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Chapter 7: Q85CE (page 285)

The expectation value of the electron鈥檚 kinetic energy in the hydrogen ground state equals the magnitude of the total energy (see Exercise 60). What must be the width of a cubic finite wall, in terms of a0, for this ground state to have this same energy?

Short Answer

Expert verified

The width of the cubic finite wall should be 0.29 nm, about five times the Bohr radii.

Step by step solution

01

A concept:

Energy of the cubic finite wall can be given by,

E1,1,1=3蟺2h22mL2

Where, his Planck鈥檚 constant, Lis the width of the wall, and mis the mass.

02

Width of the cubic finite wall:

Rewrite the equation for energy as below.
13.6eV=3蟺2h22mL2L=蟺h32m脳13.6eV=3.14(1.055脳10-34J.s)2(9.11脳10-31kg(13.6脳1.6脳1019J)/3=0.29nm

Hence, width of the wall is 0.29 nm .

03

Width in terms of a0:

As you know that,a0=0.05nm

Where,a0 is the radius of the hydrogen atom.

Hence, the obtained width of the wall is about five times that of the a0.

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Most popular questions from this chapter

In hydrogen鈥檚 characteristic spectra, each series - the Lyman, the Balmer, and so on 鈥� has a 鈥渟eries limit,鈥� where the wavelengths at one end of the series tend to bunch up, approaching a single limiting value. Is it at the short-wavelength or the long-wavelength end of the series that the series limit occurs, and what is it about hydrogen鈥檚 allowed energies that leads to this phenomenon? Does the infinite well have series limits?

Here we Pursue the more rigorous approach to the claim that the property quantized according to ml is Lz,

(a) Starting with a straightforward application of the chain rule,

鈭�鈭�蠁鈥�=鈥�鈭�x鈭�蠁/鈭�鈭�x+鈭�y鈭�蠁鈭�鈭�y+鈭�z鈭�蠁鈭�鈭�z

Use the transformations given in Table 7.2 to show that

鈭�鈭�蠁鈥�=鈥�-y鈭�鈭�x+x鈭�鈭�y

(b) Recall that L = r x p. From the z-component of this famous formula and the definition of operators for px and py, argue that the operator for Lz is -ih鈭�鈭�蠁..

(c) What now allows us to say that our azimuthal solutioneiml蠁 has a well-defined z-component of angular momentum and that is value mlh.

Roughly, how does the size of a triply ionized beryllium ion compare with hydrogen?

Some degeneracies are easy to understand on the basis of symmetry in the physical situation. Others are surprising, or 鈥渁ccidental鈥�. In the states given in Table 7.1, which degeneracies, if any, would you call accidental and why?

What are the dimensions of the spherical harmonics 螛l,ml(胃)桅ml(蠒)given in Table 7.3? What are the dimensions of theRn,l(r)given in Table 7.4, and why? What are the dimensions ofP(r), and why?
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