Chapter6_1 - Quantum Mechanics: Give up the idea of...

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PHY3063 R. D. Field Department of Physics Chapter6_1.doc University of Florida How Do We Interpret Pilot Waves? Probabilistic Interpretation: In 1926 Max Born suggested that we interpret the “pilot waves” as “probability amplitudes” called “wave functions” where ) , ( t r r Ψ = “probability amplitude” and 2 ) , ( ) , ( t r t r r r Ψ = ρ = “probability density” such that r d t r 3 ) , ( r is the probability of finding the particle at time t within the small region d 3 r = dxdydz about the point r r . Must require that the overall probability of finding the particle somewhere be finite: finite N r d t r r d t r allSpace allSpace = = Ψ = 3 2 3 ) , ( ) , ( r r Now normalize the probability of finding the particle somewhere in space to one: 1 ) , ( 3 2 = Ψ allSpace N r d t r r , where N t r t r N / ) , ( ) , ( r r Ψ = Ψ . However, must require that Ψ (x,y,z,t) be a “square integrable” function for all values of t. Note that the probability amplitude is a complex function , but the probability density is a real number and Probability = |probability amplitude| 2
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Unformatted text preview: Quantum Mechanics: Give up the idea of predicting the result of a single measurement! Can determine only the probability of each of the many possible outcomes. Knowing the probabilities allows one to calculate average quantities and standard deviations. Rolling the Dice: It is like rolling a dice. You know there is a one in six chance of rolling a one, so that if you roll the dice millions of times then, on the average, the value one will occur 1/6 of the time. However, on a given roll you do not know which of the six numbers will appear. Gauge Invariance: Note that the following two wave functions describe the same state: ) , ( ) , ( 1 t r t r r r = and ) , ( ) , ( 2 t r e t r i r r = . Wave Packet-3-2-1 1 2 3 x v group At fixed time t Probability amplitude! Many amplitudes correspond to the same probability! Also require to be continuous and single-valued!...
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This note was uploaded on 05/29/2011 for the course PHY 3063 taught by Professor Field during the Spring '07 term at University of Florida.

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