Lecture07 - Physics 344 Foundations of 21st Century Physics...

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Physics 344 Foundations of 21 st Century Physics: Relativistic and Quantum Systems Instructor: Dr. Mark Haugan Office: PHYS 282 [email protected] TA: Dan Hartzler Office: PHYS 7 [email protected] Grader: Fan Chen Office: PHYS 222 [email protected] 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: Chapters 6 and 8 in Six Ideas that Shaped Physics, Unit R.
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1 0 t t x x 10 1 x x = 10 1 t t The Lorentz Transformation Equations An inertial observer uses spacetime coordinates to represent where and when an event occurs in their frame. By thinking carefully about how inertial observers measure spacetime coordinates and using the Principle of Relativity to relate the outcomes of a few basic types of measurements made by observers in relative motion, we determined how the spacetime coordinates in two frames, S and S’ , with S’ in standard orientation relative to S, are related. The Lorentz transformation equations 1 1 1 ( ) x x Vt γ = - 1 1 1 2 V t t x c γ = - 1 1 y y = 1 1 z z = express the coordinates of an event measured in S’ in terms of the coordinates of the event measured in S . 10 1 t t 10 1 x x =
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1 1 1 ( ) x x Vt γ = - 1 1 1 2 V t t x c γ = - 1 1 y y = 1 1 z z = In recitation yesterday you showed that the inverse of the Lorentz transformation equations above are 1 1 1 ( ) x x Vt γ = + 1 1 1 2 V t t x c γ = + 1 1 y y = 1 1 z z = They express the coordinates representing an event in S in terms of the coordinates representing it in S’ . You verified, for example, that if the Lorentz transformation maps t 1 and x 1 to t’ 1 and x’ 1 , then the inverse transformation maps t’ 1 and x’ 1 back to t 1 and x 1 . Here is the calculation for t 1 , ( 29 1 1 1 1 1 1 1 2 2 2 2 2 2 2 1 1 1 1 1 1 2 2 2 2 = 1 V V V t t x t x x Vt c c c V V V V t x x t t t c c c c γ γ γ γ γ γ = + = - + - - + - = - =
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Velocity Transformation Equations By considering the effect of the Lorentz transformation equations on the coordinates of a pair of events on a particle’s world line, we could determine the relationship between the components of the particle’s velocity measured in S and S’ . For example, the x component measured in S is, by definition, t x 2 1 t’ x’ t 21 2 1 21 2 1 x x x x v t t t - = - which we can relate to the x’ component measured in S’ 21 2 1 21 2 1 x x x x v t t t - = - using the Lorentz transformation equations.
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2 2 2 ( ) x x Vt γ = - 2 2 2 2 V t t x c γ = - 1 1 1 ( ) x x Vt γ = - 1 1 1 2 V t t x c γ = - and Because of the common form of these equations we find the useful result that coordinate separations transform just like coordinates do,
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