prompter: Why combination of ground and first excited states of harmonic oscillator show a change in probability density with time?

assistant: The combination of the ground and first excited states of a harmonic oscillator forms a wave function that is not time-independent. This means that the probability density of finding the oscillator at a particular location changes with time.

To understand why this is the case, we can consider the time-evolution of the wave function. The wave function for a harmonic oscillator can be written as a sum of the ground state and the first excited state, where the ground state is the lowest energy state and the first excited state has slightly higher energy. The ground state has a probability density that is symmetrically distributed around the equilibrium position of the oscillator, while the first excited state has a probability density that is shifted slightly to one side.

As time passes, the wave function undergoes periodic oscillations, with the amplitude of the oscillations increasing over time. This is because the ground state and the first excited state have slightly different frequencies, and as they interfere with each other, the amplitude of the resulting wave function changes with time. This change in amplitude leads to a corresponding change in the probability density of finding the oscillator at a particular location.

In particular, the regions where the ground state and the first excited state have the same phase will experience constructive interference, leading to an increase in the probability density in those regions. Conversely, the regions where the ground state and the first excited state have opposite phase will experience destructive interference, leading to a decrease in the probability density in those regions.

Overall, this means that the probability density of finding the oscillator at a particular location will change periodically with time as a result of the interference between the ground state and the first excited state. This phenomenon is known as quantum beats and is a general feature of quantum systems that are in a superposition of states with slightly different energies or frequencies.

prompter: Imagine I wanted to explain this to a barely-literate, facebook-obsessed baby boomer. Can you give me a summary that even someone like that could understand? Write it in the format of a click-bait pseudo-news article.

assistant: #### Title: The other thing in the Box with the cat in Quantum mechanics, explained by Scientists

Follow the author on twitter: @ScienceGaryLovesCats

We often think of things as either being, or be-isn't. But scientists say that now, things just ain't that simple anymore. When we see light reflected off a surface, the electrons in the atoms in the light are becoming energized, causing them to have "different eneregy levels" according to scientists. The jumps between these energy levels create light, which we use to see. But why do atoms have "energy levels" in the first place? This, scientists say, is because the electrons can only do certain things, kinda like how chess pieces can only make certain moves. But electrons don't just take one move, they can take multiple moves at once, due to wibbly-wobbly quantum weirdness, as Dr Who would call it.

But before your head explodes, it gets even *weirder*, these states have two parts to them, we can just call them up-down and left-right. And while the combination of the up-down and left-right parts of a single state is constant, when you combine the up-down and left-right states of different states, it changes over time! wowzer! This is especially true for the particles in boxes, which scientists love. Not because there's cats in the box, but because in Quantum mechanics, the whole universe is made of them, in a way.

assistant: 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.

prompter: Thank you for the explanation. Can you provide an example of how this phenomenon is observed in a physical system?

assistant: 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: Sure! 

One example of this phenomenon can be seen in the behavior of a charged particle in a magnetic field. When a charged particle is placed in a magnetic field, it experiences a force perpendicular to its velocity, which causes it to move in a circular or helical path.

The motion of the particle can be described by a wave function, which is a superposition of different energy levels of the system. When the particle is in a superposition of the ground and first excited states of the harmonic oscillator, its wave function oscillates between the two spatial structures, resulting in a time-dependent probability density.

This oscillation between the two states affects the trajectory of the particle, causing it to spiral in a particular way. This phenomenon is known as cyclotron motion and has been observed in a variety of physical systems, including the motion of electrons in a magnetic field.

Overall, the combination of ground and first excited states of the harmonic oscillator can lead to a time-dependent probability density distribution, which is a manifestation of quantum interference. This phenomenon is crucial for understanding many physical and chemical processes at the quantum level.

I hope this answers your question. If there is anything else you need don't hesitate to ask.