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### Exercises

1. A particle of mass is placed in a finite spherical well: with and . Find the ground-state by solving the radial equation with . Show that there is no ground-state if .

2. Consider a particle of mass in the three-dimensional harmonic oscillator potential . Solve the problem by separation of variables in spherical polar 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, , , etc.) with nuclear charge has the form where and are constants, and is the distance between the nucleus and the electron. Show the following:
1. .
2. , where .
3. The energy is where .
4. The expectation values of the potential and kinetic energies are and , respectively.
5. The expectation value of is .
6. The most probable value of is .

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 which leaves the atom without perturbing the extranuclear electron, Find the probability that the resulting ion will be left in an , state. Find the probability that it will be left in a , state. What is the probability that the ion will be left in an state?

5. Calculate the wavelengths of the photons emitted from the , to , 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 to transition. What fraction of the transition energy is the recoil energy?   Next: Spin Angular Momentum Up: Rydberg Formula Previous: Rydberg Formula
Richard Fitzpatrick 2010-07-20