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Chapter 5: Q 8CQ (page 186)

Equation 5-16 gives infinite well energies. Because equation 5-22 cannot be solved in closed form, there is no similar compact formula for finite well energies. Still many conclusions can be drawn without one. Argue on largely qualitative grounds that if the walls of a finite well are moved close together but not changed in height, then the well must progressively hold fewer bound states. (Make a clear connection between the width of the well and the height of the walls).

Short Answer

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Answer:

When the walls of our infinite well are drawn closer together, the wavelengths of standing waves shorten, and the particle is no longer imprisoned in that condition.

Step by step solution

01

Theory of Infinite well energy

The wavelengths of the standing waves get shorter when the walls of our infinite well are pulled closer together, suggesting greater momentum and, in turn, kinetic energy. The kinetic energies of some states will exceed the height of the potential energy walls as the walls become closer together, and the particle will no longer be trapped in that state.

02

Theory for wall energy

When the walls of our infinite well are brought closer together, the wavelengths of the standing waves shorten, implying more momentum and, hence, kinetic energy. As the potential energy barriers move closer together, the kinetic energies of some states will exceed the height of the potential energy walls, and the particle will no longer be imprisoned in that state.

As the walls are brought together the particle will no longer be imprisoned in that state.

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Most popular questions from this chapter

Repeat the exercise 60-62 for the first excited state of harmonic oscillator.

Using equation (23), find the energy of a particle confined to a finite well whose walls are half the height of the ground-state infinite well energy, . (A calculator or computer able to solve equations numerically may be used, but this happens to be a case where an exact answer can be deduced without too much trouble.)

For a total energy of 0, the potential energy is given in Exercise 96. (a) Given these, to what region of the x-axis would a classical particle be restricted? Is the quantum-mechanical particle similarly restricted? (b) Write an expression for the probability that the (quantum-mechanical) particle would be found in the classically forbidden region, leaving it in the form of an integral. (The integral cannot be evaluated in closed form.)

In several bound systems, the quantum-mechanically allowed energies depend on a single quantum number we found in section 5.5 that the energy levels in an infinite well are given by, En=a1n2wheren=1,2,3.....andis a constant. (Actually, we known whata1is but it would only distract us here.) section 5.7 showed that for a harmonic oscillator, they areEn=a2(n鈭�12), wheren=1,2,3.....(using ann鈭�12with n strictly positive is equivalent towith n non negative.) finally, for a hydrogen atom, a bound system that we study in chapter 7,En=鈭�a3n2, wheren=1,2,3.....consider particles making downwards transition between the quantized energy levels, each transition producing a photon, for each of these three systems, is there a minimum photon wavelength? A maximum ? it might be helpful to make sketches of the relative heights of the energy levels in each case.

Consider a particle of mass mand energy E in a region where the potential energy is constant U0. Greater than E and the region extends tox=+鈭�

(a) Guess a physically acceptable solution of the Schrodinger equation in this region and demonstrate that it is solution,

(b) The region noted in part extends from x = + 1 nm to +鈭�. To the left of x = 1nm. The particle鈥檚 wave function is Dcos (109m-1 x). Is also greater than Ehere?

(c) The particle鈥檚 mass m is 10-3 kg. By how much (in eV) doesthe potential energy prevailing from x=1 nm to U0. Exceed the particle鈥檚 energy?
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