Download Chapter 15

Chapter 3
Formalism
3.1
Hilbert space
• Let’s recall for Cartesian 3D space:
• A vector is a set of 3 numbers, called components –
it can be expanded in terms of three unit vectors
(basis)




A  ex Ax  ey Ay  ez Az
• The basis spans the vector space
• Inner (dot, scalar) product of 2 vectors is defined as:
 
A  B  Ax Bx  Ay By  Az Bz
• Length (norm) of a vector

 
A  A A
3.1
Hilbert space
3.1
Hilbert space
• Hilbert space:
• Its elements are functions (vectors of Hilbert space)
• The space is linear: if φ and ψ belong to the space
then φ + ψ, as well as aφ (a – constant) also belong to
the space
David Hilbert
(1862 – 1943)
3.1
Hilbert space
• Hilbert space:
• Inner (dot, scalar) product of 2 vectors is defined as:
 ( ), ( )    * ( ) ( )d





• Length (norm) of a vector is related to the inner
product as:




 * ( ) ( )d    ( )
2

d
David Hilbert
(1862 – 1943)
3.1
Hilbert space
• Hilbert space:
• The space is complete, i.e. it contains all its limit
points (we will see later)
• Example of a Hilbert space: L 2, set of squareintegrable functions defined on the whole interval




 * ( ) ( )d  

David Hilbert
(1862 – 1943)
3.1
Wave function space
• Recall:
 2
 (r , t ) dxdydz  1
• Thus we should retain only such functions Ψ that
are well-defined everywhere, continuous, and
infinitely differentiable
• Let us call such set of functions F
• F is a subspace of L 2
• For two complex numbers λ1 and λ2 it can be shown

that if

 1 (r )  F  2 ( r )  F


1 1 (r )  2 2 (r )  F
3.1
Scalar product
• In F the scalar product is defined as:

 
 ,     * (r ) (r )dr
,    , *
, 11  2 2   1 ,1   2 , 2 
11  22 ,   1 * 1,   2 * 2 , 
• φ and ψ are orthogonal if  ,   0
• Norm is defined as  , 

 
 2 
 ,    * (r ) (r )dr    (r ) dr
• Properties of the scalar product:
3.1
Scalar product
• Schwarz inequality
 ,     ,   ,  
Karl Hermann
Amandus Schwarz
(1843 – 1921)
Orthonormal bases

• A countable set of functions ui (r )


• is called orthonormal if: ui ( r ), u j ( r )    ij
• It constitutes a basis if every function in
 F can be 
expanded in one and only one way:  (r ) 
ci ui (r )

i


u j ,    u j ,  ciui    u j , ciui    ci u j , ui 
i
i

 i

 
  ci ij  c j
c j  u j ,   u j * (r ) (r )dr

i
• Recall for 3D vectors:
 
ei  e j   ij


C   ci ei
i
 
ci  C  ei
Orthonormal bases
• For two functions




 (r )   bi ui (r ); (r )   c j u j (r )
i
• a scalar product is:
j


,     biui ,  c j u j    bi * c j ui , u j 
j
 i
 i, j
  bi * c j ij   bi * ci
 ,    bi * ci
i, j
i
i
 ,    ci * ci   ci
2
 i
• Recall for 3D vectors: B  C   bi ci
i
i
Orthonormal bases



 (r )   ci ui (r )   ui , ui (r )
i

i

i


 


  

 ui * (r ' ) (r ' )dr ' ui (r )   i ui * (r ' )ui (r ) (r ' )dr '


  

 (r )    ui * (r ' )ui (r ) (r ' )dr '
 i

• This means that
• Closure relation


 
 ui * (r ' )ui (r )   (r  r ' )
i
Orthonormal bases
• A set of functions labelled by a continuous index α

w (r )


• is called orthonormal if: w (r ), w ' (r )    (   ' )
• It constitutes a basis if every function in F can be
expanded in one and only one way:


 (r )   c( ) w (r )d
w ,   w ,  c( ' ) w 'd '   w , c( ' ) w ' d '
  c( ' )w , w ' d '   c( ' )    'd '  c ( )

 
c( )  w ,    w * (r ) (r )dr
Orthonormal bases
• For two functions




 (r )   b( ) w (r )d ; (r )   c( ' ) w ' (r )d '
• a scalar product is:



