PHY209-Chap5-Formalism of QT-MRA
Engineering Physics II
(Quantum Physics)
Chapter 5: Formalism of Quantum Mechanics
5.1
5.2
5.3
5.4
5.5
5.6
5.7
Hilbert Space: Ket and Bra Notation of Vectors
The Schrödinger Equation in Bra-Ket Notation
Unitary Transformation: Change of Basis
Eigenvalue Related Theorems
Postulates of Quantum Mechanics
Physical Significance of Bra-Ket Vectors
Relevant Problems
PHY209-Chap5-Formalism of QT-MRA
5.1 Hilbert Space: Ket and Bra Notation of Vectors
In this section of Chapter 5, we are going to study the properties of a multi-dimensional linear vector
space. Quantum mechanics was formulated by Schrödinger, Heisenberg and Dirac. Here we shall use
mainly the Dirac’s approach which requires the knowledge of complex linear vector space, sometimes
called Hilbert space. Let us begin the brief mathematical properties of this space.
State Vectors in a Hilbert Space
Dirac introduced the symbol a to denote the state of a quantum mechanical system, known as the state
vector or ket vector. We call a the ket a. The state vectors a , b , c , …. etc. constitute a complex
vector space. The linear combination c1 a + c2 b + c3 c + LL is also a state vector of the space, where
c1 , c2 , c3 LL , etc are complex constants. All vectors c a , c ≠ 0 , which has the same direction as a ,
corresponds to the same physical state.
†
Corresponding to every vector a is defined an adjoint vector a , for which Dirac used the notation
 a | , called a bra vector. The adjoint is nothing but the transpose along with the complex conjugate
elements:
 a | ≡ | a † ,
 a |† ≡ | a .
(5.1)
If | b = c | a , then
 b | ≡ | b † = (c | a )† = | a † c † =  a | c*,
(5.2)
where * indicates complex conjugation. It is important to mention here that the ket vector a is usually
represented by a column vector and the bra vector  a | is represented by a row vector. Therefore, if
 a1 
 
 a2 
a   a3  ,
 
 M 
 
 aN 
then
(
 a | ≡ | a †  a1*
a2*
)
a3* L a *N .
The inner product, sometimes called scalar product of the ket vectors a and | b is defined as
 a | b = b | a * .
(5.3)
The inner product is linear in the second vector:
2
PHY209-Chap5-Formalism of QT-MRA
 a | (c1 | b + c2 | c ) = c1  a | b + c2  a | c .
(5.4)
The squared length or norm of the vector a is denoted by  a || a ≡ a | a . By definition
 a | a ≥ 0 .
(5.5)
Any vector space (in general infinite dimensional) such that between any two vectors in the space an
inner product with the above properties, Eqs. (5.3)-(5.5) is defined, is called a Hilbert space. The space
of ket vectors (or bra vectors) representing quantum mechanical states is thus a Hilbert space. Bra–ket
notation of Dirac is a notation for linear algebra and linear operators on complex vector spaces together
with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically
designed to ease the types of calculations that frequently come up in quantum mechanics. Its use in
quantum mechanics is quite widespread. Many phenomena that are explained using quantum mechanics
are explained using the bra–ket notation.
Basis in Hilbert Space
A Hilbert space must have a set of basis vectors and these vectors should be linearly independent and any
general ket vector of the space must be expressed in terms of these basis vectors. Let
{| eα  },
α = 1, 2, 3,LL N be the set of basis vectors of a Hilbert space where N is the dimension of the
space. Then a general ket vector | a can be expanded in terms of the set of basis vectors {| eα  }αN=1 :
| a = a1 | e1  + a2 | e2  + a3 | e3  + LL + a N | e N  ≡
N
 aα | e α  ≡  aα | e α  ,
α =1
(5.6)
α
where aα are the components of the vector | a and usually complex.
A set of N vectors {| eα  }αN=1 is said to be a linearly independent set if
 λα | e α  = 0

λ1 = λ2 = LL = λ N = 0 .
(5.7)
α
The maximum number of such linearly independent vectors possible in a linear vector space gives the
dimensionality of the space. As mentioned earlier, the dimension of a Hilbert space may both be finite or
infinite. It should be mentioned here that there are some methods to find the set of basis vectors for a
Hilbert space.
The basis vectors should have the following orthonormality property:
1, α = β
 eα | e β  = δ αβ = 
 0 α ≠ β.
3
(5.8)
PHY209-Chap5-Formalism of QT-MRA
Completeness Relation
We can expand the ket vector | a in terms of the basis vectors {| eα  }αN=1 as
| a =  aα | eα  .
α
Taking inner product of | a above with | e β  from the left, we obtain
 e β | a =  e β |  aα | eα 
α
=  aα  e β | eα 
α
=  aα δ αβ
α
= aβ ,
that is
a β =  e β | a ,
or
aα =  eα | a .
(5.9)
Thus, the ket vector | a =  aα | eα  becomes
α
| a =  eα | a | eα 
α
=  | eα   eα | a
α

=   | eα   eα

α

|  | a .


Therefore, we have

| a =   | eα   eα

α

|  | a ,


from which we can write
 | eα eα | ≡ Iˆ ,
(5.10)
α
where Î is a unit operator of N × N dimension. The relation given by Eq. (5.10) is known as the
completeness relation.
4
PHY209-Chap5-Formalism of QT-MRA
Representation of Basis Vectors
A general state vector | a can be represented by a column vector in the N-dimensional Hilbert space:
 a1 
 
 a2 
| a =  aα | eα    a3  .
 
α
 M 
 
 aN 
(5.11)
In our N-dimensional Hilbert space, the basis vectors are represented by the following column vectors to
satisfy the orthonormality property:
1
 
0
| e1  ≡  0  ,
 
M
 
0
0
 
1
| e2  ≡  0  ,
 
M
 
0
0
 
0
| e3  ≡  1  ,
 
M
 
0
L LL
0
 
0
| eN  ≡  0 .
 
