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?

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

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

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?

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?

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

A parallel beam of electrons of a given momentum pass through a screen S1 containing a slit and then produces a diffraction pattern on a screen S2 placed behind it. The width of the central maximum observed on the screen S2 can be increased by

An exact measurement of the position of a Simple Harmonic Oscillator (SHO) is made with the result x = x0, [The SHO has energy levels En (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?

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, a0 is a constant. The quantum numbers of the state are

A spinless particle moves in a central potential V(r).