Lecture 2: Dirac notation and a review of linear algebra
Read Sakurai chapter 1, Baym chatper 3
1
State vector space and the dual space
Space of wavefunctions The space of wavefunctions is the set of all the possible wavefunctions of a given system. Ψ1 (ξ), Ψ2 (ξ), ..., are wavefunctions in the representation of ξ. ξ
can be general. For example, in the coordinate representation ξ = (~q, σ), where ~q represents
the set of coordinates (~q1 , ...~qN ), and σ represents the set of particle spin (σ1,z , ..., σ2,z ). The
inner product between two wavefunctions are defined as
Z
(ΨA , ΨB ) =
∗
Z
dξΨ (ξ)Ψ(ξ) =
∑
σ1,z ,...,σN,z
d~q1 ...d~qN Ψ∗σ1 σ2 ...σN (~q1 , ..~qN )Ψσ1 σ2 ...σN (~q1 , ..,~qN ). (1)
Space of state vectors (right-vectors); the Hilbert space More conveniently, we use the
notation of the state vector |Ψi (right/ket-vector) to represent a wavefunction Ψ(ξ) of a given
quantum system. The advantage of the state vector notation is that it does not depend on
concrete representations. All the state vectors |Ψi span the linear space denoted as the Hilbert
space H of a given quantum system. The following correspondence between a wavefunction
and a state vector is defined as:
1) Ψ(ξ) ←→ |Ψi;
2) c1 Ψ1 (ξ) + c2 Ψ2 (ξ) ←→ c1 |Ψ1 i + c2 |Ψ2 i;
3) The inner product: (Ψ1 , Ψ2 ) ←→ (|Ψ1 i, |Ψ2 i).
The orthonormal complete bases for the space of state vectors
complete bases Ψα , we have
∑ Ψα(ξ)Ψ∗α(ξ´) = δ(ξ − ξ´).
(Ψα , Ψ´α ) = δ(α, α´ ),
For an orthonormal
(2)
α
For any wavefunction Ψ, we can expand it as
Ψ(ξ) = ∑ Ψα (ξ)(Ψα (ξ), Ψ),
(3)
α
and the inner product as
(Ψ´ , Ψ) = ∑(Ψα , Ψ´ )∗ (Ψα , Ψ),
(4)
α
Using the formalism of right-vectors, we use |Ψα i to represent the wavefunction of
1
Ψα (ξ), and rewrite the above equations as
(|Ψα i, |Ψ´α i = δ(α, α´ ),
|Ψi = ∑ |Ψα i(|Ψα i, |Ψi),
α
´
(|Ψ i, |Ψi) =
´
∗
∑(|Ψα, i|Ψ i) (|Ψαi, |Ψi),
(5)
α
where (|Ψα i, |Ψi) is the coordinate of the state vector |Ψi projection to the basis |Ψα i.
The dual space (space of left-vectors) The right-vector space H is a linear space. All the
linear mappings from the right-vector space H to the complex number field C also form a
linear space, which is denoted as the dual space. We use left-vector (bra-vector) to denote an
element in the dual space. Let us consider a linear mapping denoted by hA| in the dual space,
which can be determined by its operation on the orthonormal bases Φα in the Hilbert space
H.
hA : |Φα i → a∗α ,
(6)
where a∗α is a complex number, and α is the index to mark the orthonormal bases. Then
for any right-vector |Bi = ∑α bα |Ψα i where bα = (|Bi, |Ψα i), the operation of hA| on |Bi is
represented as
hA| : |Bi → ∑ a∗α bα .
(7)
α
We can identify a one-to-one correspondence between a right-vector and a linear mapping (a
left-vector) as
hA| ←→ |Ai = aα |Φα i,
(8)
such that the operation of hA| on any right-vector |Bi is expressed as
hA| : |Bi → ∑ a∗α bα = (|Ai, |Bi).
(9)
α
Below we will simply use the notation hA|Bi to denote the mapping hA| : |Bi. Using these
notations, we can rewrite Eq. 5 as
hΨα |Ψ´α i = δ(α, α´ ),
|Ψi = ∑ |Ψα ihΨα |Ψi.
(10)
α
We define the conjugation operation for left and right vectors, and complex numbers as
|Ψi = hΨ|; hφ| = |φi; ā = a∗ ,
(11)
thus we have
a|Ai = hA|ā,
ahB| = |Biā.
2
(12)
2
Operator of an observable
We define the linear operator L acting on the right-vectors in H , which satisfies
1) L|Bi is still a right-vector in H ,
2) L(b|Bi + c|Ci) = bL|Bi + cL|Ci.
L can be determined through its operation on the orthonormal basis |Φβ i of H as
L|Φβ i = ∑ Lαβ |Φα i,
(13)
α
where Lαβ = hΨα |L|Ψβ i is the matrix element of L for the basis of Φα .
For any state-vector |Bi = ∑α bα |Φα i, the operation of L is
L|Bi = ∑ bβ L|Φβ i = ∑ Lαβ bβ |Φα i.
