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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 –axis and one wave along -axis

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

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

For an electron in the(n,l,ml)=(2,0,0) state in a hydrogen atom, (a) write the solution of the time-independent Schrodinger equation,

(b) verify explicitly that it is a solution with the expected angular momentum and energy.

How many different 3d states are there? What physical property (as opposed to quantum number) distinguishes them, and what different values may this property assume?

Classically, an orbiting charged particle radiates electromagnetic energy, and for an electron in atomic dimensions, it would lead to collapse in considerably less than the wink of an eye.

(a) By equating the centripetal and Coulomb forces, show that for a classical charge -e of mass m held in a circular orbit by its attraction to a fixed charge +e, the following relationship holds

ω=er-3/24πε0m.

(b) Electromagnetism tells us that a charge whose acceleration is a radiates power P=e2a2/6ε0c3. Show that this can also be expressed in terms of the orbit radius as P=e696π2ε03m2c3r4. Then calculate the energy lost per orbit in terms of r by multiplying the power by the period T=2π/ωand using the formula from part (a) to eliminate .

(c) In such a classical orbit, the total mechanical energy is half the potential energy, or Eorbit=-e28πε0r. Calculate the change in energy per change in r : dEorbit/dr. From this and the energy lost per obit from part (b), determine the change in per orbit and evaluate it for a typical orbit radius of 10-10m. Would the electron's radius change much in a single orbit?

(d) Argue that dividing dEorbit/dr by P and multiplying by dr gives the time required to change r by dr . Then, sum these times for all radii from rinitial to a final radius of 0. Evaluate your result for rinitial=10-10m. (One limitation of this estimate is that the electron would eventually be moving relativistically).

Calculate the electric dipole moment p and estimate the transition time for a hydrogen atom electron making an electric dipole transition from the

(n,l,m)=(3,2,0)to the (2,1,0) state.

In general, we might say that the wavelengths allowed a bound particle are those of a typical standing wave,λ=2L/n , where is the length of its home. Given that λ=h/p, we would have p=nh/2L, and the kinetic energy, p2/2m, would thus be n2h2/8mL2. These are actually the correct infinite well energies, for the argumentis perfectly valid when the potential energy is 0 (inside the well) and is strictly constant. But it is a pretty good guide to how the energies should go in other cases. The length allowed the wave should be roughly the region classically allowed to the particle, which depends on the “height” of the total energy E relative to the potential energy (cf. Figure 4). The “wall” is the classical turning point, where there is nokinetic energy left: E=U. Treating it as essentially a one-dimensional (radial) problem, apply these arguments to the hydrogen atom potential energy (10). Find the location of the classical turning point in terms of E , use twice this distance for (the electron can be on both on sides of the origin), and from this obtain an expression for the expected average kinetic energies in terms of E . For the average potential, use its value at half the distance from the origin to the turning point, again in terms of . Then write out the expected average total energy and solve for E . What do you obtain

for the quantized energies?
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