,    b( )w (r )d ,  c( ' )w ' (r )d '


  b * ( )c( ' )w (r ), w ' (r ) dd '

  b * ( )c( ' ) (   ' )dd '   b * ( )c( )d
 ,    b * ( )c( )d
 ,    c * ( )c( )d   c( ) d
2
Orthonormal bases



 (r )   c( ) w (r )d   w , w (r )d

   w



 
* (r ' ) w (r )d  (r ' )dr '



 
 (r )    w * (r ' ) w (r )d  (r ' )dr '


 

 w * (r ' ) (r ' )dr ' w (r )d




• This means that


 
 w * (r ' )w (r )d   (r  r ' )
• Closure relation


w (r ), w ' (r )   (   ' )
Example of an orthonormal basis
• Let us consider a set of functions:

 
 r0 (r )   (r  r0 )

• The set is orthonormal:



 
  
 
 r0 (r ),  r0 ' (r )    (r  r0 ) (r  r0 ' )dr   (r0  r0 ' )
• Functions in F can be expanded:

 

 
 
 (r )    (r  r0 ) (r0 )dr0    r0 (r ) (r0 )dr0

 



 r0 ,   r0 ,   r0 ' (r ) (r0 ' )dr0 '    r0 , (r0 ' ) r0 ' (r ) dr0 '







  (r0 ' )  r0 ,  r0 ' dr0 '   (r0 ' ) r0  r0 'dr0 '   (r0 )

 

 




 


 (r0 )   r0 ,    r0 (r )dr

Example of an orthonormal basis
• For two functions

  


 


 (r )    r0 (r ) (r0 )dr0 ; (r )    r0 ' (r ) (r0 ' )dr0 '
• a scalar product is:

  

 
,     r0 (r ) (r0 )dr0 ,   r0 ' (r ) (r0 ' )dr0 '



  
   * (r0 ) (r0 ' )  r0 (r ),  r0 ' (r ) dr0 dr0 '




 


   
   * (r0 ) (r0 ' ) (r0  r0 ' )dr0 dr0 '    * (r0 ) (r0 )dr0

 
 ,     * (r0 ) (r0 )dr0

 
 2 
 ,    * (r0 ) (r0 )dr0    (r0 ) dr0
Example of an orthonormal basis






 
 (r )    r0 (r ) (r0 )dr0    r0 (r )  r0 , dr0

   


 
 
(r ' ) (r )dr  (r ' )dr '


 
 
 (r )     (r ' ) (r )dr  (r ' )dr '


 
 
 ( r ' ) ( r ' ) dr '   ( r ) dr

r
r0
0
 0

r0

r0

r0
0

r0
0
• This means that

 
 
  r0 (r ' ) r0 (r )dr0   (r  r ' )
• Closure relation




 
 r0 (r ),  r0 ' (r )   (r0  r0 ' )
State vectors and state space
• The same function ψ can be represented by a
multiplicity of different sets of components,
corresponding to the choice of a basis
• These sets characterize the state of the system as
well as the wave function itself
• Moreover, the ψ function appears on the same
footing as other sets of components
State vectors and state space
• Each state of the system is thus characterized by a
state vector, belonging to state space of the system Er
• As F is a subspace of L 2, Er is a subspace of the
Hilbert space
3.6
Dirac notation
• Bracket = “bra” x “ket”
• < > = < | > = “< |” x “| >”
Paul Adrien
Maurice Dirac
(1902 – 1984)
3.6
Dirac notation
• We will be working in the Er space
• Any vector element of this space we will call a ket
vector
• Notation:

• We associate kets with wave functions:

 (r )  F    Er
• F and Er are isomporphic
• r is an index labelling components
Paul Adrien
Maurice Dirac
(1902 – 1984)
3.6
Dirac notation
• With each pair of kets we associate their scalar
product – a complex number
 ,


• We define a linear functional (not the same as a
linear operator!) on kets as a linear operation
associating a complex number with a ket:

