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Chapter 5: Q20E (page 187)

A comet in an extremely elliptical orbit about a star has, of course, a maximum orbit radius. By comparison, its minimum orbit radius may be nearly 0. Make plots of the potential energy and a plausible total energyversus radius on the same set of axes. Identify the classical turning points on your plot.

Short Answer

Expert verified

The graph is given as,

Step by step solution

01

Given data

A comet has an extremely elliptical orbit about a star. The minimum value of the orbit radius is nearly zero as compared to the maximum orbit radius.

Write the expression for the potential energy of the comet.

Uc=-GMmr

Here, Uc is the potential energy of the comet, G is the gravitational constant, M is the mass of the star, m is the mass of the comet and r is the distance of the comet from the star.

02

Concept of potential energy

The potential energy of the comet varies inversely with its position.

03

Step 3:

The potential energy of the comet varies inversely with its position.

The diagram given below shows the variation of the potential energy and the total energy of the comet.

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Most popular questions from this chapter

Quantum-mechanical stationary states are of the general form Ψ(x,t)=ψ(x)e-iωt. For the basic plane wave (Chapter 4), this is Ψ(x,t)=Aeikxe-iωt=Aei(kx-ωt), and for a particle in a box it is Asinkxe-iωt. Although both are sinusoidal, we claim that the plane wave alone is the prototype function whose momentum is pure-a well-defined value in one direction. Reinforcing the claim is the fact that the plane wave alone lacks features that we expect to see only when, effectively, waves are moving in both directions. What features are these, and, considering the probability densities, are they indeed present for a particle in a box and absent for a plane wave?

Verify that solution (5-19) satisfies the Schrodinger equation in form (5.18).

Simple models are very useful. Consider the twin finite wells shown in the figure, at First with a tiny separation. Then with increasingly distant separations, In all case, the four lowest allowed wave functions are planned on axes proportional to their energies. We see that they pass through the classically forbidden region between the wells, and we also see a trend. When the wells are very close, the four functions and energies are what we might expect of a single finite well, but as they move apart, pairs of functions converge to intermediate energies.

(a) The energies of the second and fourth states decrease. Based on changing wavelength alone, argue that is reasonable.

(b) The energies of the first and third states increase. Why? (Hint: Study bow the behaviour required in the classically forbidden region affects these two relative to the others.)

(c) The distant wells case might represent two distant atoms. If each atom had one electron, what advantage is there in bringing the atoms closer to form a molecule? (Note: Two electrons can have the same wave function.)

Determine the particle’s most probable position.

Sketchψ(x) . Would you expect this wave function to be the ground state? Why or why not?
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