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 wave- functions 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 Z (Ψ ,Ψ ) = dξΨ∗(ξ)Ψ(ξ) = d~q ...d~q Ψ∗ (~q ,..~q )Ψ (~q ,..,~q ). (1) A B ∑ 1 N σ1σ2...σN 1 N σ1σ2...σN 1 N σ1,z,...,σN,z

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) c1Ψ1(ξ) + c2Ψ2(ξ) ←→ c1|Ψ1i + c2|Ψ2i; 3) The inner product: (Ψ1,Ψ2) ←→ (|Ψ1i,|Ψ2i).

The orthonormal complete bases for the space of state vectors For an orthonormal complete bases Ψα, we have

´ ´ ∗ ´ ´ (Ψα,Ψα) = δ(α,α ), ∑Ψα(ξ)Ψα(ξ ) = δ(ξ − ξ ). (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 |Ψαi to represent the wavefunction of

1 Ψα(ξ), and rewrite the above equations as

´ ´ (|Ψαi,|Ψαi = δ(α,α ), |Ψi = ∑|Ψαi(|Ψαi,|Ψi), α ´ ´ ∗ (|Ψ i,|Ψi) = ∑(|Ψα,i|Ψ i) (|Ψαi,|Ψi), (5) α where (|Ψαi,|Ψi) is the coordinate of the state vector |Ψi projection to the basis |Ψαi.

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 : |Φαi → aα, (6) ∗ where aα is a complex number, and α is the index to mark the orthonormal bases. Then for any right-vector |Bi = ∑α bα|Ψαi where bα = (|Bi,|Ψαi), the operation of hA| on |Bi is represented as

∗ hA| : |Bi → ∑aαbα. (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|Φαi, (8) such that the operation of hA| on any right-vector |Bi is expressed as

∗ hA| : |Bi → ∑aαbα = (|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|Ψαi = δ(α,α ), |Ψi = ∑|ΨαihΨα|Ψi. (10) α

We define the conjugation operation for left and right vectors, and complex numbers as

|Ψi = hΨ|; hφ| = |φi;a ¯ = a∗, (11)

thus we have

a|Ai = hA|a¯, ahB| = |Bia¯. (12)

2 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 |Φβi of H as

L|Φβi = ∑Lαβ|Φαi, (13) α where Lαβ = hΨα|L|Ψβi is the matrix element of L for the basis of Φα.

For any state-vector |Bi = ∑α bα|Φαi, the operation of L is

L|Bi = ∑bβL|Φβi = ∑Lαβbβ|Φαi. (14) β 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 We further extend the definition of conjugation operation be- low

hΨ1|L|Ψ2i = hΨ2|L|Ψ1i, L1L2 = 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 |Ψβi = hΨβ|L|Ψαi = Lβα. (17)

† ∗ If the operators L = L , i.e., Lαβ = Lβα, we call that L is Hermitian.

3 3 Outer product between left and right-vectors as opera- tors

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 |Ψαi, from Eq. 5, we have

I = ∑|ΨαihΨα|, (19) α where I is the identity operator. 6) Expansion of a linear operator L as

L = ∑ |ΨαihΨα|L|Ψα´ ihΨα´ | = ∑ |ΨαihΨα´ |Lαα´ , (20) αα´ αα´ where Lαα´ = hΨα|L|Ψα´ i is the matrix element under the bases of |Ψαi.

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)

thus Z d~r|~rih~r|Ψi = |Ψi, (22)

and Z d~r´h~r|~r´ih~r´|Ψi = h~r|Ψi = Ψ(~r). (23)

4 Similarly, for an orthonormal basis Ψα, we have

∗ ´ ∗ ´ ´ ´ ∑Ψα(~r)Ψα(~r ) = ∑h~r|Ψαi h~r |Ψαi = ∑h~r |ΨαihΨα|~ri = h~r |{∑|ΨαihΨα}|~ri α α α α = h~r´|~ri = δ(~r −~r´). (24)

4 Representations and transformation of representations

When we fix a set of orthonormal bases |Ψαi 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 = ∑ |ΨαihΨα|hΨα|L|Ψαi, (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 |Ψαi, |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 = ∑AαBα, hA|L|Bi = ∑AαLαβBβ. (27) α αβ

Let us choose another set of orthonormal basis |ϕλi, which satisfy ∑λ |ϕλihϕλ| = I. The transformation matrix U between these two sets of bases is defined as

|ϕλi = ∑|ΨαihΨα|ϕλi = ∑|ΨαiUαλ, (28) α α where Uαλ = hΨα|ϕλi. 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 ex- pressed in terms of its coordinates in the |ϕi representation through the transformation matrix

5 U as

hΨα|Ai = ∑hΨα|ϕλihϕλ|Ai = ∑Uαλhϕλ|Ai. (30) λ λ

And for the matrix element Lαα´ in the |Ψi-representation can also be related to that in the in the |ϕi representation as

† h |L| ´ i = h | ih |L| ´ ih ´ | ´ i = U h |L| ´ iU . (31) Ψα Ψα ∑ Ψα ϕλ ϕλ ϕλ ϕλ Ψα αλ ϕλ ϕλ λα´ λλ´

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