851HW6_09Solutions - PHYS851 Quantum Mechanics I Fall 2009...

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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 6 1. [10 points] The quantum state of a free-particle of mass, M , at time t is a wave-packet of the form ψ ( x,t ) = 1 radicalbig Γ(5 / 4) σ e − ( x- x ) 4 2 σ 4 + ip x/ planckover2pi1 , We can safely predict that the width of the wave packet will grow in time. Clearly, the spreading velocity v s must be determined by the initial conditions x , p , and σ , as well as the mass, M , and Planck’s constant, planckover2pi1 , as there are no other parameters around to use. (a) [2 pts] If there are no external forces acting on the particle, which parameters can we rule out based on symmetry arguments ? The parameters x and p are depend on choice of coordinate system and inertial frame. The spread velocity of the wavefunction is not frame dependent. The basic equations for a free particle are invariant under boost and translation, therefore the spreading dynamics should not depend on frame-dependant parameters. Thus we conclude v s does not depend on x and p . (b) [2 pts] Of the remaining parameters, how many unique ways are there to combine them to make an object with units of a velocity? The remaining parameters are planckover2pi1 , M , and σ . Thus units of planckover2pi1 are kgm 2 s − 1 , thus any combination giving a velocity must depend only on planckover2pi1 /M , so that kg is cancelled. planckover2pi1 /M has units m 2 s − 1 , and the only parameter left is σ , which has units of length. Since we need a s − 1 for velocity, the only possibility is v = planckover2pi1 Mσ . (1) (c) [2 pts] Based on this result alone, give a units-based estimate for the velocity at which the wave-packet should spread. The spread velocity must be v s ∼ planckover2pi1 Mσ , (2) as there are no other possibilities. (d) [2 pts] Again, by considering units alone, what energy scale, E s , would you associate with a wave-packet of width σ ? From the width σ , and the constants planckover2pi1 and M , the only energy we can form is E s = planckover2pi1 2 Mσ 2 . (3) (e) [2 pts] We can assign a temperature to the wave-packet by setting E s = k B T . Solve this equation for σ as a function of temperature, T . This is known as the thermal de Broglie wavelength, or the thermal coherence length, usually denoted as λ coh . It gives the length-scale on which a particle at temperature T exhibits spatial coherence (quantum superposition). If we set k B T = planckover2pi1 2 / ( Mσ 2 ), we find λ coh = planckover2pi1 √ Mk B T . (4) This increases as the temperature decreases, which makes some kind of intuitive sense. 1 2. [10 points] The mass of a small virus is about 10 − 21 kg. What is the thermal coherence length of the virus at room temperature? What is it a liquid Helium temperature?...
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This note was uploaded on 12/04/2011 for the course PHY 7070 taught by Professor Smith during the Spring '11 term at University of Wisconsin.

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851HW6_09Solutions - PHYS851 Quantum Mechanics I Fall 2009...

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