Chemistry 221A Fall, 2014. Problem Set 1

Due Thursday Sept. 4 2014. 1. Consider a particle in a box of length L. Using the general form of the eigenfunctions (with

quantum number n), evaluate the following expectation values to show that: (a) x = L 2

(b) x ! x( )2 = L2

12 1!6

" 2n2#$%

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You may want to think a little about how the large n value compares with classical mechanics.

2. Consider the problem of a potential barrier of finite width, d, and height V. Consider a traveling wave originating from the left of the barrier and propagating towards the barrier with energy E < V. (a) Evaluate the probability of the particle tunneling through to the other side, as a function

of mass m, V, E, and d. Defining !! = 2m V " E( ) , you should obtain:

P = 1+ e!d " e"!d( )2

16 1" E V( ) E V( )#

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(b) Suggest how this model problem might be applied to model a scanning tunneling microscope and suggest roughly what the corresponding physical parameters might be (and therefore the corresponding tunneling probability).

Problems from the textbook: Ch. 1, problem number 2, 6