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Chapter 7: Q83E (page 284)

Exercise 81 obtained formulas for hydrogen like atoms in which the nucleus is not assumed infinite, as in the chapter, but is of massm1, whilem2is the mass of the orbiting negative charge. In positronium, an electron orbits a single positive charge, as in hydrogen, but one whose mass is the same as that of the electron -- a positron. Obtain numerical values of the ground state energy and “Bohr radius” of positronium.

Short Answer

Expert verified

(a) The ground State energy of positronium is -6.8 eV .

(b) Bohr radius of positronium is 0.106 nm .

Step by step solution

01

Energy of ground-state of positronium

As you know from that, the ground state energy is,
Eground=Z2μmE1n2

Where, z is the atomic number, μis the Reduced mass, n is the principal quantum number, m is the mass, E1is the Energy of ground state of hydrogen atom.

You also know that the electron and positron have same masses.

Hence, the reduced mass will be half of the mass of electron
Eground=Z2μmE1n2=1212mmE112=-6.8eV

Hence, ground state energy of the positronium is -6.8 eV .

02

Bohr Radius of the positronium

As you know that,

The Bohr’s Radius
rn=mzμa0

Where, a0is radius of hydrogen atom

If the reduced mass will be half of the mass of electron,

rn=mZμa0=mZ12ma0=0.106nm

Hence, Bohr Radius of the positronium is 0.106 nm .

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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.

Question: The kinetic energy of hydrogen atom wave functions for which lis its minimum value of 0 is all radial. This is the case for the 1s and 2s states. The 2 p state has some rotational kinetic energy and some radial. Show that for very large n, the states of largest allowed lhave essentially no radial kinetic energy. Exercise 55 notes that the expectation value of the kinetic energy (including both rotational and radial) equals the magnitude of the total energy. Compare this magnitude with the rotational energy alone,L2/2mr2
,assuming that n is large. That lis as large as it can be, and thatr≡n2a0.

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?

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).

Doubly ionized lithium, Li2+absorbs a photon and jumps from the ground state to its n=2level. What was the wavelength of the photon?
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