M
 
1
(5.12)
Adjoint of vector given by Eq. (5.11) is
(
adj | a = | a † =  a |  a1*
a2*
a3*
LL
)
a*N .
(5.13)
Thus, the adjoint of a ket vector is a bra vector and is represented by a row vector.
Linear Operators in Hilbert Space
An operator  is defined as a mapping of a vector | φ onto another vector | ψ  of the Hilbert space:
| ψ  = Aˆ | φ .
(5.14)
A linear operator  has the property that it maps a linear combination of input vectors to the linear
combination of the corresponding maps:
Aˆ (c1 | φ1  + c2 | φ2  ) = c1 Aˆ | φ1  + c2 Aˆ | φ2 
= c1 | ψ1  + c2 | ψ2  ,
(5.15)
for c1 , c2 ∈ C .
The sum of two linear operators is defined by
( Aˆ + Bˆ ) | φ = Aˆ | φ + Bˆ | φ .
(5.16)
The product of two operators defined as
Aˆ Bˆ | φ = Aˆ (Bˆ | φ ) .
(5.17)
5
PHY209-Chap5-Formalism of QT-MRA
The null-element and 1-element of the operators is denoted as 0̂ and 1̂ . These two operators are defined
by their action on arbitrary vectors of the Hilbert space:
0̂ | φ = 0 ,
(5.18)
1̂ | φ = | φ ,
(5.19)
for all vectors | φ .
In general, the multiplication of two operators is not commutative:
Aˆ Bˆ ≠ Bˆ Aˆ .
(5.20)
[ Aˆ , Bˆ ] = Aˆ Bˆ − Bˆ Aˆ
(5.21)
The expression
is called the commutator between the operators  and B̂ . If [ Aˆ , Bˆ ] = 0 , the operators commute. The
following rules for commutators satisfy:
[ Aˆ , Bˆ ] = −[Bˆ , Aˆ ] ,
(5.22)
[ Aˆ , Aˆ ] = 0 ,
(5.23)
[ Aˆ , 1̂] = 0 ,
(5.24)
[ Aˆ , Aˆ −1 ] = 0 ,
(5.25)
[ Aˆ , cBˆ ] = c[ Aˆ , Bˆ ] ,
(5.26)
[ Aˆ 1 + Aˆ 2 , Bˆ ] = [ Aˆ 1 , Bˆ ] + [ Aˆ 2 , Bˆ ] ,
(5.27)
[ Aˆ 1 Aˆ 2 , Bˆ ] = Aˆ 1 [ Aˆ 2 , Bˆ ] + [ Aˆ 1 , Bˆ ] Aˆ 2 ,
(5.28)
[ Aˆ , Bˆ 1 Bˆ 2 ] = Bˆ 1 [ Aˆ , Bˆ 2 ] + [ Aˆ , Bˆ 1 ]Bˆ 2 .
(5.29)
Often the anticommutator is also used. It is defined as
[ Aˆ , Bˆ ]+ = Aˆ Bˆ + Bˆ Aˆ .
(5.30)
If [ Aˆ , Bˆ ]+ = 0 , the operators are called anti-commuting.
Hermitian Operators
If an operator  satisfies the following condition
Aˆ † = Aˆ ,
(5.31)
then it is called a Hermitian operator. The Hermitian conjugate sign (†) indicates that the operator will be
transpose and the elements will be complex conjugate. It is to be mentioned here that operators in
quantum mechanics are represented by Hermitian matrices.
6
PHY209-Chap5-Formalism of QT-MRA
Matrix Representation of Vector Equation
We suppose that we are given the following equation:
| b = Aˆ | a ,
(5.32)
where  is a Hermitian operator in the Hilbert space of the ket vectors | a and | b. We expand the kets
| a and | b in terms of the basis vectors of the space {| eα  }αN=1 :
| a =  aα | eα  ,
α
and
| b =  bα | eα  .
α
Substituting the above expansions of | a and | b in Eq. (5.32), we obtain
 bβ | e β  = Aˆ  aγ | eγ  .
β
(5.33)
γ
Taking inner product of Eq. (5.33) with the basis vector | eα  from the left, we get
 eα |  bβ | e β  =  eα | Aˆ  aγ | eγ 
β

γ
 bβ eα | e β  =  aγ eα | Aˆ | eγ  ,
β

γ
 bβ δ αβ =  eα | Aˆ | eγ  aγ
β

γ
()
bα =  Aˆ αγ aγ ,
(5.34)
γ
where
(Aˆ )αβ = eα | Aˆ | eβ 
(5.35)
is the matrix element of the operator (matrix) Â in the basis {| eα  }αN=1 .
Equation (5.34) can be written as
()
()
()
()
bα = Aˆ α 1 a1 + Aˆ α 2 a2 + Aˆ α 3 a3 + LL + Aˆ αN a N ,
which is a set of the following N linear equations for α = 1, 2, 3,LL, N :
()
()
()
()
b2 = (Aˆ )21 a1 + (Aˆ )22 a2 + (Aˆ )23 a3 + LL + (Aˆ )2 N a N ,
b3 = (Aˆ )31 a1 + (Aˆ )32 a2 + (Aˆ )33 a3 + LL + (Aˆ )3 N a N ,
b 1 = Aˆ 11 a1 + Aˆ 12 a2 + Aˆ 13 a3 + LL + Aˆ 1 N a N ,
M
M
()
()
()
()
bN = Aˆ N 1 a1 + Aˆ N 2 a2 + Aˆ N 3 a3 + LL + Aˆ NN a N .
7
(5.36)
PHY209-Chap5-Formalism of QT-MRA
Thus, Eq. (5.36) can be written as a matrix equation as shown below:
 b1  
  
 b2  
b =
 3 
 M  
  
 bN  
(Aˆ )11 (Aˆ )12 (Aˆ )13
(Aˆ )21 (Aˆ )22 (Aˆ )23
(Aˆ )31 (Aˆ )32 (Aˆ )33
M
ˆA
N1
M
ˆA
N2
()
()
L
L
L
M
M
ˆA
N3 L
()
(Aˆ )1N   a1 
(Aˆ )2N   a2 
(Aˆ )3N   a3  .
(5.37)
M  M 
  
