Question

2. Electron overlap with nucleus (very important for electron capture): Since the possible position of the electron is smudged out, it even may overlap with the nucleus. a) What is the probability of an electron in the (1,0,0) state being between r-0 and ao? b) What is the probability of an electron in the (1,0,0) state being between r-0 and 1.25 fm? (remember how to solve integrals for a very small interval, see example 5.3) Example 5.3 Consider again an electron trapped in a one-dimensional (b For this wide interval, we must use the integration region of length 1.00 x 10-10 m 0.100 nm. (a) In the method to find the probability: ground state, what is the probability of finding the electron in the region from x = 0.0090 nm to 0.0110 nm? (b) In the first excited state, what is the probability of finding the electron between x = 0 and x = 0.025 nm? Solution (a) When the interval is sma is often simpler to use Eq. 5.7 to find the probability, instead of using the integration method. The width of the sma inter- val is dr= 0.01 10 nm-0.0090 nm= 0.0020 nm. Evalu- ating the wave function at the midpoint of the interval (x= 0.0 100 nrn), we can use the n= 1 wave function with Eq. 5.7 to find sin- dx T X 4π L Evaluating this expression using the limits x =0 and x,-0.025 nm gives a probability of 0.25 or 25%. This result is of course what we would expect by inspection of the graph of for n = 2 in Figure 5.11 . The interval from x 0 to x = L/4 contains 25% of the total area under the T(0.0100nm(0.002 nm) curve 0.100 nm 0.100 nm = 0.0038 = 0.38%

Answer 1

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