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Unformatted text preview: ket Notation | If , | | flipped and starred, i.e. transpose and conjugate ∗ | ∗ expression equals same expression starred – only true if real 4 9/16/2013 Postulate 2 pq qp commutator h 2 i from original paper Zeitschrift für Physik, 43 (1927), 172-198 multiply by 1 in the form of i/i ̂, In classical physics, p and q are variables and the commutator is zero In QM, they are operators as indicated with the “hats”. If we choose , then ̂ Whenever the operators of two variables do not commute (don’t equal zero), then an uncertainty principle applies ̂, ̂, pick x-component ̂ ̂ ̂, ̂ hard to work with operators by themselves, operate on an arbitrary function ̂ ̂ ̂ ̂ ̂ operators operate from right to left ̂ assume position representation ̂ see what happens when ̂ differentiate by parts 1st and 3rd term cancel satisfies commutator relation Whenever the operators of two variables do not commute (don’t equal zero), , then then an uncertainty principle applies, such as ̂ , xp x 2 (1.26) How do we get the uncertainties? • choose form for position operator, get operator for momentum • with operators, calculate expectation values for <x>,<x2>,<p>,<p2> • uncertainties are x x2 x 2 p p2 p 2 5 9/16/2013 from Postulate 2 Operators, so far … x-component of kinetic energy position x-component of momentum ̂ 2 potential energy i energy (Hamiltonian) x-component of momentum squared 2 ̂ Postulate 3 In any measurement of the observable ˆ associated with the operator A , the only values that will ever be observed are the eigenvalues a n , which satisfy the ˆ eigenvalue equation A n an n Particle-in-a-Box Example energy is so important in QM, that the operator for time independent Schrödinger Eq. energy has it’s own name, the , 1,2, … Hamiltonian wavefunctions or eigenfunctions eigenvalues ( x) 2 nx sin 0 x a n 1,2,... a a Orthonormal set of functions for solving problems with other operators, , with Postulate 4 ∗ Postulate 4 If a system is in a state described by a normalized wave function , then th...
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