Why combination of ground and first excited states of harmonic oscillator show a change in probability density with time?
user
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.
assistant
Thank you for the explanation. Can you provide an example of how this phenomenon is observed in a physical system?
user
One example of this phenomenon is observed in the behavior of electrons in a magnetic field. When a magnetic field is applied to a system of electrons, their energies become quantized, and they occupy different energy levels. The lowest energy level corresponds to the ground state, which has a spherically symmetric probability density distribution.

When an external electromagnetic field is applied to the system, the electrons can be excited to higher energy levels, such as the first excited state, which has a ring-shaped probability density distribution. The combination of these two states can lead to a time-dependent probability density distribution, which oscillates between the two spatial structures.

This phenomenon is known as magnetic resonance, and it has important applications in magnetic resonance imaging (MRI) technology, which is used in medical diagnosis. In MRI, a strong magnetic field is used to excite the electrons in the atoms of the patient's body, and the resulting time-dependent probability density distribution is used to create a detailed image of the internal structures of the body.
assistant