ssp_1y - 0.2 Quantum mechanical axioms and operators...

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0.2 Quantum mechanical axioms and operators Classical mechanics tt r p i i = 0 :, of all particles define state Equation of motion () rt p t i i , for t t > 0 System determined for ever ! Note: r i and p i of a particle can be measured simultaneously and with infinite precision. Quantum mechanics Completely determined physical system: Not all quantities have sharp values! E.g.: Position of an electron in an atom Decay of a radioactive nucleus Statements can only be made in terms of probability! 8

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Axioms of quantum mechanics cf.: Newton axioms classical mechanics Maxwell axioms electrodynamics 1. State of a physical system state function ψ 2. Physical quantity operator A linear Hermitian 3. State in which a physical quantity has a sharp value f a with = a f 4. If ψ= cf ii than contribution of i to measurable quantities from contribution to 2 9
Average or expected value of a physical quantity A f i normalized eigenfunction of A : A f i = a i f i ψ= cf ii i ψ 2 2 = c i i c i : generalized Fourier coefficient Measurement of A : System is in a state f i with eigenvalue a i () aa c i i i == 2 ψψ A { δ AA ∑∑ ccf f ccaff i j ij i j ij ji j ij ** General case: a rr d r d r ψ A 2 3 3 * * A Dirac notation: ψ ψ nm n m or A 10

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0.2 Important operators Operators to be chosen in such a way that classical behavior in classical limit 1) Position operator: operator of the position x of a particle X ψ ψ = x 2) Momentum operator: p x i xi x ψ ψ ψ =− = h h Hermitian h == = −− hh J s e V s 2 6 626 10 4 136 10 34 15 π ,, , Eigenstates (states with sharp momentum) p xx i x p ψ ψψ = h ψ 00 ee ip x x ik x x h harmonic wave with wave number k p = h λ π 2 k h p de Broglie wave length 11
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ssp_1y - 0.2 Quantum mechanical axioms and operators...

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