Ch3_dubson_version

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

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SJP QM 3220 Formalism 1 Page F-1 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 200 The Formalism of Quantum Mechanics : Our story so far … State of physical system: normalizable Ψ ( x, t ) Observables: operators ˆ x , ˆ p = i x , ˆ H Dynamics of Ψ : TDSE i ∂Ψ t = ˆ H Ψ To solve, 1 st solve TISE: ˆ H ψ = E Solutions are stationary states ψ n (x), E n ═► special solutions of TDSE: Ψ n ( x , t ) = n ( x ) e iE n t TDSE linear ═► any linear combo. of solutions is also a solution. Discrete case: Ψ ( x , t ) = c n e iE n t / n ( x ) n ( n = 1,2,3, ) Continuum case: Ψ ( x , t ) = dk φ ( k ) e i ω ( k ) t k (x) ( k any real number) e + ikx π ψ n 's , ψ k 's form complete, orthonormal sets: dx m * n = δ mn dx k ' * k = 1 2 dx e i ( k k ') x = ( k k Discrete Discrete Continuum Discrete + Continuum
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SJP QM 3220 Formalism 1 Page F-2 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 200 Notice similarity of Ψ 's to vectors; Vector V // complex function Ψ ( x, t ) Scalar real number a // complex number c Any linear combination of vectors is a vector C = a A + b B // Ψ = α Ψ 1 + β Ψ 2 Orthonormal basis vectors ˆ x ˆ x = 1 , ˆ x ˆ y = 0 // ψ m * n dx = δ mn V = V x ˆ x + V y ˆ y + V z ˆ z // Ψ = c n n n V x = ˆ x V // c n = dx n * Ψ Inner product A B = A i B i i = x , y , z // dx Ψ * Φ = dx d m m m * c n n n = d m * m . n c n m * n dx mn = d n * n c n The space of all complex, square-integrable functions Ψ (x) is called Hilbert Space. Norm V V = V 2 // Ψ * Ψ dx < Hilbert Space is an infinite-dimensional vector space with complex scalars and normalizable vectors.
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SJP QM 3220 Formalism 1 Page F-3 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 200 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 i θ 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 ˆ Q is hermitean (or hermitian, both spellings are common) if f ˆ Q g = ˆ Q f g for all f, g in Hilbert space (H-space). Which can be written (in position representation) as dx f * ( ˆ Q g ) = dx ( ˆ Q f ) * g Question: Is the operator ˆ Q = d dx ( ) Hermitian? (The answer will be no.) Let’s see why! dx f * dg dx parts ? = dx df dx * g f * ( x ) g ( x ) | −∞ + 0 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 ˆ Q g = ˆ Q f g .
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