ˆA
NN   a N 
()
The matrix equation, Eq. (5.37) can also be written in compact form as
b = Aa.
(5.38)
Equation (5.37) or Eq. (5.38) is the matrix representation of Eq. (5.32).
Principle of Superposition
ψ2 , L L, ψ N
If the ket vectors ψ1 ,
are eigenvectors of the linear operator  with the same
eigenvalue λ, then the linear combination ψ = c1 ψ1 + c2 ψ 2 + L L + c N ψ N
of the eigenvectors
will be an eigenvector of  with the same eigenvalue λ. The constants c1 ,
ψ1 , ψ 2 , L L , ψ N
c2 , L L , c N are complex scalars.
Proof
We have
Aˆ ψ = Aˆ (c1 ψ1 + c2 ψ 2 + L L + c N ψ N )
= Aˆ (c1 ψ1 ) + Aˆ (c2 ψ 2 ) + L L + Aˆ (c N ψ N )
= c Aˆ ψ + c Aˆ ψ + L L + c Aˆ ψ
1
1
2
2
N
N
= c1 λ ψ1 + c2 λ ψ2 + L L + c N λ ψ N
= λ(c1 ψ1 + c2 ψ 2 + L L + c N ψ N )
=λψ .
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PHY209-Chap5-Formalism of QT-MRA
5.2 The Schrödinger Equation in Bra-Ket Notation
The Schrödinger equation in bra-ket notation is
∂
(5.39)
| ψ (t ) = Hˆ (t ) | ψ (t ) ,
∂t
where Hˆ (t ) is the operator for the observable H(t) associated with the total energy of the system.
ih
•
Hˆ (t ) is called the Hamiltonian operator or simply the Hamiltonian of the system.
•
Given an initial state | ψ (t 0 ) , the state at any subsequent time is determined ( i.e. not
probabilistic). Probability only enters when a physical quantity is measured, upon which the state
vector undergoes a probabilistic change.
The solution of the Schrödinger equation is
ˆ
| ψ (t ) = e − i(t − t 0 ) H / h | ψ (t 0 ) .
(5.40)
If the Hilbert space is of infinite dimension and we consider continuous configurational basis vectors
| x and a general ket vector | ψ  can be expanded as
| ψ  =  ψ ( x) | x dx ,
(5.41)
where ψ ( x) is the component of the state vector | ψ  along the basis vector | x .
We can return to the wave mechanics of Schrödinger by writing the abstract Schrödinger equation in the
eigenbase | x of the position operator. For simplicity of notation, we only consider one-dimensional case
and define that there exists the following eigenvalue equation:
xˆ | x' = x'| x' ,
(5.42)
with the following orthogonality relation:
 x | x' = δ ( x − x' ) .
(5.43)
We note that the position operator x̂ has a continuous spectrum of eigenvalues, the orthogonality relation
is a Dirac delta function rather than a delta symbol as used for the discrete spectrum case.
We have
 x'| ψ  =  x'  ψ ( x) | x dx =  ψ ( x)  x' | x dx =  ψ ( x) δ ( x'− x) dx = ψ ( x' ) .
Thus,
ψ( x) =  x | ψ  .
(5.44)
Substituting the expression for ψ (x) from Eq. (5.44) into Eq. (5.41), we obtain
(
)
| ψ  =   x | ψ  | x dx =  | x x | dx | ψ  ,
from which we can write
9
PHY209-Chap5-Formalism of QT-MRA
 | x x | dx ≡ Iˆ ,
(5.45)
which is the completeness relation for the continuous basis set | x , Î is the unity operator.
Schrödinger Equation in the usual Form, the x-Representation
We have the Schrödinger equation in the abstract form:
ih
∂
| ψ (t ) = Hˆ ( xˆ , pˆ ) | ψ (t ) .
∂t
Taking inner product with | x from the left
ih
∂
 x | ψ (t ) =  x | Hˆ ( xˆ , pˆ ) | ψ (t ) ,
∂t
which can be written as
ih
∂
ψ ( x, t ) =  x | Hˆ ( xˆ , pˆ )Iˆ | ψ (t )
∂t
=  x | Hˆ ( xˆ , pˆ ) | x' x'| dx' | ψ (t )
=   x | Hˆ ( xˆ , pˆ ) | x' x | ψ (t ) dx'
=   x | Hˆ ( xˆ , pˆ ) | x' ψ ( x', t ) dx'
=  Hˆ ( xˆ , pˆ ) x | x' ψ ( x', t ) dx'
=  Hˆ ( xˆ , pˆ ) δ ( x −x' )ψ ( x', t ) dx'
= Hˆ ( xˆ , pˆ )ψ ( x, t ) ,
That is
ih
∂
ψ ( x, t ) = Hˆ ( x, pˆ )ψ ( x, t ) ,
∂t
(5.46)
which the Schrödinger equation in the configurational space. The Hamiltonian operator in the
configurational space is
pˆ 2
h2 2
Hˆ ( x, pˆ ) =
+ V ( x) = −
∇ + V ( x).
2m
2m
The quantity V(x) is the potential energy and p̂ ≡ −ih∇ .
10
(5.47)
PHY209-Chap5-Formalism of QT-MRA
5.3 Unitary Transformation: Change of Basis
The most important thing in a Hilbert space is its set of basis vectors. Different representations of vectors
and operators in a Hilbert space are possible due to the different set of basis vectors. Let us now consider
the transformation of a basis vector set from one representation to another representation. Let us suppose
that in a particular representation, the basis vector set is given by {| eα  }αN=1 . Therefore, from the
orthonormality condition of the basis vectors, we have
 eα | e β  = δ αβ .
(5.48)
Let us assume that the operator Uˆ when acts on vector set {| eα  }αN=1 gives another set of basis vectors
{| e'  }
α
N
α =1
in a different representation:
| e' α  = Uˆ | eα  .
(5.49)
Since the new set | e' α  has to be a basis, then
δ αβ = ( Iˆ ) αβ =  e' α | e' β 
=  eα | Uˆ †Uˆ | e β 
=  eα | Uˆ † Iˆ Uˆ | e β 
=  eα | Uˆ †  | e γ   eγ | Uˆ | e β 
γ
[Q
 | eγ eγ |= Iˆ ]
γ
( ) ()
= (Uˆ Uˆ ) .
=  Uˆ † αγ Uˆ γβ
γ
†
αβ
Thus, we get
Uˆ †Uˆ = Iˆ .
(5.50)
Similarly, it can be shown that
Uˆ Uˆ † = Iˆ .
(5.51)
Uˆ † = Uˆ −1 .
(5.52)
It is also easy to show
Therefore, we see that Û is a unitary operator, which transforms a basis {| eα  }αN=1 in a representation to
{
}
N
another basis in a different representation | e' α  α =1 . Equation (5.49) can be regarded as a unitary
transformation.
11
PHY209-Chap5-Formalism of QT-MRA
Representation of an Operator in a Transformed Co-ordinate System
Let us consider an operator Ô in the representation { | eα  }αN=1 . What will be the form of the operator Ô
{
}
N
in the new representation | e' α  α =1 which is obtained from the unitary transformation | e' α  = Uˆ | eα  ?
Since the operator Ô is represented in the old representation { | eα  }αN=1 , therefore, the αβ − th matrix
element of Ô is
(Oˆ )αβ = eα | Oˆ | eβ  .
{
(5.53)
}
N
Therefore, in the new basis | e' α  α =1 , the operator Ô can be expressed as
(Oˆ ')αβ = e'α | Oˆ | e' β 
[Q | e'
=  eα | Uˆ † Oˆ Uˆ | e β 
α
 = Uˆ | eα 
]
=  eα | Uˆ † IˆOˆ IˆUˆ | e β 
=  eα | Uˆ †  | ek  ek | Oˆ  | el  el |Uˆ | e β 
k
l


