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Problem 1:

The ground state wavefunction for a particle in a one-dimensional box of length L is

\psi & = & \sqrt{\frac{2}{L}} sin \frac{\pi x}{L}

Suppose the box is 10.0nm long. Calculate the probability that the particle is between x=4.95 nm and 5.05 nm.


The probability of finding the particle between x and x + dx is given by \begin{eqnarray*}
\vert\psi\vert^{2}dx & &
\end{eqnarray*}. Therefore the required probability is

\int_{x=4.95}^{5.05} {\vert\psi\vert^{2}}dx & = & \int_{x=4.95...
...}\int_{x=4.95}^{5.05} (1-cos \frac{2\pi x}{L}) dx \\
& = & 0.02

Problem 2:

The ground state wave function of the hydrogen atom is given by

\psi & = & (\frac{1}{\pi a_{0}^{3}})^{1/2}e^{-r/a_0}

a0 = 0.53A0 is the Bohr radius. Calculate the probability that the electron would be found somewhere within a small sphere of radius 1pm centred at the nucleus.


The required probability is

\int_{r=0}^{1pm} \int_{\theta=0}^{\pi} \int_{\phi=0}^{2\pi} {\...
...ta d\theta \int_{\phi=0}^{2\pi} d\phi \\
& = & \\
& = & 9.0E-6

Abhijit Poddar