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05_04_18 - 146 Normalization and probability density...

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[email protected] 146 04/18/2005 Normalization and probability density x d & 3 Probability to find a particle in the volume element is given by Probability to find the particle somewhere is one: Examples: x d t x & & 3 2 | ) , ( | Ψ 1 | ) , ( | 3 2 = Ψ x d t x & & 1 | ) , ( | 2 = Ψ ∫ ∫ ∫ - - - dxdydz t x & 1 sin | ) , , , ( | 0 0 2 0 2 2 = Ψ ∫∫ ∫ dr d d r t r ϑ ϕ ϑ ϕ ϑ π π Spherical symmetric wave functions: Probability to find a particle with a radius between r and r+dr : For a stationary state this is 1 | ) , ( | 4 0 2 2 = Ψ dr r t r π dr r t r 2 2 | ) , ( | 4 Ψ π dr r u dr r r 2 2 | ) ( | 4 | ) ( | 4 π π = Φ is not normalized to 1 but to 1/4 S±² ) ( r u π 4 1 0 2 | ) ( | = dr r u
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[email protected] 147 04/18/2005 Expectation values The most probable value of r is given by 0 ) ( = r u r The average radius after many measurements in identically prepared states: After N measurements one might have measured a radius r 1
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