β
(14)
αβ
The operation of L on the left vectors can also be defined. For a given hA|, the operation
of hA|L is defined through the following equation
hA|L : |Bi = hA|L|Bi,
(15)
where |Bi is an arbitrary right-vector. Thus hA|L is a linear mapping from the right-vector
space to complex numbers, thus it should be represented by a left-vector hφ| = hA|L, such that
hA|L : |Bi = hφ|Bi.
The conjugation operations
low
We further extend the definition of conjugation operation behΨ1 |L|Ψ2 i = hΨ2 |L|Ψ1 i,
L1 L2 = L2 L1 ,
L|Ψi = hΨ|L,
hA|Bi = hB|Ai.
(16)
In the following, we denote L as L† .
In the orthonormal basis Ψα , the matrix elements of L† reads
†
∗
Lαβ
= hΨα |L† |Ψβ i = hΨβ |L|Ψα i = Lβα
.
∗ , we call that L is Hermitian.
If the operators L = L† , i.e., Lαβ = Lβα
3
(17)
3
Outer product between left and right-vectors as operators
We define |AihB| as a linear operator. When acting on a right-vector |Ψi, it behaves as
(|AihB|)|Ψi = |AihB|Ψi.
(18)
Corollary:
1)hΨ|(|AihB|) = hΨ|AihB|,
2) |AihB| = |BihA|,
3) |AihA| is Hermitian.
4) |AihB|Ψi = hΨ|BihA| = |Ψi |AihB|.
5) For a set of orthonormal bases |Ψα i, from Eq. 5, we have
I = ∑ |Ψα ihΨα |,
(19)
α
where I is the identity operator.
6) Expansion of a linear operator L as
L = ∑ |Ψα ihΨα |L|Ψα´ ihΨα´ | = ∑ |Ψα ihΨα´ |Lαα´ ,
αα´
(20)
αα´
where Lαα´ = hΨα |L|Ψα´ i is the matrix element under the bases of |Ψα i.
Examples:
1)For a single spinless particle, we denote |~ri as the eigenstate of the coordinate operator
~r, which satisfy the orthonormal condition h~r|~r´ i = δ(~r −~r´ ). We have
Z
d~r|~rih~r| = I,
(21)
d~r|~rih~r|Ψi = |Ψi,
(22)
d~r´ h~r|~r´ ih~r´ |Ψi = h~r|Ψi = Ψ(~r).
(23)
thus
Z
and
Z
4
Similarly, for an orthonormal basis Ψα , we have
∑ Ψ∗α(~r)Ψα(~r´)
α
=
∑h~r|Ψαi∗h~r´|Ψαi = ∑h~r´|ΨαihΨα|~ri = h~r´|{∑ |ΨαihΨα}|~ri
α
α
´
α
´
= h~r |~ri = δ(~r −~r ).
4
(24)
Representations and transformation of representations
When we fix a set of orthonormal bases |Ψα i for the Hilbert space, it means that we are
using a specific representation. We can express a state vector |Ψi and a linear operator L as
matrices as
|Ai =
∑ |ΨαihΨα|Ai,
α
L =
∑ |ΨαihΨ´α|hΨα|L|Ψ´αi,
(25)
αα´
and
hA|Bi =
∑hA|ΨαihΨα|Bi
α
hA|L|Bi =
∑hΨα|Ai∗LαβhΨα|Bi.
(26)
α´
Using the matrix notation, we denote Aα = hΨα |Ai, then in the representation of |Ψα i, |Ai
is represented by a column vector of Aα , and L is represented by a matrix Lαβ . In the matrix
notation, we have
hA|Bi = ∑ A∗α Bα , hA|L|Bi = ∑ A∗α Lαβ Bβ .
α
(27)
αβ
Let us choose another set of orthonormal basis |ϕλ i, which satisfy ∑λ |ϕλ ihϕλ | = I. The
transformation matrix U between these two sets of bases is defined as
|ϕλ i = ∑ |Ψα ihΨα |ϕλ i = ∑ |Ψα iUαλ ,
α
(28)
α
where Uαλ = hΨα |ϕλ i. U is an unitary matrix, which satisfies the following relation
U †U = UU † = I.
(29)
For an arbitrary state vector |Ai, its coordinate hΨα |Ai in the |Ψi representation can be expressed in terms of its coordinates in the |ϕi representation through the transformation matrix
5
U as
hΨα |Ai = ∑hΨα |ϕλ ihϕλ |Ai = ∑ Uαλ hϕλ |Ai.
λ
(30)
λ
And for the matrix element Lαα´ in the |Ψi-representation can also be related to that in the in
the |ϕi representation as
†
hΨα |L|Ψα´ i = ∑hΨα |ϕλ ihϕλ |L|ϕλ´ ihϕλ´ |Ψα´ i = Uαλ hϕλ |L|ϕλ´ iUλα
´.
λλ´
6
(31)