 Q  | eα  eα | = Iˆ 


α
=   eα | U | ek  ek | Oˆ | el   el | Uˆ | e β 
ˆ†
k
l
( ) ()()
=  Uˆ † αk Oˆ kl Uˆ lβ
k
l


=  Uˆ † αk  Oˆ kl Uˆ lβ 
 l

k
Oˆ Uˆ
=  Uˆ †
( ) ()()
( ) ( )
= (Uˆ Oˆ Uˆ ) ,
αk
kβ
k
†
αβ
from which, we obtain
Oˆ ' = Uˆ †Oˆ U .
(5.54)
12
PHY209-Chap5-Formalism of QT-MRA
Important Properties of Unitary Transformation
(A) Normalization of a Ket Vector is Invariant under a Unitary Transformation.
Proof:
Let us assume that
| a =  aα | eα  .
α
The normalization of the ket vector a is defined as
a a = 1 ,
which is equivalent to
 aα*
eα
 aβ |eβ
α

=1
β
 aα* aβ
eα e β = 1
α β

 aα* aβ δαβ = 1 ,
α β
which leads to
 aα
2
=1.
α
We consider the unitary transformation
| a'  = Uˆ | a ,
where Û is the unitary operator.
Thus, we have
a' | a'  =  a | Uˆ †Uˆ | a
=  a | Iˆ | a
=  a | a
= 1,
That is,
 a' | a'  = 1 .
Thus, we see that the normalization of a ket vector is invariant under unitary transformation.
13
PHY209-Chap5-Formalism of QT-MRA
(B) Inner Product of Two Ket Vectors is Invariant under a Unitary Transformation.
Proof:
Let us consider any two ket vectors a and b . The inner product between these two ket vectors is  a b .
Let us consider the following unitary transformation by the help of the unitary operator Û :
| a'  = Uˆ | a
and
| b'  = Uˆ | b .
The inner product between these two kets is
 a' | b'  =  a | Uˆ †Uˆ | b
=  a | Iˆ | b
=  a | b ,
that is,
 a' | b'  =  a | b .
Thus, we see that the inner product of the two kets a and b is invariant under a unitary transformation.
(C) The Trace of an Operator is Invariant under a Unitary Transformation.
Proof:
Let us consider an operator Ô . We know that, due to the unitary transformation, an operator Ô is
transformed to Ô' , where Oˆ ' = Uˆ † Oˆ Uˆ . The αβ − th matrix element of Ô' with Uˆ † = Uˆ −1 is
(Oˆ ')αβ = (Uˆ †Oˆ Uˆ )αβ = (Uˆ −1Oˆ Uˆ )αβ .
We know that the trace of an operator is the sum of the diagonal elements. Therefore,


