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Chapter 7: Q11CQ (page 278)

Question: Explain to your friend. who has just learned about simple one-dimensional standing waves on a string fixed at its ends, why hydrogen's electron has only certain energies, and why, for some of those energies, the electron can still be in different states?

Short Answer

Expert verified

Answer:

If the energy levels are limited, still waves can have different orientations and may result in a different state but of the same energy.

Step by step solution

01

Significance of the one-dimensional standing waves

One-dimensional standing waves can be observed when a medium is having its opposite ends fixed, and nodes are located at the endpoints. The simplest example of the one-dimensional standing wave will be the one having only one antinode in the middle. This will be half of the wavelength.

02

Identification of reason for electrons having the same energy but different states

The electron of hydrogen behaves as a wave of the bound state, that is why they have only certain energy or certain allowed standing waves. But when multiple dimensions are introduced in space, it is possible to have different standing waves but still have the same frequency/energy.

Assume a square wave of two dimensions for instance, the energy/frequency of the wave will be the same for both the following conditions,

(i) when the wave contains two waves along 鈥揳xis and one wave along -axis

(ii) when the wave contains two waves along -axis and one wave along 鈥揳xis.

Thus, if the energy levels are limited, still waves can have different orientations and may result in a different state but of the same energy.

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Most popular questions from this chapter

If the constantCxinequation(7-5)were positive, the general mathematical solution would be

Ae+cxx+Be-cxx

Show that this function cannot be 0 at two points. This makes it an unacceptable solution for the infinite well, since it cannot be continuous with the wave functions outside the walls, which are 0.

For the more circular orbits, 鈩�=n-1and

P(r)鈭�r2ne-2r/na0

a) Show that the coefficient that normalizes this probability is

localid="1660047077408" (2na0)2n+11(2n)!

b) Show that the expectation value of the radius is given by

r炉=n(n+12)a0

and the uncertainty by

螖r=na0n2+14

c) What happens to the ratio螖r/r炉in the limit of large n? Is this large-n limit what would be expected classically?

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鈭�鈭�r(r鈭�鈭�r)蠄(r,胃)-h2m1r2鈭�2鈭�胃2蠄(r,胃)+U(r)蠄(r,胃)=E蠄(r,胃)

(a) Separate variables by trying a solution of the form 蠄(r,胃)=R(r)鈯�(胃), then dividing byR(r)鈯�(胃) . Show that the 胃equation can be written

d2d胃2鈯�(胃)=C鈯�(胃)

Here,(C) is the separation constant.

(b) To be physically acceptable,鈯�(胃) 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鈯�(胃) .

(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.

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

Imagine two classical charges of -q, each bound to a central charge of. +q One -q charge is in a circular orbit of radius R about its +q charge. The other oscillates in an extreme ellipse, essentially a straight line from it鈥檚 +q charge out to a maximum distance rmax.The two orbits have the same energy. (a) Show thatrmax=2r. (b) Considering the time spent at each orbit radius, in which orbit is the -q charge farther from its +q charge on average?
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