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