Phy107Spr07Lect26 - R e m in d e r e s s a y to p ic a n d...

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1 Mon. Mar. 26, 2007 Phy107 Lect 26 1 From Last Time… Matter particles are waves with wavelength λ = h / p . Matter waves can behave as particles by making a superposition (wave packet) Leads to uncertainty principle: Spread in position and spread in momentum ‘trade off’ Reminder: essay topic and paragraph due Wed. March 28 Mon. Mar. 26, 2007 Phy107 Lect 26 2 Making a particle out of waves 440 Hz + 439 Hz 440 Hz + 439 Hz + 438 Hz 440 Hz + 439 Hz + 438 Hz + 437 Hz + 436 Hz Mon. Mar. 26, 2007 Phy107 Lect 26 3 Spatial extent of localized sound wave Δ x = spatial spread of ‘wave packet’ • Spatial extent decreases as the spread in included wavelengths increases. -8 -4 0 4 8 -15 -10 -5 0 5 10 15 J Δ x Mon. Mar. 26, 2007 Phy107 Lect 26 4 Same occurs for a matter wave • Construct a localized particle by adding together waves with slightly different wavelengths. • Since de Broglie says λ = h / p , each of these components has slightly different momentum. – We say that there is some ‘uncertainty’ in the momentum • And still don’t know exact location of the particle! – Wave still is spread over Δ x (‘uncertainty’ in position) – Can reduce Δ x, but at the cost of increasing the spread in wavelength (giving a spread in momentum). Mon. Mar. 26, 2007 Phy107 Lect 26 5 Interpreting • For sound, we would just say that the sound pulse is centered at some position, but has a spread. • Can’t do that for a quantum-mechanical particle. • Many measurements indicate that the electron is indeed a point particle. • Interpretation is that the magnitude of electron ‘wave- pulse’ at some point in space determines the probability of finding the electron at that point. -8 -4 0 4 8 -15 -10 -5 0 5 10 15 J Mon. Mar. 26, 2007 Phy107 Lect 26 6 Heisenberg Uncertainty Principle • Using Δ x = position uncertainty Δ p = momentum uncertainty • Heisenberg showed that the product ( Δ x ) ( Δ p ) is always greater than ( h / 4 π ) Often write this as where is pronounced ‘ h -bar’ Planck’s constant " x ( ) " p ( ) ~ h /2 h = h 2 "
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2 Mon. Mar. 26, 2007 Phy107 Lect 26 7 Example: ‘Particle in a box’ Particle confined to a fixed region of space e.g. ball in a tube- ball moves only along length L Classically , ball bounces back and forth in tube. No friction, so ball continues to bounce back and forth,
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This note was uploaded on 01/05/2012 for the course PHYS 107 taught by Professor N/a during the Spring '08 term at University of Wisconsin.

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Phy107Spr07Lect26 - R e m in d e r e s s a y to p ic a n d...

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