8.E: Central Potentials (Exercises)
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- A particle of mass m is placed in a finite spherical well: V(r)={−V0 for r≤a0 for r>awith 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<π2ℏ2/(8m).
- 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.
- 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:
- A2=β3/π.
- β=Z/a0, where a0=(ℏ2/me)(4πϵ0/e2).
- The energy is E=−Z2E0 where E0=(me/2ℏ2)(e2/4πϵ0)2.
- The expectation values of the potential and kinetic energies are 2E and −E, respectively.
- The expectation value of r is (3/2)(a0/Z).
- The most probable value of r is a0/Z.
- 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?
- Calculate the wavelengths of the photons emitted from the n=2, l=1 to n=1, l=0 transition in hydrogen, deuterium, and positronium.
- 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)