Ch3_dubson_version - SJP QM 3220 Formalism 1 The Formalism...

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SJP QM 3220 Formalism 1 The Formalism of Quantum Mechanics : Our story so far … State of physical system: normalizable Ψ( x, t ) Observables: operators    x , p = h i x , H Dynamics of Ψ: TDSE    i h Y t = H Y To solve, 1 st solve TISE:    H y = Ey Solutions are stationary states ψ n (x), E n ═►special solutions of TDSE: Ψ n ( x , t ) = y n ( x ) e - iE n t h TDSE linear ═►any linear combo. of solutions is also a solution. Discrete case: Ψ ( x , t ) = c n e - iE n t / h y n ( x ) n ( n =1,2,3,L ) Continuum case: Ψ ( x , t ) = dk f ( k ) e - iw ( k ) t y k (x) { ( k any real number) e + ikx / 2 p ψ n 's , ψ k 's form complete, orthonormal sets: dx y m * y n = d mn dx y k ' * y k = 1 2 p dx e i ( k - k ') x = d ( k - k ') Page F-1 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 Discrete Discrete Continuum Discrete + Continuum
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SJP QM 3220 Formalism 1 Notice similarity of Ψ 's to vectors; Vector ρ V // complex function Ψ( x, t ) Scalar real number a // complex number c Any linear combination of vectors is a vector ω C = a v A + b v B // Ψ = α Ψ 1 + b Y 2 Orthonormal basis vectors x x =1 , x y = 0 // y m * y n dx = d mn ρ V = V x x + V y y + V z z // Ψ = c n y n n V x = x r V // c n = dx y n * Y Inner product ρ A r B = A i B i i = x , y , z // dx Y * F = dx d m y m m * c n y n n = d m * m . n c n y m * y n dx d mn 1 2 4 3 4 = d n * n c n The space of all complex, square-integrable functions Ψ(x) is called Hilbert Space. Norm ρ V r V = r V 2 // Y * Y dx < Hilbert Space is an infinite-dimensional vector space with complex scalars and normalizable vectors. Page F-2 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008
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SJP QM 3220 Formalism 1 Postulate 1: Every possible physical state of a system corresponds to a normed vector in Hilbert Space. The correspondence is 1-to-1 except that vectors that differ by a phase factor (scalar of modulus 1) corresponds to the same state Ψ( x, t ) ◄═► e Ψ( x, t ) Dirac Notation : dx - + f * ( x ) g ( x ) = f g = complex number => f g = g f * f f is real, non - negative c any complex number: f c g = c f g c f g = c * f g Postulate 2: (to be stated shortly!) associates with every observable a linear, hermitean operator. But first, a little background: Definition: An operator Q is hermitean (or hermitian, both spellings are common) if f Qg = Qf g for all f, g in Hilbert space (H - space). Which can be written (in position representation) as dx f * ( Qg ) = dx ( Q f ) * g Question: Is the operator Q = d dx ( ) Hermitian? (The answer will be no.) Let’s see why! dx f * dg dx parts 1 2 4 3 4 ? = dx df dx * g f * ( x ) g ( x ) | - + 0 1 2 4 4 3 4 4 - d dx ( f * ) g ( x ) dx = - df dx * g ( x ) dx So the answer is NO, there’s an extra unwanted minus sign that cropped up. It is NOT the case that for this particular operator, that f Qg = Qf g . Instead, we found here f Qg = - Qf g and that means Q is NOT hermitian.
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