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Physics LibreTexts

8.E: Central Potentials (Exercises)

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  1. A particle of mass m is placed in a finite spherical well: V(r)={V0 for ra0 for r>a
    with V0>0 and a>0. Find the ground-state by solving the radial equation with l=0. Show that there is no ground-state if V0a2<π22/(8m).
  2. Consider a particle of mass m in the three-dimensional harmonic oscillator potential V(r)=(1/2)mω2r2. Solve the problem by separation of variables in spherical coordinates, and, hence, determine the energy eigenvalues of the system.
  3. The normalized wavefunction for the ground-state of a hydrogen-like atom (neutral hydrogen, He+, Li++, et cetera.) with nuclear charge Ze has the form ψ=Aexp(βr),
    where A and β are constants, and r is the distance between the nucleus and the electron. Show the following:
    1. A2=β3/π.
    2. β=Z/a0, where a0=(2/me)(4πϵ0/e2).
    3. The energy is E=Z2E0 where E0=(me/22)(e2/4πϵ0)2.
    4. The expectation values of the potential and kinetic energies are 2E and E, respectively.
    5. The expectation value of r is (3/2)(a0/Z).
    6. The most probable value of r is a0/Z.
  4. An atom of tritium is in its ground-state. Suddenly the nucleus decays into a helium nucleus, via the emission of a fast electron that leaves the atom without perturbing the extranuclear electron, Find the probability that the resulting He+ ion will be left in an n=1, l=0 state. Find the probability that it will be left in a n=2, l=0 state. What is the probability that the ion will be left in an l>0 state?
  5. Calculate the wavelengths of the photons emitted from the n=2, l=1 to n=1, l=0 transition in hydrogen, deuterium, and positronium.
  6. To conserve linear momentum, an atom emitting a photon must recoil, which means that not all of the energy made available in the downward jump goes to the photon. Find a hydrogen atom’s recoil energy when it emits a photon in an n=2 to n=1 transition. What fraction of the transition energy is the recoil energy?

Contributors and Attributions

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)


This page titled 8.E: Central Potentials (Exercises) is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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