  Er    

 1  1  2  2   1  1   2   2

• Such functionals form a vector space
• We will call it a dual space Er*
Paul Adrien
Maurice Dirac
(1902 – 1984)
3.6
Dirac notation
• Any element of the dual space we will call a bra
vector

• Ket | φ > enables us to define a linear functional that
associates (linearly) with each ket | ψ > a complex
number equal to the scalar product:
    , 

• For every ket in Er there is a bra in Er*
Paul Adrien
Maurice Dirac
(1902 – 1984)
3.6
Dirac notation
 ,     
    ,      ,          
     ,     *   ,    *   ,  
• Some properties:
1
1
1
1
2
1
2
2
2
1
1
1
2
2
1
1
2
2
2
2
 1 * 1   2 * 2 
  ,     *   ,     *  
  * 
    *
Paul Adrien
Maurice Dirac
(1902 – 1984)
Linear operators
Aˆ    '
Aˆ 1  1  2  2   1 Aˆ  1  2 Aˆ  2
• Linear operator A is defined as:
• Product of operators:

( Aˆ Bˆ )   Aˆ Bˆ 
• In general:
Aˆ Bˆ  Bˆ Aˆ
• Commutator:
[ Aˆ , Bˆ ]  Aˆ Bˆ  Bˆ Aˆ
• Matrix element of operator A:

 Â 


Linear operators
• Example:
 ,  
• What is

       
 
 

?
         
• It is an operator – it converts one ket into another
Linear operators
• Example:
  1
• Let us assume that
• Projector operator

P̂   
         
2
ˆ
P  Pˆ Pˆ    



         P̂
• It projects one ket onto another
Linear operators
• Example:
• Let us assume that
i  j   ij
i, j  1,2,..., q
• These kets span space Eq, a subspace of E
q
Pˆq   i i
• Subspace projector operator
q
i 1
 q

P̂q     i i     i i 
i 1
 i 1

q
q




2
Pˆq  Pˆq Pˆq    i i    j  j 
q
 i 1 q
 j 1
q

  i i  j  j   i  ij  j   i i  P̂q
i , j 1
i , j 1
i 1
• It projects a ket onto a subspace of kets
Linear operators
• Recall matrix element of a linear operator A:

 Â 
• Since a scalar product depends linearly on the ket,
the matrix element depends linearly on the ket
• Thus for a given bra and a given operator we can
associate a number that will depend linearly on the
ket
• So there is a new linear functional on the kets in
space E, i.e., a bra in space of E *, which we will
denote
 Â
• Therefore

 

 Aˆ    Aˆ    Â 

Linear operators
• Operator A associates with a given bra a new bra
 Aˆ   '
• Let’s show that this correspondence is linear
  1 1  2 2
  Aˆ 
   1 1   2 2 
     Aˆ   
    Aˆ      Aˆ 
  Aˆ 
1
1
1
1
2
2

2 Aˆ 

2
 Aˆ  1 1  2 2 Aˆ  1 1 Aˆ  2 2 Aˆ
Q.E.D.
Linear operators
• For each ket there is a bra associated with it
  
 '  Â 
'  '
• Hermitian conjugate (adjoint) operator:
 '   Aˆ †
• This operator is linear (can be shown)
 '  '  *
 Aˆ    Aˆ †  *
Charles Hermite
(1822 – 1901)
Linear operators
 
Aˆ †
• Some properties:

†
 
†
 Aˆ
†
ˆ
ˆ
A   * A

†
ˆ
A  Bˆ  Aˆ †  Bˆ †
  Aˆ Bˆ 
 
   Aˆ Bˆ
†
   B̂
   Â   Bˆ Aˆ
†
  Â 
  B̂ 
†
†
†
   †
 
 Bˆ Aˆ  Aˆ Bˆ
†
†
†
Charles Hermite
(1822 – 1901)
Hermitian conjugation
• To obtain Hermitian conjugation of an expression:
• Replace constants with their complex conjugates
• Replace operators with their Hermitian conjugates
• Replace kets with bras
• Replace bras with kets
• Reverse order of factors
 *
†
ˆ
ˆ
A A
 
