Consider the potential V(x) = (x³/3)-x². Whatare itsextrema? Which of them describes a stable situationinclassical physics and in quantum mechanics? Explain youranswer.

To find the extrema I found the where the derivative wasequalto zero and they were 0 and 2. The part I was confusedaboutwas which of them describes a stable situation in classicalphysicsand quantum mechanics.

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anuradha.muthunayakeLv9

23 Aug 2023

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The Sun is powered by nuclear fusion. In each chain reaction, four hydrogen nuclei fuse together to become a single helium nucleus. The total combined mass of the individual hydrogen nuclei before the reaction is greater than the mass of the helium nucleus. The mass lost as a result of this chain reaction is turned into energy according to Einstein’s famous equation, E = mc² (Chapter 39). Consider the first part of the chain reaction in which two hydrogen nuclei (each a single proton) fuse. Figure P8.56A shows the potential energy curve of a two-proton system. The energy in this graph is given in keV, where “eV” is the abbreviation for the unit electron-volt and 1 eV = 1.6 × 10−19 J. When the protons are more than about 2 fm (2 × 10−15 m) apart, the potential energy is positive. When the protons are close together, the potential energy is negative, and the protons are fused. Figure P8.56B shows a close-up of the potential energy curve when the protons are far apart.

a. What is the minimum mechanical energy required for the two protons to fuse? Give your answer in keV.
b. The mechanical energy of an average two-proton system in the Sun is around 2 keV. What is the minimum distance between the protons in such a system?
c. Explain why it seems impossible for fusion to take place in the Sun. Note: Of course, fusion must take place in the Sun. Although it seems impossible according to classical mechanics, quantum mechanics (Part VI) makes fusion possible. According to quantum mechanics, the seemingly impossible is merely improbable, and even events with a low probability of occurrence may still happen. Because there are so many protons in the Sun, there is a good chance that at any moment many of them will fuse.

erinherring569

Part 2 is here :

http://www.cramster.com/answers-nov-10/physics/rotating-pendulum-part-2-rotating-pendulum-part-ii-note-part-httpww_1015121.aspx

This is the same rotating pendulum of the previous two problems.Now it is time to analyze the diï¬erent types of motion andequilibrium (according to the initial conditions).
Note: The âeï¬ective potentialâ method is a powerful tool to analyzeand solve problems where the Hamiltonian is conserved. A similartechnique is used for central potentials, including the Keplerproblem of celestial mechanics (two-body gravitationalproblem)âwill see that later.
(a) Let us call Eeï¬ â¡ H the âeï¬ective energy.â To make it look likea real energy, deï¬ne Eeï¬ = T2 + Veï¬ , where Veï¬ should include therelevant non-quadratic piece of kinetic energy (which eï¬ectivelybehaves, with an appropriate sign, as a potential energycontribution). In short, the eï¬ective potential is deï¬ned so thatthe eï¬ective kinetic energy left in the Hamiltonian is justquadratic.
Your task: Find the corresponding eï¬ective potential Veï¬ (Î¸), i.e.,write it down explicitly as a function of Î¸.
Hint: Using the resolution T = T0 + T1 + T2 , it should follow thatVeï¬ = V â T0 (using
Eulerâs theorem for homogeneous functions).
(b) Analyze all possible shapes of the graph of the eï¬ectivepotential Veï¬ (Î¸) (versus Î¸). For this purpose, simply use calculusand the techniques for energy graphs that we studied withone-dimensional systems. Speciï¬cally, you should plot Veï¬ (Î¸) andï¬nd the extrema.
Hint: There should be two diï¬erent regimes (or two âfamiliesâ ofshapes).
(c) At what rotation rate Ïc does the stationary point at Î¸ = 0become unstable?
(d) For Ï > Ïc , what is the stable equilibrium value ofÎ¸?
Explain the physical meaning of this stable âdynamic equilibriumâvalue. Also explain in simple terms why Î¸ = 0 is a stable âdynamicequilibriumâ position for Ï < Ïc .
(e) What is the frequency Î© of small oscillations about theâdynamic equilibriumâ points (for each of the two regimesdiscovered above)? Compute this frequency in two diï¬erentways:
(i) using the eï¬ective potential; and (ii) linearizing the equationof motion.