1. A particle is placed in a one-dimensional box of size L along the X-axis, (0 < x < L). Which of the following is true?
2. If the probability that x lies between x and x + dx is P(x) dx = ae-ax dx, where 0 < x < $$\infty $$ , a > 0, then the probability that x lies between x1 and x2 (x2 > x1) is
3. A Michelson interferometer is illuminated with monochromatic light. When one of the mirrors is moved through a distance of 25.3 μm, 92 fringes pass through the cross-wire. The wavelength of the monochromatic light is
4. An electron is in a statewith spin wavefunction \[{\phi _s} = \left[ {\begin{array}{*{20}{c}}
{\frac{{\sqrt 3 }}{2}} \\
{\frac{1}{2}}
\end{array}} \right]\] in the sz representation. What is the probability
of, finding the z-component of its spin along the $$ - {\bf{\hat Z}}$$ direction?
5. If the wave function of a particle trapped in space between x = 0 and x = L is given by $$\psi \left( x \right) = A\sin \left( {\frac{{2\pi x}}{L}} \right),$$ where A is a constant, for which value(s) of x will the probability of finding the particle be the maximum?
6. The normalized eigen states of a particle in a one-dimensional potential well \[V\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
{0,}&{{\text{if }}0 \leqslant x \leqslant a} \\
{\infty ,}&{{\text{otherwise}}}
\end{array}} \right.\] are given by $${\psi _n}\left( x \right) = \sqrt {\frac{2}{a}} \sin \left( {\frac{{n\pi x}}{a}} \right)$$ where, n = 1, 2, 3, . . .
The particle is subjected to a perturbation $$\eqalign{
& V'x = {V_0}\cos \left( {\frac{{\pi x}}{a}} \right),\,{\text{for }}0 \leqslant x \leqslant \frac{a}{2} \cr
& \,\,\,\,\,\,\,\,\,\,\, = 0,\,\,\,{\text{otherwise}} \cr} $$
The shift in the ground state energy due to the perturbation, in the first order perturbation theory, is
The particle is subjected to a perturbation $$\eqalign{ & V'x = {V_0}\cos \left( {\frac{{\pi x}}{a}} \right),\,{\text{for }}0 \leqslant x \leqslant \frac{a}{2} \cr & \,\,\,\,\,\,\,\,\,\,\, = 0,\,\,\,{\text{otherwise}} \cr} $$
The shift in the ground state energy due to the perturbation, in the first order perturbation theory, is