Answer:
(a) 4.98x10⁻⁵
(b) 7.89x10⁻⁶
(c) 1.89x10⁻⁴
(d) 0.5
(e) 2.9x10⁻²
Step-by-step explanation:
The probability (P) to find the particle is given by:
[tex] P=\int_{x_{1}}^{x_{2}}(\Psi\cdot \Psi) dx = \int_{x_{1}}^{x_{2}} ((2/L)^{1/2} Sin(\pi x/L))^{2}dx [/tex]
[tex] P = \int_{x_{1}}^{x_{2}} (2/L) Sin^{2}(\pi x/L)dx [/tex] (1)
The solution of the intregral of equation (1) is:
[tex] P=\frac{2}{L} [\frac{X}{2} - \frac{Sin(2\pi x/L)}{4\pi /L}]|_{x_{1}}^{x_{2}} [/tex]
(a) The probability to find the particle between x = 4.95 nm and 5.05 nm is:
[tex] P=\frac{2}{100} [\frac{X}{2} - \frac{Sin(2\pi x/100)}{4\pi /100}]|_{4.95}^{5.05} = 4.98 \cdot 10^{-5} [/tex]
(b) The probability to find the particle between x = 1.95 nm and 2.05 nm is:
[tex] P=\frac{2}{100} [\frac{X}{2} - \frac{Sin(2\pi x/100)}{4\pi /100}]|_{1.95}^{2.05} = 7.89 \cdot 10^{-6} [/tex]
(c) The probability to find the particle between x = 9.90 nm and 10.00 nm is:
[tex] P=\frac{2}{100} [\frac{X}{2} - \frac{Sin(2\pi x/100)}{4\pi /100}]|_{9.90}^{10.00} = 1.89 \cdot 10^{-4} [/tex]
(d) The probability to find the particle in the right half of the box, that is to say, between x = 0 nm and 50 nm is:
[tex] P=\frac{2}{100} [\frac{X}{2} - \frac{Sin(2\pi x/100)}{4\pi /100}]|_{0}^{50.00} = 0.5 [/tex]
(e) The probability to find the particle in the central third of the box, that is to say, between x = 0 nm and 100/6 nm is:
[tex] P=\frac{2}{100} [\frac{X}{2} - \frac{Sin(2\pi x/100)}{4\pi /100}]|_{0}^{16.7} = 2.9 \cdot 10^{-2} [/tex]
I hope it helps you!
