Vector Spaces

A.1 Vector Spaces

Basic results are presented here without explanation. See your PHYS20602 notes for more details. The ﬁrst ﬁve points are deﬁnitions (except for the Schwarz inequality in the 4th point); the rest can be proved.

We will consider a complex vector space, with vectors (kets) written $| α ⟩$ . (What we write inside is just a label; it might (anticipating) be the corresponding wave function, or an eigenvalue, or just a number if we have some natural ordering principle.)
For complex scalars $a$ and $b$ , the ket $| a α + b β ⟩ = a | α ⟩ + b | β ⟩$ is also in the space. The null vector is written 0 (not $| 0 ⟩$ ): $0 | α ⟩ = 0$ .
The corresponding bras $⟨ α |$ are taken from the dual space. $⟨ a α + b β | = a^{*} ⟨ α | + b^{*} ⟨ β |$ . (We use the same labels for bras and kets.)
There is a ﬁnite inner product $⟨ α | β ⟩$ ; $⟨ α | α ⟩ (\equiv {|α|}^{2})$ is real and positive (unless $⟨ α | α ⟩ = 0$ which implies $| α ⟩ = 0$ ). $⟨ β | α ⟩ = {⟨ α | β ⟩}^{*}$ . For normalised states, $⟨ α | α ⟩ = 1$ and $|⟨ α | β ⟩| \leq 1$ (with equality only for $| β ⟩ = | α ⟩$ .)
Operators transform one ket into another: $\hat{Ω} | α ⟩ = | \hat{Ω} α ⟩$ . (It may be necessary to specify a domain of validity.) The adjoint operator ${\hat{Ω}}^{†}$ is deﬁned by $⟨ α | {\hat{Ω}}^{†} = ⟨ \hat{Ω} α |$ . If ${\hat{Ω}}^{†} = \hat{Ω}$ the operator is self-adjoint or Hermitian.
The eigenvalues $ω_{i}$ of the Hermitian operator $\hat{Ω}$ are real. The corresponding normalised eigenfunctions $| ω_{i} ⟩$ are orthogonal $⟨ ω_{i} | ω_{j} ⟩ = δ_{i j}$ and form a complete set: $| α ⟩ = \sum_{i} a_{i} | ω_{i} ⟩$ for any $| α ⟩$ , where $a_{i} = ⟨ ω_{i} | α ⟩$ . (If some of the $ω_{i}$ are degenerate the orthogonalisation within the corresponding set of eigenkets will need to be done by hand, but it can be done.) If $| β ⟩ = \sum_{i} b_{i} | ω_{i} ⟩$ , then $⟨ β | α ⟩ = \sum_{i} b_{i}^{*} a_{i}$ .
If $\hat{Ω}$ is Hermitian, the matrix element can be formed with $\hat{Ω}$ “acting either way”: $⟨ β | \hat{Ω} | α ⟩ = ⟨ β | \hat{Ω} α ⟩ = ⟨ \hat{Ω} β | α ⟩ (= \sum_{i} ω_{i} b_{i}^{*} a_{i})$ .
The object $| β ⟩ ⟨ α |$ is an operator, since acting on a ket it gives another ket: $(| β ⟩ ⟨ α |) | ϕ ⟩ = (⟨ α | ϕ ⟩) | β ⟩$ ; the adjoint operator is $| α ⟩ ⟨ β |$ . By completeness, $\sum_{i} | ω_{i} ⟩ ⟨ ω_{i} | = \hat{I}$ (the identity operator).
In the basis ${| ω_{i} ⟩}$ , an operator $\hat{Θ}$ is characterised by its matrix elements $θ_{i j} \equiv ⟨ ω_{i} | \hat{Θ} | ω_{j} ⟩$ , and we have $⟨ β | \hat{Θ} | α ⟩ = \sum_{i} \sum_{j} b_{i}^{*} θ_{i j} a_{j} = (b_{1}^{*}, b_{2}^{*}, b_{3}^{*}, \dots) (\begin{matrix} θ_{11} & θ_{12} & θ_{13} & . & . \\ θ_{21} & θ_{22} & θ_{23} & . & . \\ θ_{31} & θ_{32} & θ_{33} & . & . \\ . & . & . & . & . \\ . & . & . & . & . \end{matrix}) (\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \\ . \\ . \end{matrix})$
The matrix elements of ${\hat{Θ}}^{†}$ are $⟨ ω_{i} | {\hat{Θ}}^{†} | ω_{j} ⟩ = θ_{j i}^{*}$ . (Thus the matrix of coeﬃcients is transposed as well as complex conjugated.) If $\hat{Γ}$ is another operator with matrix elements in this basis $γ_{i j}$ , the matrix elements of $\hat{Θ} \hat{Γ}$ are $\sum_{k} θ_{i k} γ_{k j}$ . Also, $\hat{Θ} \equiv \sum_{i j} | ω_{i} ⟩ θ_{i j} ⟨ ω_{j} |$ .

See section 1.2 for the aplication to quantum mechanics, including states such as $| x ⟩$ and $| p ⟩$ .

References

Shankar 1.1-6

A.1.1 Direct Products

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