The harmonic oscillator is a fundamental concept in quantum mechanics, and it has a set of quantized energy levels. When the system is in its ground state, its wave function is at its lowest energy level and has a spherically symmetric probability density distribution.

When the system is excited to its first excited state, it gains energy and its wave function becomes more complex. The first excited state has one nodal plane and a ring-shaped probability density distribution.

The combination of these two states can lead to a time-dependent probability density distribution. This is because the ground and first excited states have different spatial structures and different energies. When the system is in a superposition of these two states, the wave function oscillates between the two spatial structures, resulting in a time-dependent probability density.

This oscillation between the two states is a manifestation of quantum interference, a key feature of quantum mechanics. The exact time dependence of the probability density will depend on the relative amplitudes and phases of the two states.

This phenomenon has been observed in a variety of physical systems, such as in the behavior of electrons in a magnetic field, the motion of molecules in a chemical reaction, and the dynamics of Bose-Einstein condensates. The time-dependent behavior of these systems is crucial for understanding many physical and chemical processes at the quantum level.
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