(a) What is the time-reversed state corresponding to D(R)|j,m)? (b) Using the properties of time reversal and rotations, prove DU (R)=(-1)-D (R). (c) Provej,m)=²m|j₁-m).
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- This problem is based on problems from Carroll & Ostlie and uses data from Chabrier et al. (2000, ApJ, 542, 464) and Schaller et al. (1992, AASS, 96, 269). The lower end of the main = 3.23 and log1o(L/Lo) = -4.3. On the other sequence occurs near 0.072M0, with log10 Teff hand, an 85MO star near the upper end of the main sequence has an effective temperature and luminosity of log10 Teff = 4.705 and log10(L/Lo) = 6.006, respectively. Assume that the 0.072M. star is entirely convective so that, through convective mixing, all of its hydrogen, rather than just the inner 10%, becomes available for burning. (a) ends of the main sequence. If the less massive star is Star A and the more massive star is Star B, what is the ratio tA/tB of their lifetimes? How do these lifetimes compare to the Estimate the hydrogen-burning lifetimes of stars near the lower and upper hydrogen-burning lifetime of the Sun? (b) RB/RA of their radii? Calculate the radii of both stars in units of solar radii. What is…(b) Write a necessary condition for a transformation (q,p) to (Q,P) to be connonical. Prove that P-2(1+√qcosp)√q sinp:Q-log(1+√qcosp)Consider the length of the simple pendulum is 1 and the mass of the pendulum bob is m . Obtain the equation for the Lagrangian for the simple pendulum when its kinetic energy is m² ø? and the potential energy is mg1(1-cos ø) . 1 (A) - mi g* – mg}(1– cos ø) 2 1 (В) — mi ф - mgl (1-cos ф) 1 (С) — mі ф + mgl(1-сos ) 1 (D) – ml ở – 2mgl (1– cos ø)
- 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).One-dimensional harmonic oscillators in equilibrium with a heat bath (a) Calculate the specific heat of the one-dimensional harmonic oscillator as a function of temperature (b) Plot the T -dependence of the mean energy per particle E/N and the specific heat c. Show that E/N → kT at high temperatures for which kT > hw In this limit the energy kT is large in comparison to hw , the separation between energy levels. Hint: expand the exponential function 1 ē = ħw + eBhwA particle experiences a potential energy given by U(x) = (x² - 3)e-x² (in SI units). (a) Make a sketch of U(x), including numerical values at the minima and maxima. (b) What is the maximum energy the particle could have and yet be bound? (c) What is the maximum energy the particle could have and yet be bound for a considerable length of time? (d) Is it possible for a particle to have an energy greater than that in part (c) and still be "bound" for some period of time? Explain. Responses
- Let F = (z^2 cos y, −xz^2 sin y, 2xz cos y − cos z).a) Show that F is irrotational.b) Find a potential function f (x, yz) such that F = ∇f , and f (0, π, π/3) = 2The Lagrangian equation of motion for a system of spring and a mass m connected by a pulley(disk) of radius a as shown in the figure is given by: (a) ë- 28 , 2g , 2ke = 0 За Зт (b) ӧ+ За 2g , 2ko = 0 3m (с) ө— За 2g , ke = 0 2g, ko (d) ô + За m 0 =(5) When variables p, V,T are related by F(p, V,T) = 0, show the following: (a) ),- = 1/), ƏT V др (b) ),+), ), = 0 ƏT ),H,O, = -1 (Chain relation) (c) T