(a) for a function f() that can be expanded in a taylor series, show that where p is an arbitrary angle. for this reason, lz/h is called the generator of rotations about the z axis. hint: use eq. (4.129) and refer to problem 3.39 more generally, l n/h is the generator of rotations about the direction n, in the sense that explil-mp/h] produces a rotation of the angle in the right- hand sense about the n axis. for spins, the generator of rotations is s·n/h in particular, for spin 1/2, tells us how spinors rotate (b) construct the 2 × 2 matrix representing a rotation by 180° about the x-axis and show that it converts "spin up" (x+ into "spin down" ( as one would expect classically (c). construct the matrix representing a rotation by 90° about the y-axis, and check what it does to x+. (d). construct the matrix representing a rotation by 360° about the z-axis. if the answer is not exactly what you would have expected, discuss its implications (e). show that e'(a-n)v/2 cosp/2 ) + i(σ . n) sin(p/2) - cos

Aplane flies a distance d from west to east at a constant speed vp , with respect to the air in a time twe . on the return trip, the plane flies a distance d from east to west at a constant speed vp , with respect to the air in a time tew . on both trips the wind blows from west to east at a constant speed va , with respect to the ground 1) what is tew in terms of vp, va, and d, as needed? tew=dvp tew=dvp+va tew=dva tew=dvpâ’va tew=12dvp tew=12dvp+va tew=12dvpâ’va your submissions: b submitted: saturday, january 26 at 6: 49 pm feedback: feedback will be given once all questions have been attempted and the grade cluster button has been pressed. 2) what is twe in terms of vp, va, and d, as needed? twe=dvp twe=dvp+va twe=dva twe=dvpâ’va twe=12dvp twe=12dvp+va twe=12dvpâ’va your submissions: d submitted: saturday, january 26 at 6: 49 pm feedback: feedback will be given once all questions have been attempted and the grade cluster button has been pressed. 3) assuming d = 2300 miles, vp = 400 miles/hr, and twe = 4 hours, what is va, the speed of the wind with respect to the ground? va = 100 miles/hr va = 575 miles/hr va = 200 miles/hr va = 175 miles/hr va = 975 miles/hr 4) once again, assuming d = 2300 miles, vp = 400 miles/hr, and twe = 4 hours, what is tew, the time it takes the plane to fly a distance d miles from east to west? tew = 5.75 hours tew = 11.5 hours tew = 7.67 hours tew = 8 hours tew = 10.2 hours

The force acting on a beam was measured under the same operating conditions and a sample of 5 data points was collected. this process was repeated by 3 observers. the pooled standard deviations of these 3 data sets was 1.21. determine the true mean force using all pooled data (with a 95% probability). the result should look like this: true mean = mean +/- margin of error

Write the function getkthdigit(n, k) that takes a possibly-negative int n and a non-negative int k, and returns the kth digit of n, starting from 0, counting from the right