The potential energy of the rotation of one \(\mathrm{CH}\), group relative to its neighbour in ethane can be expressed as \(V(\Psi)=V_{0} \cos 3 \phi\). Show that for small displacements the motion of the group is harmonic and calculate the energy of excitation from \(v=0\) to \(v=1\). What do you expect to happen to the energy levels and wavefunctions as the excitation increases?

In the context of quantum mechanics, potential energy represents the stored energy of a system due to its position. It's a fundamental concept that determines how particles or groups of particles interact with each other. For the quantum harmonic oscillator, the potential energy is a function of the position of the particle within the system. The potential energy can be visualized as a well or a valley, where a particle experiences a force drawing it towards the lowest point or the equilibrium position.

When we analyze the motion of a CH group in ethane, the potential energy is given by a function of the angle, and the goal is to express it in terms familiar to harmonic motion. By expanding the potential energy function using a Taylor series for small angular displacements and keeping only terms up to the second power, we arrive at an expression akin to that of a classic harmonic oscillator. This expression, which relates directly to the force exerted on the particle, is critical for understanding the harmonic nature of molecular vibrations and rotations.

Harmonic motion is a type of periodic oscillatory motion where the restoring force is directly proportional to displacement from an equilibrium position. In classical physics, it's typically characterized by the motion of a mass attached to a spring obeying Hooke's law. In quantum systems like the rotation of a CH group in ethane, the conditions for harmonic motion are identified by finding a quadratic dependence of potential energy on displacement.

This quadratic relationship implies a restoring force that pulls the system back toward its equilibrium state, leading to oscillatory motion. When the motion of our system is approximated as harmonic, we can employ elegant mathematical solutions that provide deep insights into the system's behavior, such as energy quantization and symmetrical wavefunctions. It's important to note that this is an approximation, valid only for small excitations close to the equilibrium position.

The vibrational quantum number, often denoted by v, is an integer that specifies the state of vibrational energy in a quantum system. In the case of the quantum harmonic oscillator, energy levels are quantized, meaning that they can only take on certain discrete values. These energy levels are indexed by the vibrational quantum number. The energy associated with a particular vibrational state is given by the formula E_v = (v + 1/2)ℏω, where ℏ is reduced Planck's constant, and ω is the natural angular frequency of oscillation.

The vibrational quantum number is vital for understanding transitions between energy levels. For instance, the exercise provided investigates the energy required to move from the ground state (v=0) to the first excited state (v=1). As vibrational quantum number increments, we move to higher energy levels, representing more energetic states of the system.

Energy levels in quantum systems are analogous to steps on a ladder that a particle can occupy - these are the allowable energy states of the system. For the quantum harmonic oscillator, these levels are equally spaced, which is a special feature of such systems. The energy difference between adjacent levels is given by ℏω, this regular spacing simplifies calculations and predictions about the system's behavior.

However, the real potential energy profile of molecular rotations, like in our exercise, can become anharmonic as we consider higher energies or larger displacements. In these cases, the spacing between energy levels may change, and the system's behavior becomes more complex. This deviation from the harmonic ideal provides insight into more realistic molecular dynamics. In summary, accounting for the vibrational quantum number and the true form of potential allows us to predict the properties of molecules in different states of excitation.