 
Aˆ    Aˆ †
Charles Hermite
(1822 – 1901)
3.2
Hermitian operators
Aˆ  Aˆ †
• For a Hermitian operator:
 Aˆ    Aˆ  *
• Hermitian operators play a fundamental role in
quantum mechanics (we’ll see later)
• E.g., projector operator is Hermitian:
P̂   
• If:
 
†
ˆ
P    
†
ˆ
A  Aˆ
†
ˆ
B  Bˆ
† ˆ†
ˆ
ˆ
ˆ
AB  B A  Bˆ Aˆ  Aˆ Bˆ
†
 
†

[ Aˆ , Bˆ ]  0
Aˆ Bˆ   Aˆ Bˆ
†
Charles Hermite
(1822 – 1901)
Representations in state space
• In a certain basis, vectors and operators are
represented by numbers (components and matrix
elements)
• Thus vector calculus becomes matrix calculus
• A choice of a specific representation is dictated by
the simplicity of calculations
• We will rewrite expressions obtained above for
orthonormal bases using Dirac notation
Orthonormal bases
• A countable set of kets
• is called orthonormal if:
ui
ui u j   ij
• It constitutes a basis if every vector in E can be
expanded in one and only one way:
   ci ui
i
u j    ci u j ui   ci ij  c j
i
i
cj  uj 
Orthonormal bases
   ci ui   ui  ui
i
i
  ui ui     ui ui  
i
 i



    ui ui    1̂ 
 i

Pˆ{ui }   ui ui  1̂
i
• Closure relation
• 1 – identity operator
Orthonormal bases
• For two kets
   bi ui ;    ci ui
i
i
bi  ui  ; ci  ui 
• a scalar product is:
    1̂ 


    ui ui      ui ui 
i
 i

    bi * ci
i
  bi * ci
i
    ci * ci   ci
i
i
2
Orthonormal bases
• A set of kets labelled by a continuous index α
• is called orthonormal if:
w
w w '   (   ' )
• It constitutes a basis if every vector in E can be
expanded in one and only one way:
   c( ) w d
w   w
 c( ' ) w
'

d '   c( ' ) w w ' d '
  c( ' )    'd '  c ( )
c( )  w 
Orthonormal bases
   c( ) w d   w  w d
  w w  d 
 
 w

 w


w d   1̂ 
Pˆ{w }   w w d  1̂
• Closure relation
• 1 – identity operator

w d 
Orthonormal bases
• For two kets
   b( ) w d ;    c( ) w d
b( )  w  ; c( )  w 
• a scalar product is:
 
 w

    1̂ 

w d     w w  d
  b * ( )c( )d
    b * ( )c( )d
    c( ) d
2
Representation of kets and bras
• In a certain basis, a ket is represented by its
components
• These components could be arranged as a columnvector:
 u1  


 u2  
 ... 


 ui  


 ... 
 ...

 w 

 ...





Representation of kets and bras
• In a certain basis, a bra is also represented by its
components
• These components could be arranged as a rowvector:
 u
1
 u2
...
...
 ui
w 
...

...
Representation of operators
• In a certain basis, an operator is represented by
matrix components:
 A11

 A21
 ...

 Ai1
 ...

Aij  ui Aˆ u j
A12 ... A1 j
A22 ... A2 j
...
...
...
Ai 2
...
Aij
...
...
...
...

...
...

...
...
A( ,  ' )  w Aˆ w '
...
... 
 ...


  ... A( ,  ' ) ...
  ...
...
... 
'
ui Aˆ Bˆ u j  ui Aˆ1̂Bˆ u j

ˆ
ˆ
 ui A  uk uk  B u j   ui Aˆ uk uk Bˆ u j
k
 k

Representation of operators
 '  Â 
ci '  ui  '  ui Aˆ   ui Â1̂


 ui Aˆ   u j u j     ui Aˆ u j u j    Aij c j
j
j
 j

 c1 '   A11
  
 c2 '   A21
 ...    ...
  
 ci '   Ai1
 ...   ...
  
A12
...
A1 j
A22 ... A2 j
...
...
...
Ai 2
...
Aij
...
...
...
... c1 
 
... c2 
... ... 
 