tr Oˆ ' =  Oˆ ' αα =  Uˆ −1Oˆ Uˆ αα =    Uˆ −1 αk Oˆ kl Uˆ lα 


α
α
α  k l



=    Uˆ lα Uˆ −1 αk  Oˆ kl =  Uˆ Uˆ −1 lk Oˆ kl


k l  α
k l



=    Uˆ Uˆ −1 lk Oˆ kl  =  Uˆ Uˆ −1Oˆ ll


l  k
 l
ˆ
ˆ
=  IˆO =  O
( )
(
( )
)
( ) ()()
( ) ( ) ()
(
( )ll
= tr (Oˆ ),
l
)()
(
(
) ()
)
( )ll
l
That is,
( ) ()
tr Oˆ ' = tr Oˆ .
Thus, we see that the trace of an operator is invariant under a unitary transformation.
14
PHY209-Chap5-Formalism of QT-MRA
5.4 Eigenvalue Related Theorems
When an operator acts on a vector in a Hilbert space, it gives another vector in the same space. Let us
consider the equation
Lˆ | a = | b ,
(5.55)
where L̂ is a linear operator and | a and | b are two ket vectors in the same Hilbert space. Usually, in a
Hilbert space, there exists some vectors upon which if a linear operator acts, give the same vector with a
scalar multiple:
Lˆ | a = λ | a ,
(5.56)
where λ is a scalar. The vectors which satisfy Eq. (5.56), are called the eigenvectors of the operator L̂ .
The scalar number λ is called the eigenvalue of the operator L̂ and the vector | a is called the
eigenvector. Equation (5.56), is known as an eigenvalue equation. Eigenvalue related theorems and their
proofs are given below.
Theorem 1
Eigenvalues of a Hermitian operator are all real
Proof:
Let us consider the following eigenvalue equation
Aˆ | am  = λm am ,
(5.57)
where  is a Hermitian operator and λm is its eigenvalue corresponding to the eigenvector am . We
know that for Hermitian operator  , we have
Aˆ † = Aˆ .
(5.58)
ˆ†
This means that the adjoint of  is equal to A . This is a special property of a Hermitian operator.
Hermitian operators are self-adjoint. Now the bra form of Eq. (5.57) is am Aˆ † = λ*m am , which
 am Aˆ = λ*m am . Therefore, we have
am Aˆ am = λ* am am .
But taking the inner product of Eq. (5.57) with am from the left, we obtain
am Aˆ am = λm am am .
(5.59)
(5.60)
Thus, by equating Eqs. (5.59) and (5.60), we obtain
am Aˆ am = λ*m am am = λm am am ,
which gives
( λm − λ*m ) am am = 0 .
But am am ≠ 0 , therefore, ( λm − λ*m ) = 0, which implies λm = λ*m , and thus the eigenvalues are real.
15
PHY209-Chap5-Formalism of QT-MRA
Theorem 2
Eigenvectors corresponding to non-degenerate eigenvalues of a Hermitian operator are
orthogonal to each other
Proof
Let us consider a Hermitian operator  and eigenvectors | a and | b corresponding to non-degenerate
eigenvalues λa and λb , where λa ≠ λb :
Aˆ | a = λa | a ,
(5.61)
Aˆ | b = λb | b .
(5.62)
and
Taking inner product of Eq. (5.61) with | b from the left, we obtain
 b | Aˆ | a =  b | λa | a = λa  b | a .
(5.63)
Now, taking the adjoint of Eq. (5.62),
b | Aˆ † = λ*b b | ,

