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Chapter 7: Q85CE (page 285)

The expectation value of the electron’s 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=3π2h22mL2

Where, his Planck’s 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=3π2h22mL2L=πh32m×13.6eV=3.14(1.055×10-34J.s)2(9.11×10-31kg(13.6×1.6×1019J)/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

What are the dimensions of the spherical harmonics Θl,ml(θ)Φml(ϕ)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?

Residents of flatworld-a two-dimensional world far, far away-have it easy. Although quantum mechanics of course applies in their world, the equations they must solve to understand atomic energy levels involve only two dimensions. In particular, the Schrodinger equation for the one-electron flatrogen atom is

-h2m1r∂∂r(r∂∂r)ψ(r,θ)-h2m1r2∂2∂θ2ψ(r,θ)+U(r)ψ(r,θ)=Eψ(r,θ)

(a) Separate variables by trying a solution of the form ψ(r,θ)=R(r)⊝(θ), then dividing byR(r)⊝(θ) . Show that the θequation can be written

d2dθ2⊝(θ)=C⊝(θ)

Here,(C) is the separation constant.

(b) To be physically acceptable,⊝(θ) must be continuous, which, since it involves rotation about an axis, means that it must be periodic. What must be the sign of C ?

(c) Show that a complex exponential is an acceptable solution for⊝(θ) .

(d) Imposing the periodicity condition find allowed values ofC .

(e) What property is quantized according of C .

(f) Obtain the radial equation.

(g) Given thatU(r)=-b/r , show that a function of the formR(r)=er/a is a solution but only if C certain one of it, allowed values.

(h) Determine the value of a , and thus find the ground-state energy and wave function of flatrogen atom.

Consider a cubic 3D infinite well.

(a) How many different wave functions have the same energy as the one for which (nx,ny,nz)=(5,1,1)?

(b) Into how many different energy levels would this level split if the length of one side were increased by 5% ?

(c) Make a scale diagram, similar to Figure 3, illustrating the energy splitting of the previously degenerate wave functions.

(d) Is there any degeneracy left? If so, how might it be “destroyed”?

A hydrogen atom electron is in a 2p state. If no experiment has been done to establish a z-component of angular momentum, the atom is equally likely to be found with any allowed value of LZ. Show that if the probability densities for these different possible states are added (with equal weighting), the result is independent of both ϕandθ

Knowing precisely all components of a nonzero L→would violate the uncertainty principle, but knowingthat L→is precisely zerodoes not. Why not?

(Hint:For l=0 states, the momentum vector p→ is radial.)
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