Question

The quantized energy levels in the infinite well get farther apart as $n$ increases, but in the harmonic oscillator they are equally spaced. (a) Explain the difference by considering the distance "between the walls" in each case and how it depends on the particle's energy. (b) A very important bound system, the hydrogen atom, has energy levels that actually get closer together as $n$ increases. How do you think the separation between the potential energy "walls" in this system varies relative to the other two? Explain.

Short answer

In the infinite well potential, the energy levels get farther apart as $n$ increases because the particle has more states to occupy with more energy. In the harmonic oscillator, the energy levels are equally spaced due to the proportional relationship between potential energy and the square of displacement. Whereas in a hydrogen atom, the energy levels get closer together as $n$ increases due to the Coulombic (inverse square) potential.

Step-by-step solution

Infinite well potential

In an infinite well, the particle is confined to a region and can never escape, regardless of the level of energy it has. The distance between the quantum energy levels or 'walls' widens as the energy ($n$) increases, because with more energy, the particle can occupy more states. The energy levels can be represented as $E_n=n^2$ (where $n$) is the quantum number, so the energy levels get farther apart as $n$ increases.

Harmonic oscillator potential

The potential energy in a harmonic oscillator is proportional to the square of the displacement from the equilibrium position. This means that the force needed to extend or compress the system increases linearly with displacement. As a result, unlike the infinite well, the energy levels in a quantum harmonic oscillator are equally spaced. This is represented as $E_n=n+\frac{1}{2}$, and thus energy levels difference is constant regardless of $n$.

Hydrogen atom system

In the case of a hydrogen atom, the separation between energy levels gets smaller as $n$ increases. This is because the potential energy is a Coulombic (inverse square) potential. As the energy increases, the electron is more likely to be found at higher radii, thus effectively increasing the 'distance between the walls'. However, this potential also gives rise to the property that the difference between successive energy levels decreases as $n$ increases. Therefore, unlike the former two potentials, the more energy the electron in the H-atom has, the less difference in energy is required to move to next higher state.

Key concepts

Quantum Mechanics

Quantum mechanics reveals the behavior of particles at the smallest scales. Unlike classical mechanics, where objects move in a continuous and predictable fashion, quantum mechanics describes phenomena where certain properties, like energy and position, are quantized. This means that particles, such as electrons in atoms, can only occupy specific energy levels. These levels are discrete, which is similar to how a ladder has distinct rungs, and a person can only stand on these rungs, not in between.

The rules governing this quantization are fundamental to understanding how microscopic systems, like atoms and particles in a well, behave. Quantum mechanics forces us to abandon the notion of particles traveling in smooth trajectories. Instead, we work with probabilities and wave functions, mathematical tools that give us the likelihood of finding a particle in a certain state or location.

Infinite Well Potential

Imagine confining a particle within an 'infinite well,' which means strong walls trapping the particle so tightly that it can never escape. In quantum mechanics, this is modeled by the infinite potential well, where the potential energy inside the well is zero, and it is infinite outside. Due to the quantization of energy, the particle can only have specific energy values. As the quantum number, represented by $n$, increases, the particle's wavelength becomes shorter, and thus, the allowed energy levels spread out further apart following the relationship $E_n=n^2$

The spacing of energy levels getting larger with higher $n$ illustrates a particle gaining more energy and being able to occupy more quantum states within the confines of this well. It also helps visualize quantization, as there are finite and distinct energies that a particle can have.

Harmonic Oscillator

In the realm of quantum mechanics, a harmonic oscillator is another fundamental model, often representing the behavior of atoms in a crystal lattice or the vibrational motion of molecules. The classic image of this is a mass on a spring. The potential energy is a parabolic shape, meaning it is symmetric and increases as the particle moves away from the center.

The energies of a quantum harmonic oscillator are given by $E_n=(n+\frac{1}{2})\hslash \omega$, where $n$ is the quantum number, $\hslash$ is the reduced Planck constant, and $\omega$ is the angular frequency of the oscillator. The spacing of energy levels is uniform—the difference between them remains constant. This leads to an interesting dynamic, where higher energy states are no harder to reach from their immediate next lower state than the ground state is from the first excited state.

Hydrogen Atom

The hydrogen atom is a simple yet essential system in quantum mechanics. It consists of a single electron orbiting a proton, and the forces between them are described by the Coulombic potential. This is because the attractive force between the electron and proton is electromagnetic in nature and falls off with the square of the distance between the particles.

The energy levels in a hydrogen atom decrease as $n$ increases, but the spacing between these levels gets closer. The energy of an electron in a hydrogen atom is described by $E_n=-\frac{\kappa }{n^2}$, where $\kappa$ is a constant derived from the electron mass, electron charge, and electric constant. This relationship reflects the fact that far away from the nucleus, the electron feels less of the attractive force and hence has higher potential 'walls' to jump between energy levels.

Coulombic Potential

The Coulombic potential describes the force between two charged particles, like the electron and the proton in the hydrogen atom. It's an inverse square law, meaning the force decreases as the square of the distance between the charges. Even more fascinating in quantum mechanics is that the Coulombic potential shapes the behavior of the electron in the atom.

This potential leads to unique properties of quantum systems; the most significant is the varying spacings between energy levels. The further away an electron is from the nucleus, the less energy is required to move it to an even higher orbit. Conversely, close to the nucleus, the gaps between energy levels are more extensive due to a stronger attractive force exerted on the electron.

Quantization of Energy

At the very heart of quantum mechanics lies the concept of quantization of energy. It is the principle that energy only exists in specific, set amounts, much like the pre-set rungs on a ladder. A particle, like an electron in an atom, cannot have just any random energy; it must have one of an allowed set of values.

This characteristic is observed across various quantum mechanical systems, such as electrons bound in atoms or molecules, and it has profound implications for the behavior of all microscopic particles. Quantum energy levels allow for the existence of distinct atomic states, the stability of atoms, and the myriad of phenomena that stem from this orderly structure of the microscopic world.