 b | Aˆ = λb  b | .
(5.64)
Taking inner product of the above equation, Eq. (5.64), with | a from the right, we obtain
 b | Aˆ | a = λb  b | a .
(5.65)
Now equating Eqs. (5.63) and (5.65), we obtain
( λa − λb )b | a = 0 ,
which for λa ≠ λb (non-degenerate), gives
 b | a = 0 ,
(5.66)
showing that the vectors | a and | b are orthogonal to each other.
16
PHY209-Chap5-Formalism of QT-MRA
Theorem 3
Every Hermitian operator can be brought into a diagonal form by a unitary
transformation
Proof
Let us consider a Hermitian operator  in the basis vector set {| eα  }αN=1 . Now, if | uα  be the
eigenvector corresponding to the eigenvalue λα of the operator  , then the eigenvalue equation is
Aˆ | u  = λ | u  .
(5.67)
α
α
α
Let us now consider the unitary operator Û which transforms the basis vector set {| eα  }αN=1 into another
basis vector set {| uα  }αN=1 , then from a unitary transformation, we have
| uα  = Uˆ | eα  .
(5.68)
Since, we know that for non-degenerate eigenvalues of a Hermitian operator, the corresponding
eigenvectors are orthogonal to each other, therefore, if the eigenvectors {| uα  }αN=1 are normalized and
from the unitary property Uˆ †Uˆ = Uˆ Uˆ † = Iˆ of the unitary operator Û , we can write
uα | u β  = δ αβ .
(5.69)
Due to this unitary transformation, the Hermitian operator  is transformed to  ' , where
Aˆ ' = Uˆ † Aˆ Uˆ .
(5.70)
Taking the αβ − th matrix element of  ' :
( Aˆ ' )αβ =  eα | Aˆ ' | e β 
=  eα | Uˆ † Aˆ Uˆ | e β 
[ by using Eq. (5.70)]
=  eα | Uˆ † Aˆ | u β 
[ by using Eq. (5.68)]
=  eα | Uˆ † λ β | u β 
[ by using Eq. (5.67)]
= λ β eα | Uˆ †Uˆ | e β 
[ by using Eq. (5.68)]
= λ β eα | Iˆ | e β 
[Q Uˆ †Uˆ = Iˆ ]
= λ β  eα | e β 
[from the orthonormality condition]
= λ β δ αβ ,
that is,
( Aˆ ' )αβ = λβ δ αβ ,
which shows that the transformed operator Â' is diagonal.
17
PHY209-Chap5-Formalism of QT-MRA
5.5 Postulates of Quantum Mechanics
An important distinction needs to be made between quantum mechanics and quantum physics. Quantum
mechanics is a mathematical language, much like calculus. Just as classical physics uses calculus to
explain nature, quantum physics uses quantum mechanics to explain nature. There are four postulates to
quantum mechanics, which will form the basis of quantum computers:
•
•
•
•
Postulate 1: The State Space: Definition of a quantum bit, or qubit.
Postulate 2: Evolution of Quantum States: How qubit(s) transform (evolve).
Postulate 3: Quantum Measurement: The effect of measurement.
Postulate 4: Composite Systems: How qubits combine together into systems of multi-qubits.
Postulate 1: State Space
Postulate 1 defines “the setting” in which quantum mechanics take place, which is the Hilbert space
(inner product space which satisfy the condition of completeness).
Postulate 1 states that
“Any isolated physical system is associated with a complex vector space with inner product
called the State Space of the system. The system is completely described by its state vector,
which is a unit vector in the system’s state space.”
•
•
•
The system is completely described by a state vector, a unit vector, pertaining to the state space.
The state space describes all possible states the system can be in.
Postulate 1 does not tell us either what the state space or state vector is.
A Qubit: The Simplest State Space
The simplest quantum system is a state space with 2 dimensions - - there are two possible states the
system can be in! A qubit is defined by
ψ =a 0 +b1 .
(5.71)
If this will be a unit vector then from the normalization condition:
ψ ψ =| a |2 + | b |2 = 1.
(5.72)
A linear combination of states is called a superposition of states - qualitatively a new feature: a qubit can
be a mixture of two classical bits.
Requirement for Unit Norm
At this point, it becomes clear why quantum states must be unit vectors. If we write our quantum state
using any orthonormal basis:
ψ = c1 φ1 + c2 φ2 + L + c N φ N =  c j φ j ,
j
then the modulus of ψ is given by
18
(5.73)
PHY209-Chap5-Formalism of QT-MRA
ψ =
ψ ψ = | c1 |2 + | c2 |2 + L + | c N |2 .
(5.74)
Since | c j |2 is the probability of measuring the state ψ in the state φ j , then the norm of a quantum
state is simply the sum of the probability of measure each φ j . Since our measurement must give us
something, then the sum of the probabilities must be 1. Hence the need for a unit vector.
Postulate 2: Evolution of Quantum States
“The evolution of an isolated quantum system is described by a unitary transformation. That is
the state vector ψ 0 of the system at time t 0 is related to the state vector ψ at time t by a
unitary operator Uˆ = Uˆ (t , t 0 ) through ψ = Uˆ ψ 0 .”
Example:
Let ψ 0 is a quantum bit,
 a
ψ 0 = a 0 + b 1 ≡  
 b
and
0 1
 .
Uˆ = 
1 0
We have
0 1 a b
   =   = b 0 + a 1 .
ψ = Uˆ ψ 0 = Uˆ (a 0 + b 1 ) = 
1 0 b a
Example:
Let ψ 0 is a quantum bit
1
ψ 0 = 1 0 + 0 1 = 0 ≡  
0
and
1 1 1 
 .

Uˆ =
2 1 − 1
We have
1 1 1   1  1 1 1  1  1  0  1

   =
  =
  +
  =
( 0 + 1 ).
ψ = Uˆ ψ 0 = Uˆ 0 =
2 1 − 1  0 
2 1
2 0
2 1
2
Important: Û must be unitary, that is Uˆ †Uˆ = Iˆ.
19
PHY209-Chap5-Formalism of QT-MRA
Example:
Let
1 1 1 
 ,

Hˆ = Uˆ =
2 1 − 1
We have
1 1 1 
 .

then Hˆ † = Uˆ † =
2 1 − 1
1 1 1  1 1 1  1  2 0   1 0  ˆ

 = 
 = 
 = I .


Uˆ †Uˆ =
2 1 − 1 2 1 − 1 2  0 2   0 1 
Postulate 3: Quantum Measurement
{ }
“Quantum measurements are described by a collection M̂ m of measurement operators. These
are operators acting on the state space of the system being measured. The index m refers to the
measurement outcomes that may occur in the experiment. If the state of the quantum system is
ψ immediately before the measurement then the probability that result m occurs is given by:
ˆ †M
ˆ
p(m) = ψ M
m m ψ
and the state of the system after measurement is:
ˆmψ
M
ψ' =
.
ˆ †M
ˆ ψ
ψM
m m
(5.75)
(5.76)
The measurement operators satisfy the completeness equation:
 Mˆ m† Mˆ m = Iˆ. The completeness
m
equation expresses the fact that probabilities sum to one:
 p(m) =  ψ Mˆ m† Mˆ m ψ
m
= 1. ”
m
ˆ = 0 0 and M
ˆ = 1 1.
Some important measurement operators are the followings: M
0
1
ˆ 0 = 0 0 =  1  (1 0 ) =  1 0  ,
ˆ 1 = 1 1 =  0  (0 1) =  0 0  .
M
M
0
0 0
1
0 1
 


 


