# Please explain the following graph for a quantum mechanical harmonic oscillator

Graphs such as the above keep coming up when talking about harmonic oscillators in a quantum mechanical sense. However, I simply cannot make sense of them. What does each line represent why are they waves and what is the parabola? Also, I see sometimes graphs such as:

These seem to follow on from the initial graph. It's something to do with the recovery of classical behavior but I don't understand the first graph so stand no chance of getting the second ones at high n - perhaps you could explain the relation between the first and second graph please? Thanks

Background

Let's start by considering the case of 2 hydrogen atoms separated by a large distance. If they happen to be on a trajectory that brings them closer together they can interact with one another. If they get close enough together they can form a bond and create a hydrogen molecule. This process is pictured in the following graph.

image source

This "potential well" is often referred to as a "Morse potential". Out on the right edge of the curve (say 5 angstroms), we have our 2 separated hydrogen atoms. As the distance between them lessens, stabilization occurs, they can fall into the well and form a hydrogen molecule.

The red lines in the Morse potential represent different vibrational states of the hydrogen molecule. Initially, the hydrogen atom is formed in one of the higher vibrational states. It can either absorb more energy through collisions and go back to 2 dissociated atoms or it can give away some excess energy through collisions and fall to a lower vibrational level.

The classical view of the hydrogen atom is two weights connected by a spring. The more energy in the system, the wider the oscillation of the spring. With lower energy the oscillations are smaller. The energy of the spring system (y-axis) as a function of separation (x-axis) is given by Hooke's Law as

$$E(x)={1\over2}kx^2$$

This is really an equation that describes a parabola, so the lower part of a Morse potential can be accurately described by a parabola. Your top drawing represents the lower part of the Morse potential diagram.

While we have used hydrogen as our model in the above explanation, the same reasoning applies to any bond formation (and subsequently bond stretch) process. For example, we could describe a $\ce{C-H}$ bond stretch in a completely analogous manner.

Each energy state (vibrational level) is described by its own wavefunction ($\Psi_n$). The square of the wavefunction (${\Psi_{n}^2}$) describes the probability of finding the electron at a certain point on the x-coordinate for that specific vibrational energy level. From the solution to the wavefunction we find that as we go to higher and higher energy levels, the number of nodes (a point where the wavefunction = 0) in our wave continually increases. The square of the wavefunction produces positive values (the "upside-down" parabolas). We always have the same number of nodes in the wavefunction and the probability distribution. For example, in the lowest vibrational energy level (n=0) there is no node in either the wavefunction or the probability distribution. In the n=1 vibrational level there is one node in the wavefunction and probability distribution.