There is a set of functions that are simultaneous eigenfunctions of both operators,
 the two observables can be obtained simultaneously with full precision
 an eigenvalue can be obtained in both observable measurements
if  Aˆ , Bˆ 
the A  B product cannot be zero  both  cannot be zero simultaneously.
 the two observables cannot be obtained simultaneously with full precision
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: 
Born:
The probability of finding the particle in the volume d at the point in space r is
y  r  d
2
Remember? Probability of an event is given by |f|2 where f is a complex number
(probability amplitude)
Born equated the y  r  with an amplitude, and thus the equivalent intensity is
y  r   probability density
2
y  r  has no direct physical meaning, it can have any value
y r 
2
describes a physical property, and it is always
y  r   y *  r y  r 
2

and +
, , ,  
the probability of finding the particle in one point in space at a given time
y r ,t 
2
Normalization: The probability of finding the particle SOMEWHERE in space and
time MUST be =1
  y  r , t  dx dy dz dt
2

d
y *  r , t y  r , t  dx dy dz dt
 1
d
Wavefunctions which obey the eigenvalue equation are called eigenfunctions
Pˆ
 r 
eigenfunction
and
p
 r 
eigenfunction
represent wavefunctions, the scalar product of bras and kets
must be the same as the scalar product of the wavefunctions, that is
A B  y *A y B d
for eigenfunctions we have
a a    a  a d  1 (normalization)
a b    a b d  0 (orthogonality)