... ci 
... ... 
Representation of operators
 '  Â 
c' ( )  w  '  w Aˆ   w Â1̂ 


 w Aˆ  w ' w ' d '    w Aˆ w ' w '  d '
  A( ,  ' )c( ' )d '
Representation of operators
bi  ui  ; ci  ui 
 Aˆ  
 A11

 A21
 b1 * b2 * ... b j * ... ...

 Ai1
 ...

A12
...
A1 j
A22 ... A2 j
...
...
...
Ai 2
...
Aij
...
...
...
...  c1 
 
...  c2 
...  ... 
 
...  ci 



...  ... 
Representation of operators
 c1 
 
 c2 
    ... c1 * c2 *
 
 ci 
 ... 
   c1c1 *

 c2 c1 *
  ...

 ci c1 *
 ...

ci  ui 
... c j * ...
c1c2 * ... c1c j * ... 

c2 c2 * ... c2 c j * ... 
...
...
...
... 

ci c2 * ... ci c j * ... 
...
...
...
... 
Representation of operators
A 
†
ij
†
ˆ
 ui A u j  u j Aˆ ui *  A ji *
†
ˆ
A ( ,  ' )  w A w '  w ' Aˆ w *  A * ( ' ,  )
†
• For Hermitian operators:
Aij  A ji *
A( ,  ' )  A * ( ' ,  )
Aii  Aii *
A( ,  )  A * ( ,  )
• Diagonal elements of Hermitian operators are
always real
Change of representations
• How do representations change when we go from
one basis to another?
u t
i
• Let’s denote
Sik  ui tk
• Some properties:
S S    S
†
kl
†
ki
Sil  
i
i
 t k 1̂ tl
SS 
†
ij
k
S   S  * 
†
ki
ik
tk ui


tk ui ui tl  t k   ui ui  tl
 i

 tk tl   kl

  Sik S   ui t k t k u j  ui   t k t k
k
k
 k
 ui 1̂ u j  ui u j   ij
S †S  SS †
†
kj

 uj

1
Change of representations


tk   tk 1̂  tk   ui ui     t k ui ui 
 i

i
  S ki† ui 
i


ui   ui 1̂  ui   tk tk     ui tk tk 
 k

k
  Sik tk 
k


 tk   1̂ tk     ui ui  tk    ui ui tk
 i

i
   ui Sik
i
Change of representations


 ˆ
ˆ
tk A tl  tk   ui ui  A  u j u j  tl
 i
  j

  tk ui ui Aˆ u j u j tl   S ki† Aij S ji
i, j
i, j

 ˆ

ˆ
ui A u j  ui   t k t k  A  tl tl  u j
 k
  l

  ui tk tk Aˆ tl tl u j
k ,l
  Sik Akl S
k ,l
†
ij
Eigenvalue equations
• A ket is called an eigenvector of a linear operator if:
    
• This is called an eigenvalue equation for an operator
• This equation has solutions only when λ takes
certain values - eigenvalues
• If:
    
• then:
 Aˆ †    *
Eigenvalue equations
• The eigenvalue is called nondegenerate (simple) if
the corresponding eigenvector is unique to within a
constant
• The eigenvalue is called degenerate if there are at
least two linearly independent kets corresponding to
this eigenvalue
• The number of linearly independent eigenvectors
corresponding to a certain eigenvalue is called a
degree of degeneracy
Eigenvalue equations
• If for a certain eigenvalue λ the degree of
ˆ  i    i ; i  1,2,...g
degeneracy is g: A
• then every eigenvector of the form
   ci 
i
i
• is an eigenvector of the operator A corresponding to
the eigenvalue λ for any ci:
i
i
i
i
ˆ
ˆ
ˆ


c


c
A


c


A   A ci 
i
i
i
i
i

i
i
• The set of linearly independent eigenvectors
corresponding to a certain eigenvalue comprises a gdimensional vector space called an eigensubspace
Eigenvalue equations
• Let us assume that the basis is finite-dimensional,
with dimensionality N
    
