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- prove that LzPnem (r,0,0) = mhynem (r, 0, 0)Your video analysis of the motion of a marble gives it position in frame 23 as (x23, Y23) = (0.134 m, 0.120 m) and its position in frame 24 as (x24, 324) = (0.122 m, 0.112 m). You esti- mate that you can measure the r and y positions with uncertainty +0.003 m. The frame rate of the video is 30 frames/s, which means the time interval between frames is At = 0.033 333 s. The uncertainty of the frame rate of a video camera is VERY small. For the sake of this prob- lem, use 8(At) = 1 x 10-6 s. The mass of the marble is (2.031 +0.001) x 10-2 kg. Calculate the following quantities: 4. The value of the momentum component p = mv, and its uncertainty, 5. The value of the momentum component py = muy and its uncertainty, and 6. The value of the kinetic energy K = mv² = m(v² + v²) and its uncertainty.(a) The cross-section for scattering of muons with air at atmospheric pressure is 0.1 barns, and the natural lifetime of muons 2.2 x 10-6 s. Explain what is meant by the terms elastic scattering and lifetime. Which of these fac- tors limits the distance over which a beam of muons can travel in air in a laboratory, if the muon velocity is 106 ms-¹? (Assume number density of air at STP = 2.69 × 1025 m−³) (b) Given your answer to (a), why is it possible to detect showers of muons at ground level caused by the impact of primary cosmic ray particles with air at around 12 km altitude? Given that the mean energy of muons detected at ground level is 4 GeV, calculate the distance (in air) over which the number of such muons in a beam would reduce by a factor of e. What kind of interactions contribute to the scattering cross-section for these particles? [1 barn = 10-28 m²; mass of muon = 105 MeV/c²; number density of air at STP = 2.69 x 1025 m-³]
- A particle is measured to be in the interval 0 < x < x0. According to the many-worlds interpretation, what is the best way to describe where the particle was immediately before the measurement was performed?Show that the spherical harmonics Y2,2(θ,φ)= ((15/32π)^1/2)*sin(2θ)*e^∓2iφ and Y3,3(θ,φ)= ((35/64π)^1/2)*sin(3θ)*e^∓3iφ are normalized.Find a formula for the temperature of an Einstein solid in the limit q « N . Solve for the energy as a function of temperature to obtain U = N€e-€/kT (where € is the size of an energy unit).
- Determine the probability distribution function in the phase space for a relativistic particle in a volume V and with energy ε(p) = √√√/m²c²+p²c², where p is the ab- solute value of the momentum, m the mass, and c the speed of light. Give the final result in terms of the modified Bessel functions r+∞ Ky (z) = ™ (v-1)! 2 -zcosht e cosh (vt) dt Ky(z) ~ Check what happens in the limit ² →0. mc² kT z 0.If k of n independent, randomly chosen test patterns are misclassified, then as given in Eq. 38 k has a binomial distribution Prove that the maximum likelihood estimate for p is then ˆp = k/n, as given in Eq. 39.The velocity olistribution is Ve 2xî - ayj + (3t-bz k). Is it steady? Find the eg. of the stream lines through the pt. (1,1,3) at t=o,1.
- A free neutron has a mean lifetime of 15 minutes when at rest. How fast must it moves to have a mean lifetime of 60 minutes?Consider a potential barrier represented as follows: U(x) = 0 if x < 0; εx if 0 < x < a; 0 if x > a Determine the transmission coefficient as a function of particle energy.(d) Prove that for a classical particle moving from left to right in a box with constant speed v, the average position = (1/T) ff x(t) dt = L/2, where T L/v is the time taken to move from left to right. And = : (1/T) S²x² (t) dt L²/3. Hint: Only consider a particle moving from left x = 0 to right x = L = and do not include the bouncing motion from right to left. The results for left to right are independent of the sense of motion and therefore the same results apply to all the bounces, so that we can prove it for just one sense of motion. Thus, the classical result is obtained from the Quantum solution when n >> 1. That is, for large energies compared to the minimum energy of the wave-particle system. This is usually referred to as the Classical Limit for Large Quantum Numbers.