Answer:
P = 1/8
Explanation:
The wave function of a particle in a one-dimensional box is given by:
[tex] \psi = \sqrt \frac{2}{L} sin(\frac{n \pi x}{L}) [/tex]
Hence, the probability of finding the particle in the one-dimensional box is:
[tex] P = \int_{x_{1}}^{x_{2}} \psi^{2} dx [/tex]
[tex] P = \int_{x_{1}}^{x_{2}} (\sqrt \frac{2}{L} sin(\frac{n \pi x}{L}))^{2} dx [/tex]
[tex] P = \frac{2}{L} \int_{x_{1}}^{x_{2}} (sin^{2}(\frac{n \pi x}{L}) dx [/tex]
Evaluating the above integral from x₁ = 0 to x₂ = L/8 and solving it, we have:
[tex] P = \frac{2}{L} [\frac{L}{16} (1 - 4\frac{sin(\frac{n \pi}{4})}{n \pi})] [/tex]
[tex] P = \frac{1}{8} (1 - 4\frac{sin(\frac{n \pi}{4})}{n \pi}) [/tex]
Solving for n=4:
[tex] P = \frac{1}{8} (1 - 4\frac{sin(\frac{4 \pi}{4})}{4 \pi}) [/tex]
[tex] P = \frac{1}{8} (1 - \frac{sin (\pi)}{\pi}) [/tex]
[tex] P = \frac{1}{8} [/tex]
I hope it helps you!
The total probability of finding a particle in this one-dimensional box is [tex]\frac{1}{8}[/tex]
Given the following data:
To determine the total probability of finding a particle in a one-dimensional box:
A particle in a one-dimensional box describes the translational motion of a particle that is trapped inside an infinitely deep well, from which it is unable to escape.
Mathematically, the wave function of a particle in a one-dimensional box is given by this formula:
[tex]\psi = \sqrt{\frac{2}{L} } sin\frac{n\pi}{L} x[/tex] ...equation 1.
Where:
In a one-dimensional box, the probability of finding a particle is given by the formula:
[tex]P=\int\limits^{x_2}_{x_1} {\psi^2} \, dx[/tex] ...equation 2.
Substituting eqn. 1 into eqn. 2, we have:
[tex]P=\int\limits^{x_2}_{x_1} {(\sqrt{\frac{2}{L} } sin\frac{n\pi}{L} x)^2} \, dx \\\\P = \frac{2}{L} \int\limits^{x_2}_{x_1} {( sin^2(\frac{n\pi}{L} x))} \, dx\\\\P=\frac{2}{L} [\frac{L}{16} (1-4(\frac{sin\frac{n\pi}{4} }{n\pi} ))]\\\\P=\frac{1}{8} (1-4(\frac{sin\frac{n\pi}{4} }{n\pi} ))[/tex]
Substituting the value of n, we have:
[tex]P=\frac{1}{8} (1-4(\frac{sin\frac{4\pi}{4} }{4\pi} ))\\\\P=\frac{1}{8} (1-(\frac{sin\pi }{\pi} ))\\\\P=\frac{1}{8} (1-0)\\\\P=\frac{1}{8}[/tex]
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