ui Aˆ  u j u j    ui 

ui Aˆ    ui 

j
j
ui Aˆ u j u j    ui 
 A
ij
  ij c j  0

A c
ij
j
 ci
j
j
• This is a system of N linear homogenous equations
for N coefficients cj
• Condition for a non-trivial solution:
A  1  0
Eigenvalue equations
• This equation is called the characteristic equation
A11  
A21
A12
...
A1N
A22   ...
A2 N
...
...
AN 1
AN 2
...
...
0
... ANN  
• This is an Nth order equation in and it has N roots –
the eigenvalues of the operator
• Condition for a non-trivial solution:
A  1  0
Eigenvalue equations
• Let us select λ0 as one of the eigenvalues
 A
ij
 0 ij c j  0
A  0 1  0
j
• If λ0 is a simple root of the characteristic equation,
then we have a system of N – 1 independent
equations for coefficients cj
• From linear algebra: the solution of this system (for
one of the coefficients fixed) is
c j   c ;  1
0
j 1
0
1
 0   c j u j   0j c1 u j  c1  0j u j
j
j
j
Eigenvalue equations
• Let us select λ0 as one of the eigenvalues
 A
ij
 0 ij c j  0
A  0 1  0
j
• If λ0 is a multiple (degenrate) root of the
characteristic equation, then we have less than N – 1
independent equations for coefficients cj
• E.g., if we have N – 2 independent equations then
(from linear algebra) the solution of this system is
c j   c   c ;     1;     0
0
j 1
0
j 2
0
1
0
2
0
2
0
1
 0  c1   u j  c2   u j
0
j
j
0
j
j
3.2
Eigenproblems for Hermitian operators
†
ˆ
ˆ
• For: A  A
    
 Aˆ  *   Aˆ †    Â 
 Â     


Im  Aˆ   0
Im     0
• Therefore λ is a real number
 Â   
• Also:
• If:
    
• Then: 
• But:
    

              
               
   0
3.2
Observables
• Consider a Hermitian operator A whose eigenvalues
form a discrete spectrum
a ; n  1,2,...
n

• The degree of degeneracy of a given eigenvalue an
will be labelled as gn
• In the eigensubspace En we consider gn linearly
independent kets:
Aˆ  ni  an  ni ; i  1,2,..., g n
• If
an  an '
• Then
 ni  nj'  0
3.2
Observables
• Inside each eigensubspace
 ni  nj   ij
• Therefore:
 ni  nj'   ij nn'
• If all these eigenkets form a basis in the state space,
then operator A is called an observable
gn
 
n
i 1
i
n
  1̂
i
n
3.2
Observables
gn
• For an eigensubspace projector
i
i
ˆ
Pn    n  n
i 1
gn
gn
Aˆ  Aˆ1̂  Aˆ   ni  ni   an   ni  ni  a Pˆ
 nn
n
i 1
n
i 1
n
Aˆ   an Pˆn
n
• These relations could be generalized for the case of
continuous bases
• E.g., a projector is an observable
P̂   
  1
3.2
Observables
• If
• Then
[ Aˆ , Bˆ ]  0
Bˆ Aˆ   aBˆ 

