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Chapter 7: Q83CE (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 mass m1, 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 鈥淏ohr radius鈥� 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=Z2渭mE1n2

Where, zis the atomic number, 渭is 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=Z2渭mE1n2=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鈥檚 Radius

rn=mz渭a0

Where, a0is radius of hydrogen atom

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

rn=mZ渭a0=mZ12ma0=0.106nm

Hence, Bohr Radius of the positronium is 0.106 nm.

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Most popular questions from this chapter

Question: An electron is trapped in a cubic 3D well. In the states (nx,ny,nz)= (a) (2,1,1) (b) (1,2,1)(c) (1,1,2), what is the probability of finding the electron in the region 0猢�x猢�L,L/3猢�y猢�2L/3,0猢�z猢�L. Discus any difference in these results.

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?
  1. What are the initial and final energy levels for the third (i.e., third-longest wavelength) line in the Paschen series? (See Figure 7.5)
  2. Determine the wavelength of this line.

An electron confinedtoa cubic 3D infinite well 1 nu on aside.

  1. What are thethree lowest differentenergies?
  2. To how many different states do these three energies correspond?

A mathematical solution of the azimuthal equation (7-22) is 桅(蠁)=Ae-顿蠁+Be-顿蠁 , which applies when D is negative, (a) Show that this simply cannot meet itself smoothly when it finishes a round trip about the z-axis. The simplest approach is to consider 蠁=0 and 蠁=2蟺. (b) If D were 0, equation (7-22) would say simply that the second derivative 桅(蠁)of is 0. Argue than this too leads to physically unacceptable solution, except in the special case of 桅(蠁) being constant, which is covered by the ml=0 , case of solutions (7-24).
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