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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 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, z is 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

For states where l = n - t the radial probability assumes the general form given in Exercise 54. The proportionality constant that normalizes this radial probability is given in Exercise 64.

(a) Show that the expectation value of the hydrogen atom potential energy is exactly twice the total energy. (It turns out that this holds no matter what l may be)

(b) Argue that the expectation value of the kinetic energy must be the negative of the total energy.

Consider two particles that experience a mutual force but no external forces. The classical equation of motion for particle 1 is v藱1=F2on1/m1, and for particle 2 is v藱2=F1on2/m2, where the dot means a time derivative. Show that these are equivalent to v藱cm=constant, and v藱rel=FMutual/渭 .Where, v藱cm=(m1v藱1+m2v藱2)/(m1m2),FMutual=-Fion2and渭=m1m2(m1+m2).

In other words, the motion can be analyzed into two pieces the center of mass motion, at constant velocity and the relative motion, but in terms of a one-particle equation where that particle experiences the mutual force and has the 鈥渞educed mass鈥� 渭.

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.

A particular vibrating diatomic molecule may be treated as a simple harmonic oscillator. Show that a transition from that n=2state directly to n=0ground state cannot occur by electric dipole radiation.

Taking then=3states as representative, explain the relationship between the complexity numbers of nodes and antinodes-of hydrogen's standing waves in the radial direction and their complexity in the angular direction at a given value of n. Is it a direct or inverse relationship, and why?
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