†
†
†
ˆ M
ˆ =M
ˆ M
ˆ +M
ˆ M
ˆ = Iˆ and are thus complete.
We observe that  M
m
m
0
0
1
1
m
Example:
Let
ψ =a 0 +b1 .
Then
ˆ †M
ˆ
p(0) = ψ M
0 0 ψ .
ˆ †M
ˆ =M
ˆ , hence
We note that M
0
0
0
1 0 a
ˆ †M
ˆ
ˆ
   = (a *
p(0) = ψ M
0 0 ψ = ψ M 0 ψ = (a * b * )
0 0 b
a
b * )  =| a |2 .
0
Hence, the probability of measuring 0 is related to its probability amplitude a by way of | a |2 .
It is important to note that the state after measurement is related to the outcome of the measurement.
20
PHY209-Chap5-Formalism of QT-MRA
For example, let us suppose 0 was measured, then the state of the system after this measurement is
re-normalized as:
ˆ ψ
M
0
|a|
=
a
0.
|a|
Postulate 4: Composite Systems
“The state space of a composite physical system is the tensor product sometimes called
Kronecker product of the state spaces of the component physical systems. For example, let us
suppose systems 1 through n and system i is in state ψi , then the joint state of the total system is
ψ1 ⊗ ψ 2 ⊗ L ⊗ ψ n . ”
Before giving the explicit representation of the tensor/Kronecker product, let’s discuss some physical
properties of composite quantum systems. This should lead us to some abstract properties about the
mathematical operation needed to treat the behaviour of multiple quantum systems as one, bigger
quantum system. For simplicity, we’ll consider a composite system of two qubits, but the generalization
to multiple quantum systems of different dimensions is straightforward.
Composite/Joint quantum system must follow the following properties:
1. Dimensions
The first observation to make is that we should be able to see a composite system made of two
qubits (2 dimensions each) as a single quantum system with 4 dimensions. This follows from the
fact that since each qubit has two exclusive states each ( 0 2 and 1 2 ), then the full system will
have 4 distinct states namely:
00 4 = 0 2 ⊗ 0 2 ←→ qubit 1 in 0 2 and qubit 2 in 0 2 ,
01 4 = 0 2 ⊗ 1 2 ←→ qubit 1 in 0 2 and qubit 2 in 1 2 ,
10 4 = 1 2 ⊗ 0 2 ←→ qubit 1 in 1 2 and qubit 2 in 0 2 ,
11 4 = 1 2 ⊗ 1 2 ←→ qubit 1 in 1 2 and qubit 2 in 1 2 .
We have added the extra subscripts 2 and 4 to explicitly denote the fact that they are vectors of
dimensions 2 and 4 respectively.
It would make sense to write explicitly that
1
0
0
 
 
 
0
1
0
00 4 =   , 01 4 =   , 10 4 =   , and
0
0
1
 
 
 
0
0
0
 
 
 
21
0
 
0
11 4 =   .
0
 
1
 
PHY209-Chap5-Formalism of QT-MRA
2. Measurement Probabilities
Now, we will introduce some arguments about probability. Let us assume that qubit 1 is in state
ψ1 2 and qubit 2 is in state ψ 2 2 , such that the composite, 4-dimensional state is abstractly
given by Ψ 4 = ψ1ψ 2 4 = ψ1 2 ⊗ ψ2 2 . Since the composite system can be seen as a single,
larger system of higher dimension, Born’s Rule still applies. Therefore, the probability of
measuring qubit 1 in φ1 and qubit 2 in φ2 , that is, measuring the composite system in
Φ 4 = φ1φ2 4 = φ1 2 ⊗ φ2 2 will be given by
2
P(Φ) = Φ 4 Ψ 4
= φ1φ2 4 ψ1ψ 2 4
2
=
[ φ1 2 ⊗ φ2 2 ][ ψ1 2 ⊗ ψ2 2 ] 2 .
But, if we think of each qubit as their own separate system, then the probability of measuring
qubit 1 in φ1 2 and the probability of measuring qubit 2 in φ2 2 is given by
P(φ1 ) = φ1 ψ1
2
and
P( φ 2 ) = φ 2 ψ 2
2
respectively.
Basic probability theory tells us that the probability of two independent things happening is given
by the product of the individual probability, then we must have
P(Φ) = P(φ1 )P(φ2 ),
which essentially means that we must have:
[ φ1 2 ⊗ φ2 2 ][ ψ1 2 ⊗ ψ2 2 ] 2 =
φ1 ψ1
2
φ2 ψ 2
2
.
3. Composite/Joint Quantum Operations
A final physical argument has to do with quantum operations. If Û 1 is a unitary operator on qubit
1 and Û 2 is a unitary operator on qubit 2, then the composite/joint operation Uˆ 1 ⊗ Uˆ 2 must have
the property that:
(Uˆ 1 ⊗ Uˆ 2 ) ψ1ψ2
4
(
= Uˆ 1 ⊗ Uˆ 2
) ( ψ1 2 ⊗ ψ2 2 )= (Uˆ 1 ψ1 2 )⊗ (Uˆ 2 ψ2 2 ) .
22
PHY209-Chap5-Formalism of QT-MRA
In other words, the resulting composite state after applying the composite quantum operation,
(Uˆ 1 ⊗ Uˆ 2 ) ψ1ψ2
4
must be equal to the composite state of the individual state after the individual
operations, Uˆ 1 ψ1 2 and Uˆ 2 ψ 2 2 respectively.
Example:
Let us suppose
 a
c
ψ1 = a 0 + b 1 ≡   and ψ 2 = c 0 + d 1 =   ,
 b
d
then the state vector of the composite system is
ψ1 ⊗ ψ 2 = ψ1 ψ 2 = ψ1ψ 2 = (a 0 + b 1 ) (c 0 + d 1
= ac 0 0 + ad 0 1 + bc 1 0 + bd 1 1
)
11
1  0
01
0  0
= ac    + ad    + bc    + bd   
00
0 1 
10
1 1 
1
0
0
0
 
 
 
 
0
1
0
0
= ac  + ad  + bc  + bd 
0
0
1
0
 
 
 
 
0
0
0
1
 
 
 
