Analogy between vectors and functions 1 2 Vectors Vector v or wn Dot product v†.wn = number v†.v=1 means vector v is normalized wn†.wm=0 orthogonal vectors 3 Vector space with normalization 4 Complete basis set: All orthonormal vectors wn that span the vector space (usually 3-­‐D); any vector can be expressed as a sum of them. Getting coefficient cn: wn†.v =cn The dual space element: wn†. Operator: transforms a vector into another: u = A v They are called “matrices” Functions Function Ψ(x) or ϕn(x) Overlap integral ∫dx Ψ*(x)ϕn(x) = number ∫dx Ψ*(x) Ψ (x)=1 means function Ψ (x) is normalized ∫dx ϕn*(x) ϕm(x)=0 orthogonal functions Function space with normalization… Complete basis set: All orthonormal functions ϕn(x) that solve an eigenvalue equation  ϕn(x) = an ϕn(x); any function can be expressed as a sum of them. 5 Getting coefficient cn: ∫dx ϕn*(x) Ψ(x) = cn 6 The dual space element: ∫dx ϕn*(x)__ 7 Operator: transforms a function into another: χ(x) =  Ψ(x)
They are called “operators” 8 Matrix element: the number Matrix element: the number .
†.
wn A wm = Anm ∫dx ϕn*(x)  ϕm(x) = Anm is the matrix element Anm of the matrix A. is the matrix element Anm of the operator  . .
9 Eigenvalue problems: A v = a v Eigenvalue problems: Â Ψ(x) = a Ψ(x) 10 Explicit vector/matrix notation: Explicit vector/matrix notation: ⎛ A11 A12 " ⎞
⎛ c1 ⎞
⎛ A11 A12 # ⎞
⎛ c1 ⎞
⎛ 0 ⎞
⎛ 0 ⎞
w 2 = ⎜ 1 ⎟ A = ⎜⎜ A21 A22 " ⎟⎟ v = ⎜⎜ c2 ⎟⎟ ϕ 2 ! ⎜ 1 ⎟ Â ! ⎜ A21 A22 # ⎟ Ψ ! ⎜ c2 ⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜⎝ !
⎜⎝ ! ⎟⎠
⎜⎝ "
⎜⎝ " ⎟⎠
⎝ ! ⎠
⎝ " ⎠
! # ⎟⎠
" $ ⎟⎠
*
*
†
*
*
*
v i = ( c1 c 2 ! ) i ∫ dxΨ (x)__ = ( c1 c 2 ! ) i Dirac vector-­‐function notation ket |Ψ> or |n> bracket <Ψ|n> = number <Ψ|Ψ>=1 means ket |Ψ> is normalized <n|m>=0 orthogonal bracket …also called Hilbert space Complete basis set: All orthonormal kets that solve the eigenvalue equation  |n> = an |n>; any ket can be expressed as a sum of them. Getting coefficient cn: <n|Ψ > = cn The dual space element: (called ‘bra’) <n| Operator: transforms a ket into another: |χ> =  |Ψ>
They are called “operators” Matrix element: the number <n| Â |m> = Anm is the matrix element Anm of operator A. Eigenvalue problems: Â |Ψ> = a |Ψ> Explicit vector/matrix notation: ⎛ c1
⎛ 0 ⎞
⎛ A
A12 # ⎞
11
| 2 >! ⎜ 1 ⎟ Â ! ⎜⎜ A21 A22 # ⎟⎟ | Ψ >! ⎜⎜ c2
⎜
⎟
⎜
⎟
"
$ ⎠
⎝ "
⎜⎝ "
⎝ " ⎠
< Ψ |= ( c1
*
c 2 ! ) i *
⎞
⎟
⎟
⎟⎠