SJP QM 3220 Spin 1 Page S-1 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 complete orthonormal set € ψ= cnn∑n , ψ→ cn{ }=c1c2c3cn                  € u1=100            u2=010            If ket's are represented by column vectors, then bra's are represented by the transpose conjugate of column = row, complex conjugate. Operators can be represented by matrices: no hat on matrix element where {| n >} is some complete orthonormal set. Matrix Formulation of QMSJP QM 3220 Spin 1 Page S-2 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 Why is that? Where does that matrix come from? Consider the operator and 2 state vectors related by (  ) In basis {| n >} , € ψ= cnn∑n = nn∑nψcn   ϕ= dnn∑n = nn∑nϕdn   Now project equation  onto | m > by acting with bra: € mϕ = mˆ A ψ = cnn∑mˆ A ndn= Amnn∑ cn But, this is simply the rule for multiplication of matrix  column. € d1d2d3            = A11A12…A21A22            c1c2            So there you have it, that's why the operator is defined as this matrix, in this basis! Now, suppose are energy eigenstates, then A matrix operator is diagonal when represented in the basis of its own eigenstates, and the diagonal elements are the eigenvalues. Notice that in general operators don't commute . Same goes for Matrix Multiplication: A B ≠ B ASJP QM 3220 Spin 1 Page S-3 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 Claim: The matrix of a hermitian operator is equal to its transpose conjugate: Proof: € mˆ A n =ˆ A m n = nˆ A m*⇒ Amn= Anm* Similarly, adjoint (or "Hermitian conjugate") Proof: € ˆ A m n = mˆ A tn = nˆ A m* Of course, it's difficult to do calculations if the matrices and columns are infinite dimensional. But there are Hilbert subspaces that are finite dimensional. For instance, in the H-atom, the full space of bound states is spanned by the full set {n, ℓ, m} (= | nℓm>). The sub-set {n=2, ℓ=1, m = +1, 0, -1} forms a vector space called a subspace. Subspace? In ordinary Euclidean space, any plane is a subspace of the full volume. If we consider just the xy components of a vector , then we have a perfectly valid 2D vector space, even though the "true" vector is 3D. Likewise, in Hilbert space, we can restrict our attention to a subspace spanned by a small number of basis states. Example: H-atom subspace {n=2, ℓ=1, m = +1, 0, -1} Basis states are (can drop n=2, ℓ=1 in label since they are fixed.) € ˆ L zm =  m mˆ L 2m = 2( + 1) m( = 1)= 22m (for all m) € ⇒ Lz( )mn= mˆ L zn =  +1 0 00 0 00 0 −1          € L2mn= mˆ L 2n = 2 1 0 00 1 00 0 1         SJP QM 3220 Spin 1 Page S-4 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 (What about Lx? Ly?) Before seeing what all this matrix stuff is good for, let's examine spin because it's very important physically and because it will lead to 2D Hilbert space with simple 2 x 2 matrices. bracket or inner-product: Which integral you do depends on the configuration space of problem. Key defining properties of bracket:  <f | g >* = < g | f > c = constant  <f | c • g > = c < f | g > , < c • f | g > = c* < f | g >  <α | ( b| β > + c | γ >) = b < α | β > + c < α | γ > Dirac proclaims: < g | f > = < g | next to | f > bracket = "bra" and "ket" Ket | f > represents vector in H-space (Hilbert Space) "ket" "wavefunction" Both ψ and ψ(x) describe same state, but | ψ > is more general:  position -representation, momentum-rep, energy-rep. Review of Dirac Bra - Ket Notation Different "representations" of same H=space vector | ψ >SJP QM 3220 Spin 1 Page S-5 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 What is a "bra"? < g | is a new kind of mathematical object, called a "functional". insert state function here input output function: number number operator: function function functional: function numbe < g | wants to bind with | f > to produce inner product < g | f > For every ket | f > there is a corresponding bra < f |. Like the kets, the bra's form a vector space.  | c f > → < c f | = c* < f | (?)  | α f + β g > → < α f + β g | = α* < f | + β* < g |  < α f + β g | h > = α* < f | h > + β* < g | h >  Complex number  bra = another bra => bra's form any linear combo of bra's = another bra vector space The vector space of bras is called a "dual space". It's the dual of the ket vector space. is a ket. What is the corresponding bra? Definition: hermitean conjugate or adjoint € ˆ A f g ≡ f Atg for all f, g. (If is hermitean or self-adjoint.) Some properties:   € Proof : fˆ A t( )tg =ˆ A tf g =gˆ A tf*=ˆ A g f*= fˆ A f           Def'n of A†SJP QM 3220 Spin 1 Page S-6 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 The adjoint of an operator is analogous to complex conjugate of a complex number: The "ket-bra" | f > < g | is an operator. It turns a ket (function) into another ket (function): € f g ( ) h = f g h Projection Operators € ψ(x) = cnun(x) = unψ n∑n∑un(x) →ψ= cnnn∑= nψn∑ n = nn∑ nψ => "Completeness relation" (discrete spectrum case) = "projection operator" picks out portion of vector | ψ > that lies along | n > € ˆ P n ψ= n nψ= cnn |ψ > u2 = |2> u1 = |1> |2> <2|ψ> u1<u1|ψ> = |1> <1|ψ>SJP QM 3220 Spin 1 Page S-7 M. Dubson, (typeset by J. …