 
= ac 00 + ad 01 + bc 10 + bd 11 .
Why the tensor product? This is not a proof, but one would expect some way to describe a composite
system. Tensor product works for classical systems. For quantum systems tensor product captures the
essence of superposition, that is if system A is in state A , and B in state B , then there should be some
way to have a little of A and a little of B. Tensor product exposes this.
In chapter 6, we shall see the impact of these postulates when we describe the fundamentals of quantum
computing.
5.6 Physical Significance of Bra-Ket Vectors
In the early days of quantum theory, P. A. M. (Paul Adrian Maurice) Dirac created a powerful and
concise formalism for it which is now referred to as Dirac notation or bra-ket (bracket) notation. Two
major mathematical traditions emerged in quantum mechanics: Heisenberg’s matrix mechanics and
Schrödinger’s wave mechanics. These distinctly different computational approaches to quantum theory
are formally equivalent, each with its particular strengths in certain applications. Heisenberg’s variation,
as its name suggests, is based matrix and vector algebra, while Schrödinger’s approach requires integral
and differential calculus. Dirac’s notation can be used in a first step in which the quantum mechanical
23
PHY209-Chap5-Formalism of QT-MRA
calculation is described or set up. After this is done, one chooses either matrix or wave mechanics to
complete the calculation, depending on which method is computationally the most expedient.
In Dirac’s notation what is known is put in a ket, |  . So, for example, | p expresses the fact that a
particle has momentum p. It could also be more explicit: | p = 2 , the particle has momentum equal to 2.
Similarly, the symbol | x = 1.23 indicates that the particle has position 1.23. The symbol | ψ  represents
a system in the state ψ and is therefore called the state vector. The ket can also be interpreted as the
initial state in some transition or event.
The bra  | represents the final state or the language in which you wish to express the content of the ket
|  . For example,  x = 0.25 | ψ  is the probability amplitude that a particle in state ψ will be found at
position x = 0.25 . In conventional notation we write this as ψ ( x = 0.25) , the value of the function ψ at
x = 0.25 . The absolute square of the probability amplitude,  x = 0.25 | ψ 
2
is the probability density that
a particle in state ψ will be found at x = 0.25 . Thus, we see that a bra-ket pair can represent an event, the
result of an experiment. In quantum mechanics an experiment consists of two sequential observations one that establishes the initial state (ket) and one that establishes the final state (bra).
If we write  x | ψ  , we are expressing ψ in coordinate space without being explicit about the actual value
of x. The quantity  x = 0.25 | ψ  is a number, but the more general expression  x | ψ  is a mathematical
function, a mathematical function of x, or we could say a mathematical algorithm for generating all
possible values of  x | ψ  , the probability amplitude that a system in state | ψ  has position x. For the n-th
excited state of the well-known particle in a one dimensional infinite potential box of dimension L is
 x | ψ n  = ψ n ( x ) = 2 / L sin(n π x / L) . However, if we wish to express ψ in momentum space we would
write,  p | ψ  = ψ ( p) .
The major point here is that there is more than one language in which to express | ψ  . The most common
language for chemists is coordinate space (x, y, and z, or r, θ , and φ , etc.), but we shall see that
momentum space offers an equally important view of the state function. It is important to recognize that
 x | ψ  and  p | ψ  are formally equivalent and contain the same physical information about the state of the
system. One of the tenets of quantum mechanics is that if you know | ψ  you know everything there is to
know about the system, and if, in particular, you know  x | ψ  = ψ ( x ) , you can calculate all of the
properties of the system in terms of the position x. If you wish, you can transform into any other
appropriate language such as momentum space.
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PHY209-Chap5-Formalism of QT-MRA
A bra-ket pair can also be thought of as a vector projection - the projection of the content of the ket onto
the content of the bra, or the “shadow” the ket casts on the bra. For example,  φ | ψ  is the projection of
the state ψ onto the state φ. It is the probability amplitude that a system in state ψ will be subsequently
found in state φ. It is also what we have come to call an overlap integral.
5.7 Relevant Problems
Problem 5.1
Show that  a | b =  b | a * .
Problem 5.2
If φ is an eigenstate of Ĥ and  is any Hermitian operator, then show that
φ [ Hˆ , Aˆ ] φ = 0.
Problem 5.3
Consider the Hamiltonian in one-dimension
pˆ 2
Hˆ = z + Vˆ ( z),
2m *
(p5.3a)
where
pˆ z = −ih
d
dz
(p5.3b)
and
Vˆ ( z) = λzˆ n ,
where λ is a real constant and n is an integer.
(i) Prove that
  pˆ 2 

[ Hˆ , zˆpˆ z ] = ih − 2 z  + λnzˆ n  ,
  2m * 

(p5.3c)
(p5.3d)
(ii) Starting with the result of the previous step, prove that if φ is an eigenstate of Ĥ , the following
is true:
1
Tˆ = n Vˆ ,
(p5.3e)
2
where L stands for the expectation value in the state φ and T̂ is the kinetic energy operator
expressed as
h2 d2
Tˆ = −
.
2m * dz 2
Equation (p5.3e) is known as the Virial Theorem.
(p5.3f)
Problem 5.4
What are basis vectors for a Hilbert space? Discuss also the importance of basis vectors.
Problem 5.5
Establish the completeness relation and verify it.
Problem 5.6
Show that normalization of a ket vector is invariant under a unitary transformation.
Problem 5.7
Prove that the inner product of two ket vectors is invariant under a unitary transformation.
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PHY209-Chap5-Formalism of QT-MRA
Problem 5.8
Show that the eigenvalues of a Hermitian operator are all real.
Problem 5.9
Prove that the eigenvectors corresponding to non-degenerate eigenvalues of a Hermitian
operator are orthogonal to each other.
Problem 5.10 State and explain the postulates of quantum mechanics.
ˆ
Problem 5.11 Prove that if Ĥ is Hermitian, then Uˆ = eiH is unitary.
Problem 5.12 For an arbitrary V(z), show that
1 dVˆ
Tˆ =
z
,
2
dz
where the average is taken over an eigenstate φ of the Hamiltonian Hˆ = Tˆ + Vˆ ( z ).
********************
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