## 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 x_{1} and x_{2} (x_{2} > x_{1}) is

^{-ax}dx, where 0 < x < $$\infty $$ , a > 0, then the probability that x lies between x

_{1}and x

_{2}(x

_{2}> x

_{1}) 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 s_{z} representation. What is the probability
of, finding the z-component of its spin along the $$ - {\bf{\hat Z}}$$ direction?

_{z}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

## 7. A parallel beam of electrons of a given momentum pass through a screen S_{1} containing a slit and then produces a diffraction pattern on a screen S_{2} placed behind it. The width of the central maximum observed on the screen S_{2} can be increased by

_{1}containing a slit and then produces a diffraction pattern on a screen S

_{2}placed behind it. The width of the central maximum observed on the screen S

_{2}can be increased by

## 8. An exact measurement of the position of a Simple Harmonic Oscillator (SHO) is made with the result x = x_{0}, [The SHO has energy levels E_{n} (n = 0, 1,
2, . . .) and associated normalized wave functions $${\psi _{\text{n}}}$$ ]. Subsequently, an exact measurement of energy E is made, using the general notation Pr(E = E') denoting the probability that a result E' is obtained for this measurement, the following statements are written. Which one of the following statements is correct?

_{0}, [The SHO has energy levels E

_{n}(n = 0, 1, 2, . . .) and associated normalized wave functions $${\psi _{\text{n}}}$$ ]. Subsequently, an exact measurement of energy E is made, using the general notation Pr(E = E') denoting the probability that a result E' is obtained for this measurement, the following statements are written. Which one of the following statements is correct?

## 9. An atomic state of hydrogen is represented by following wave function $$\psi \left( {r,\,\theta ,\,\phi } \right) = \frac{1}{{\sqrt 2 }}{\left( {\frac{1}{{{a_0}}}} \right)^{\frac{3}{2}}}\left( {1 - \frac{r}{{2{a_0}}}} \right){e^{\frac{{ - r}}{{2{a_0}}}}}\cos \theta $$ where, a_{0} is a constant. The quantum numbers of the state are

_{0}is a constant. The quantum numbers of the state are