Chapter6_all - PHY3063 R. D. Field How Do We Interpret...

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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 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 Ψ = Ψ and ) , ( ) , ( 2 t r e t r i r r Ψ = Ψ φ . Wave Packet -3 -2 -1 0 1 2 3 x v group At fixed time t Probability amplitude! Many amplitudes correspond to the same probability! Also require Ψ to be continuous and single-valued!
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PHY3063 R. D. Field Department of Physics Chapter6_2.doc University of Florida General Principles: Probability Amplitudes I. The probability of an event is given by the square of the absolute value of a complex number Ψ , which is called the “probability amplitude”: P = probability Ψ = “probability amplitude” P = | Ψ | 2 II. When an event can occur in several alternative ways, the probability amplitude is the sum of the probability amplitudes for each way considered separately: Ψ = Ψ 1 + Ψ 2 Ψ 1 + Ψ 2 | 2 = P 1 + P 2 +2Re( Ψ 1 Ψ 2 * ) P 1 + P 2 III . If an experiment is performed which is capable of determining whether one or the other alternative is actually taken, then the probability of the event is the sum of the probabilities for each alternative: P = P 1 + P 2 Example: Double Slit Experiment Let Ψ 1 be the amplitude for an electron to go from the source S through slit 1 and arrive at the point P on the screen. Let Ψ 2 be the amplitude for an electron to go from the source S through slit 2 and arrive at the point P on the screen.
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Chapter6_all - PHY3063 R. D. Field How Do We Interpret...

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