Phy107Lect26 - From Last Time Particle can exist in...

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Mon. Nov 8 Phy107 Lecture 26 From Last Time… • Particle can exist in different quantum states, having — Different energy — Different momentum — Different wavelength • The quantum wavefunction describes wave nature of particle. • Square of the wavefunction gives probability of finding particle. • Zero’s in probability arise from interference of the particle wave with itself.
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Mon. Nov 8 Phy107 Lecture 26 Back to the particle in a box • Here is the probability of finding the particle along the length of the box. • Can we answer the question: Where is the particle? Wavefunction Probability = (Wavefunction) 2
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Mon. Nov 8 Phy107 Lecture 26 Where is the particle? • Can say that the particle is inside the box, (since the probability is zero outside the box), but that’s about it. • The wavefunction extends throughout the box, so particle can be found anywhere inside. • Can’t say exactly where the particle is , but I can tell you how likely you are to find at a particular location.
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Mon. Nov 8 Phy107 Lecture 26 How fast is it moving? • Box is stationary, so average speed is zero. • But remember the classical version • Particle bounces back and forth. – On average, velocity is zero. – But particle is still moving. – Velocity (momentum) even changes sign (direction) L
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Mon. Nov 8 Phy107 Lecture 26 Quantum momentum • Quantum version is similar. • Ground state is a standing wave, made equally of – Wave traveling right ( positive momentum + h / λ ) – Wave traveling left ( negative momentum - h / λ ) • Can’t say exactly what the momentum is either! λ = 2 L One half- wavelength p = h = h 2 L momentum L
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Mon. Nov 8 Phy107 Lecture 26 Uncertainty in Quantum Mechanics Position uncertainty = L Momentum uncertainty from h λ to + h = 2 h = h L ( Since =2L ) Reducing the box size reduces position uncertainty, but the momentum uncertainty goes up! L = 2 L One half- wavelength The product is constant: (position uncertainty)x(momentum uncertainty) ~ h
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Phy107 Lecture 26 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 ) The exact value of the product depends on the problem (particle in a box, pendulum, hydrogen atom) This is a minimum uncertainty relation. Planck’s
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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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Phy107Lect26 - From Last Time Particle can exist in...

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