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hw05key - Chem 455 Fall 2012 Homework 5 due Mon Oct 29...

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Chem 455 Fall 2012 Homework 5 due Mon Oct 29 5:00pm (in box 65 on 3 rd floor) 1. Tunneling. An electron with energy 0.200 eV runs into a potential step of height 3.50 eV. What is the penetration depth of the electron? The penetration depth is 1/ K , with . For an electron with m = m e , E = 0.200 eV and V 0 = 3.50 eV, this gives K = 9.31 nm -1 and 1/ K = 0.107 nm. 2. Tunneling. Consider an electron with 0.800 eV kinetic energy approaching a 1D potential barrier of height 1.00 eV and width 1.10 nm. What is the probability the particle tunnels across the barrier? Using the expressions we get K = 2.29 nm -1 , q = 1.25, and therefore T = 0.0165, so above one percent. 3. Tunneling. The maximum safe current in a copper wire with a diameter of 3.0 mm is about 20 A. For higher currents, the wire melts. In an STM (scanning tunneling microscope), a current of 0.50 nA passes from the tip to the surface in a filament of diameter 1.2 nm. Compare the current density (current per cross-section area, in A mm -2 ) in the copper wire with that in the STM. Copper: STM tip: The current density in the STM tip is over 150 times larger than in the copper wire. 4. Operators. Do the operators for kinetic energy ̂ and for linear momentum ̂ commute? To find out, evaluate the commutator ̂ ̂ by applying it to an arbitrary function. Let’s write out the commutator of the two operators: [ ̂ ̂] ̂ ̂ ̂ ̂ Next, we evaluate each of the two terms separately: ̂ ̂ ̂ ̂ ̂ ̂
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