Physics 322: Modern Physics Spring 2015 Problem Set #8 (Having Tunneled Through Spring Break…) Due Friday, March 27 in lecture. ASSUMED READING: Before starting this homework, you should read all of Chapter 6 of Harris’ Modern Physics (focus on Sections 1 and 2) 1. [Harris 6.1 (tweaked)] Could the situation depicted in the diagram to the right represent a particle in a bound state? Clearly explain your reasoning. HINT: How would your answer change if the potential barriers were 2x as wide? 200x as wide? 2. [Harris 6.20 (modified)] Particles of energy ! E are incident from the left, where U(x)=0, and at the origin encounter an abrupt drop in potential energy, whose depth is ! −3E . a. Sketch this potential including the total energy of the particles on the same scale. b. Classically, what would the particles do. Specifically, what would happen to their kinetic energy as they move and why? c. Apply quantum mechanics. Assuming an incident wave of the form ! ψ inc ( x ) = 1eikx (where the normalization constant has been given the simple value of 1), determine completely the wave function everywhere that describes this particle stream, including numeric values for the multiplicative constants. d. What is the probability that incident particles will be reflected? 3. [Harris 6.15 (tweaked)] Calculate the reflection probability for a 5eV electron encountering a step in which the potential drops by 2eV. Clearly explain your reasoning. HINT: If you are doing a lot of math, you are approaching the problem incorrectly. The mathematical heavy lifting was done in Harris, you just need to explain its application here. – Page 1! of 3! – Physics 322: Modern Physics Spring 2015 4. [Harris 6.9] In the wide-barrier transmission probability of equation (6-18), E ⎛ E ⎞ −2 L 2 m (U0 − E ) /! ! T ≅ 16 ⎜ 1 − ⎟ e U0 ⎝ U0 ⎠ the coefficient multiplying the exponential, ! 16 E ⎛ E⎞ 1 − , is often U 0 ⎜⎝ U 0 ⎟⎠ omitted. When is this justified, and why? NOTE: Understanding the answer to this may simplify the next problem. 5. [Harris 6.32] Jump to Jupiter. The gravitational potential energy of a 1 kg object is plotted versus position from Earth’s surface to the surface of Jupiter. Mostly it is due to the Sun, but there are downturns at each end, due to the attractions of the two planets. Make a crude approximation that this is a rectangular barrier o f w i d t h 6 x 1 011 m a n d approximate height (in terms of energy per kilogram lifted) of 4×108 J/kg. Your mass is 65 kg, and you launch yourself from Earth at an impressive 4 m/s. What is the probability that you can jump to Jupiter? MATH NOTE: Since most of your calculators can’t work with numbers this small, to convert your exponential into a power of 10, use the fact that log e N ! log10 N = . log e 10 6. [Harris 6.35] Fusion in the Sun: Without tunneling, our Sun would fail us. The source of its energy is nuclear fusion, and a crucial step is the fusion of a light-hydrogen nucleus, which is a proton, and a heavyhydrogen nucleus, which is of the same charge but twice the mass. When these nuclei get close enough, their short range attraction via the strong force overcomes their Coulomb repulsion. This allows them to stick together, resulting in a reduced total mass/internal energy and a consequent release of kinetic energy. However, the Sun’s temperature is simply too low to ensure that nuclei move fast enough to overcome their repulsion. a. By equating the average thermal kinetic energy that the nuclei 3 would have when distant, ! kBT , an the Coulomb potential energy 2 – Page 2! of 3! – Physics 322: Modern Physics Spring 2015 they would have when 2 fm apart, roughly the separation at which they stick, show that a temperature of about 109 K would be needed. For full credit, explain why this is a good “classical” guess on the temperature necessary for fusion. b. The Sun’s core is only about 107 K. If nuclei can’t make it “over the top,” they must tunnel. Consider the following model, illustrated in the figure: One nucleus is fixed at the origin, while the other approaches from far away with energy E. As r decreases, the Coulomb potential energy increases until the separation r is roughly the nuclear radius rnuc, whereupon the potential energy is Umax and then quickly drops down into a very deep “hole” as the strong-force attraction takes over. Given that E≪Umax, the point b, where the tunneling must begin, will be very large compared with rnuc, so we approximate the barrier’s width L as simply b. Its height. U0, we approximate by the Coulomb potential evaluated at b/2. Finally, for the energy E, which fixes b, 3 let us use ! 4 × kBT , which is a reasonable limit, given the natural 2 range of speeds in a thermodynamic system. Combining these approximations, show that the exponential factor in the widebarrier tunneling probability (equation 6-18) is ⎡ −e2 4m ⎤ ! exp ⎢ ⎥. ⎣ ( 4π ε0 ) ! 3kBT ⎦ c. Using the proton mass for m, evaluate this factor for a temperature of 107 K (on the order of the temperature of the Sun’s core). Then evaluate it at 3000K, the temperature of an incandescent filament or hot flame, and rather high by Earth standards. Discuss the consequences. – Page 3! of 3! –
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