Important Questions from Waves

1. Which of the following is related to Doppler effect?

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2. The velocity v(x) of a particle moving in one dimension is given by v(x) = v 0 sin \(\rm\left(\frac{\pi x}{x_0}\right) \) , where v 0  and x 0  are positive constants of appropriate dimensions. If the particle is initially at x/x 0  = ϵ, where |ϵ| ≪ 1, then, in the long time, it
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3. The position of a particle in one dimension changes in discrete steps. With each step it moves to the right, however, the length of the step is drawn from a uniform distribution from the interval \(\left[ {{\rm{λ }}\,{\rm{ - }}\,\frac{{\rm{1}}}{{\rm{2}}}{\rm{w,}}\,{\rm{λ }}\,{\rm{ + }}\,\frac{{\rm{1}}}{{\rm{2}}}{\rm{w}}} \right] \) , where λ and w are positive constants. If X denotes the distance from the starting point after N steps, the standard deviation \(\sqrt {\left\langle {{X^2}} \right\rangle \, - {{\left\langle X \right\rangle }^2}} \)  for large values of N is

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4. A particle of mass m in one dimension is in the ground state of a simple harmonic oscillator described by a Hamiltonian \(\frac{{{{\rm{P}}^{\rm{2}}}}}{{{\rm{2m}}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{m}}{{\rm{\omega }}^{\rm{2}}}{{\rm{x}}^{\rm{2}}} \) in the standard notation. An impulsive force at time t = 0 suddenly imparts a momentum P₀ =  \(\sqrt {{\rm{hm\omega }}} \) to it. The probability that the particle remains in the original ground state is

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5. In an elastic scattering process at an energy E, the phase shifts satisfy δ ₀ ≈ 30°, δ ₁≈  10°, while the other phase shifts are zero. The polar angle at which the differential cross-section peaks is closest to

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