Aˆ   a 
Aˆ Bˆ   aBˆ 
 
Aˆ Bˆ   a Bˆ 
• If a is non-degenerate then

B̂   
• so this ket is also an eigenvector of B
• If a is degenerate then
Bˆ   Ea
• Thereby, if A and B commute, each eigensubspace
of A is globally invariant (stable) under the action of B
3.2
Observables
[ Aˆ , Bˆ ]  0
• If
Aˆ  1  a1  1
• Then
Aˆ  2  a2  2
a1  a2
 1 Aˆ Bˆ  2  a1  1 Bˆ  2
 1 Bˆ Aˆ  2  a2  1 Bˆ  2
 1 Aˆ Bˆ  2   1 Bˆ Aˆ  2  a1  a2   1 Bˆ  2
 1 Bˆ  2  0
• If two operators commute, there is an orthonormal
basis with eigenvectors common to both operators
3.4
Questions QM answers
• 1) How is the state of a system described
mathematically? (In CM – via generalized coordinates
and momenta)
• 2) For a given state, how can one predict results of
measurements of various physical quantities? (In CM
– unambiguously, via the calculated trajectory in a
phase space)
• 3) For a given state of the system known at time t0,
how can one find a state of this system at an arbitrary
time t? (In CM – using Hamilton’s equations)
• Answers to these questions are given by the
postulates of QM
State of a system
• 1st postulate: At certain time t0 a state of this system
is defined by a ket belonging to the state space E
 (t0 )  E
Physical quantities
• 2nd postulate: Every measurable physical quantity is
described by an observable operator acting in E
Measurement
• 3rd postulate: Measurements of a physical quantity
result only in (real) eigenvalues of a corresponding
observable
Measurement
• 3rd postulate: Measurements of a physical quantity
result only in (real) eigenvalues of a corresponding
observable
• It is not obvious a priori whether the spectrum of the
measured quantity is continuous or discrete (e.g., a
system consisting of a proton and an electron)
3.4
Spectral decomposition
• If
Aˆ un  an un
  1
• Then the state of the system
   cn u n
n
• 4th postulate: The probability of measuring an
eigenvalue an of an observable A in a certain state of
the system is:
P an   cn  un 
2
2
   cn u n    n
n
n
 n  cn un
3.4
Spectral decomposition
 n  cn un  un  un  un un   P̂n 
 n  n  cn   Pˆn † Pˆn    Pˆn Pˆn    P̂n 
2
P an   cn  un 
2
2
  P̂n 
   cn u n    n
n
n
 n  cn un
3.4
Spectral decomposition
• The mean value of an observable:
  an u n 
n
2
Aˆ
  an  un un 

n
   an un un     Aˆ un un 
n
n


  Â un un     Â 
n

Aˆ

  Aˆ 
  an P an 
n
3.4
Spectral decomposition
Aˆ v   v
  1
• If
• Then the state of the system
   c( ) v d
• 4th postulate: The probability of measuring an
eigenvalue of an observable A between α and α+dα in
a certain state of the system is:
dP    c( ) d  v 
2
c( )  v 
2
2
2
   
• ρ – probability density
d
3.4
Spectral decomposition
• The mean value of an observable:
   v 
2
Â

  dP 
d     v v  d
    v v  d    Â v v  d


  Â  v v d    Â 
Aˆ

  Aˆ 
3.2
RMS deviation
• How can one quantify the dispersion of the
measurements around the mean value?
• Averaging a deviation from the average is not
adequate:
ˆ
ˆ
ˆ
ˆ
A A  A  A  0
• Instead, the RMS deviation is used:

 Aˆ  Aˆ
A 
Aˆ  2 Aˆ Aˆ  Aˆ



2
2
2
2
Aˆ  2 Aˆ  Aˆ 
2

Aˆ  Aˆ 
2
2
Aˆ  Aˆ
2
2
3.2
RMS deviation
• How can one quantify the dispersion of the
measurements around the mean value?
• Averaging a deviation from the average is not
adequate:
ˆ
ˆ
ˆ
ˆ
A A  A  A  0
• Instead, the RMS deviation is used:
A 
   Aˆ    d

2



     d      d 

 



2
2
Reduction via measurement
• When the measurement is performed only one
possible result is obtained
• Then the state of the system after the measurement
of an eigenvalue is:
an
  un
• We can write this as:
an
 
cn un
cn
2

Pˆn 
 Pˆn 
Reduction via measurement
• 5th postulate: If measurement of a physical quantity
in a given state of the system yields a certain
eigenvalue, the state of the system immediately after
the measurement is the normalized projection of the
initial state onto a state associated with that
eigenvalue
• The state of the system after the measurement is the
eigenvector corresponding to that eigenvlaue
an
 
cn un
cn
2

Pˆn 
 Pˆn 
Reduction via measurement
• We shall consider only ideal measurements
• This means that the perturbations the measurement
devices produce are only due to the quantummechanical aspect of measurement
• We will consider the studied system and the
measurement device together as a whole
Time evolution of the system
• 6th postulate: The time evolution of the state vector
of the system is determined by the Schrödinger
equation:
d
i  t   Hˆ t   t 
dt
• H – is the Hamiltonian operator, observable
associated with the total energy of the system
Sir William Rowan
Hamilton
(1805 – 1865)
3.5
Time evolution of the system
• How does the mean value of an observable evolve?
d  t  Aˆ t  t 
ˆ
d ˆ
d


