Lecture38

Lecture38 - Physics 344 Foundations of 21st Century...

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Physics 344 Foundations of 21 st Century Physics: Relativistic and Quantum Systems Instructor: Dr. Mark Haugan Office: PHYS 282 haugan@purdue.edu TA: Dan Hartzler Office: PHYS 7 dhartzle@purdue.edu Grader: Fan Chen Office: PHYS 222 chen926@purdue.edu Office Hours: If you have questions, just email us to make an appointment. We enjoy talking about physics! Help Session: Thursdays 2:00 – 4:00 in PHYS 154 Reading: Chapter 9 in the text. Re-read sections 8.5 and 8.6 Final Exam: Monday, December 12 at 10:20am in MATH 175
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Multiparticle Quantum Systems We developed a photon model of electromagnetic radiation using a complex- valued wavefunction that could predict and explain both the interference and photon-counting phenomena we observed. A wavefunction 2 2 2 2 2 2 2 2 2 2 1 ( , , , ) ( , , , ) ( , , , ) x y z t x y z t x y z t c t x y z Ψ = + + Ψ ≡ ∇ Ψ Photon wavefunctions satisfy the wave equation, ( ) ( , , , ) i k r t x y z t e ϖ ⋅ - Ψ a a represents a photon state with definite energy and momentum and E = p k = a a so that their energy and momenta are properly related, E 2 = p 2 c 2 . To predict and explain interference effects in electron scattering and other experiments involving massive particles we made an analogy to our photon model and began using wavefunctions to represent particle states. To impose the correct relationship between nonrelativistic particle kinetic energy and momentum, K = p 2 /2m , we require that these wavefunctions satisfy the free-particle Schrödinger equation, 2 2 2 2 2 2 2 ( , , , ) ( , , , ) 2 i x y z t x y z t t m x y z Ψ = - + + Ψ
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We touched briefly on multiparticle systems when we began to study interacting particles. For as system like the hydrogen atom with two different kinds of particles, a 2-particle wavefunction ( , , ) p e r r t Ψ a a specifies the joint probability for detecting the photon in the neighborhood of a particular location AND for detecting the electron in the neighborhood of a particular location at time t , * ( , , ) ( , , ) ( , , ) p e p e p e r r t r r t r r t ρ = Ψ Ψ a a a a a a Such 2-particle wavefunctions satisfy the Schrödinger equation ( 29 2 2 2 2 ( , , ) ( , , ) ( , , ) ( , , ) 2 2 p e p p e e p e p e p e p e i r r t r r t r r t V r r r r t t m m Ψ = - ∇ Ψ - ∇ Ψ + - Ψ a a a a a a a a a a but, as we saw in the case of vibrating diatomic molecules, we can separate such 2-particle situations into effective single-particle ones, one for the free motion of the system’s center of mass and one for the system’s internal state. ( 29
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Lecture38 - Physics 344 Foundations of 21st Century...

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