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

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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 0 e ( x - x 0 ) 4 2 σ 4 0 + ip 0 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 0 , p 0 , and σ 0 , 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 0 and p 0 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 0 and p 0 . (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 0 . Thus units of planckover2pi1 are kg m 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 0 , which has units of length. Since we need a s 1 for velocity, the only possibility is v = planckover2pi1 0 . (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 0 , (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 0 ? From the width σ 0 , and the constants planckover2pi1 and M , the only energy we can form is E s = planckover2pi1 2 2 0 . (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 0 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 / ( 2 0 ), we find λ coh = planckover2pi1 Mk B T . (4) This increases as the temperature decreases, which makes some kind of intuitive sense. 1
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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? With k B 10 23 and planckover2pi1 10 34 , we find that at room temperature, T 300K, we find λ coh (300K) 10 34 ( 21 23+2) / 2 3 10 34+21 1 10 14 m . (5) At liquid Helium temperature, T 3K, the temperature decreases by two orders of magnitude, so the coherence length increases by one order, giving λ coh (3K) 10 13 m (6)
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