A

A    Aˆ


dt
dt
dt
t
ˆ
1
1

A

 Hˆ Aˆ    Aˆ Hˆ   

 i
i
t
d Aˆ
1 ˆ ˆ
Aˆ

[ A, H ] 
dt
i
t
• Recall the CM result:
du
u
 [u , H ] 
dt
t
3.5
Compatibility of observables
• If two (observable) operators commute, there exists
a basis common to both operators
• There is at least one state that will simultaneously
yield specific eigenvalues for these two operators,
thereby these two observable can be measured
simultaneously
• Such operators are called compatible with each
other
• If, on the other hand, the operators do not commute,
a state cannot in general be an eigenvector of both
observables, thus these operators are called
incompatible
3.5
Compatibility of observables
• When two observables are compatible, the
measurement of the second does not produce any
loss of the information obtained from the
measurement of the first
• When two observables are incompatible, the
measurement of the second does produces a loss of
the information obtained from the measurement of
the first
3.5
The uncertainty principle
• Recall Schwarz inequality:
   
• In Dirac’s notation:
• Since

A 
 ,     ,   ,  

 Aˆ  Aˆ

2
 ( Aˆ  Aˆ )( Aˆ  Aˆ )  
B 
• Then:

 Bˆ  Bˆ

2
 

f f
 ( Bˆ  Bˆ )( Bˆ  Bˆ ) 

f g 
f f
g g   A B
g g
3.5
The uncertainty principle
f g   ( Aˆ  Aˆ )( Bˆ  Bˆ ) 
  ( Aˆ Bˆ  Aˆ Bˆ  Bˆ Aˆ  Aˆ Bˆ ) 
  Aˆ Bˆ   Aˆ  Bˆ   Bˆ  Aˆ   Aˆ Bˆ  
• Let us calculate:
 Aˆ Bˆ  Aˆ Bˆ  Bˆ Aˆ  Aˆ Bˆ  Aˆ Bˆ  Aˆ Bˆ
• Similarly:
g f  Bˆ Aˆ  Aˆ Bˆ
• On the other hand:
 Re  f g
f g g f  f g
  Im
2
2
 f g  g f 
 
 

2
i


f g
  Im
2
2
f g

2
 Aˆ Bˆ  Bˆ Aˆ / 2i    [ Aˆ , Bˆ ] / 2i
2
2
3.5
The uncertainty principle
• Synopsizing:
 A B  f g
• Hence:
f g
 [ Aˆ , Bˆ ]

2 2
 A B  
 2i

2




 [ Aˆ , Bˆ ]


 2i





2
2
• This is the generalized uncertainty principle
[ xˆ, pˆ x ]  i
2
 i 
2 2
• Then:  x  p x   
 2i 
• Recall:
 x p
x


2

xp x 
2
3.5
The uncertainty principle
• Synopsizing:
 A B  f g
f g
• Hence:
 [ Aˆ , Bˆ ]

2 2
 A B  
 2i

ˆ  Hˆ Bˆ  Qˆ
• If A
 [ Aˆ , Bˆ ]


 2i

2








2
2
• And operator Q doesn’t depend on time explicitly
• Then:

   [ Hˆ , Qˆ ] / 2i
2
H
2
Q

2
3.5
The uncertainty principle
• Recall:
• Hence:
d Aˆ
1 ˆ ˆ
Aˆ

[ A, H ] 
dt
i
t
d Qˆ
1 ˆ ˆ
Qˆ

[Q, H ] 
dt
i
t
[Qˆ , Hˆ ]  i
ˆ
d
Q

 H Q 
2 dt
d Qˆ
dt
 d Qˆ 
2



2
2
• Then:    [ H
ˆ , Qˆ ] / 2i 
 2 dt 
H Q




2
3.5
The uncertainty principle
• Introducing Δt as the time it takes the expectation
value of Q to change by one standard deviation:
Q 
d Qˆ
dt
t
ˆ
d
Q

 H Q 
2 dt
• Then:
H
d Qˆ
ˆ
d
Q

t 
dt
2 dt
 H  E

Et 
2