The concept of a potential energy landscape is fundamental in physics, describing the forces acting on a particle as it moves through space. The particle’s energy determines where it can exist, much like a ball rolling across a varied terrain. Within this framework, the double well potential stands out as a foundational model, particularly in the study of quantum mechanics. It provides a simplified yet powerful way to explore some of the most counter-intuitive phenomena governing subatomic particles.
Defining the Double Well Potential
The double well potential is a specific energy structure defined by two distinct valleys, or minima, separated by a hill, or maximum, in the center. This structure produces a characteristic M-shape with two low-energy regions. The two minima represent stable positions where a particle prefers to reside, while the central maximum acts as a barrier to movement between them.
To visualize this, imagine a marble rolling in a landscape shaped like a wide, shallow letter ‘W’. The marble settles into one of the two valleys, representing the low-energy minima. For a classical particle, moving from one well to the other requires enough energy to climb over the central barrier. If the particle’s energy is less than the barrier’s height, it remains trapped in its initial well.
The height and width of the central barrier control the energy required for a classical particle to switch sides. A higher, wider barrier means the two wells are more isolated. This classical description highlights the difference observed when applying the rules of quantum mechanics.
The Quantum Difference: Tunneling
The behavior of a quantum particle within the double well potential deviates sharply from the classical expectation due to quantum tunneling. A quantum particle does not need sufficient energy to surmount the central barrier. Instead, it has a finite probability of passing straight through the barrier, even if its total energy is less than the barrier height.
This effect arises because the particle is governed by a wave function, which describes the probability of finding the particle at any location. The wave function does not drop instantly to zero within the barrier region. Instead, it decays exponentially, leaving a small, non-zero probability for the particle to appear on the other side.
The probability of tunneling is highly sensitive to the barrier’s properties. As the barrier becomes taller or wider, the wave function decays more quickly, significantly reducing the tunneling probability. Tunneling is observable primarily for microscopic particles and thin barriers.
Because the particle can tunnel back and forth, it does not remain localized in a single well. Instead, the particle oscillates coherently between the two potential minima. This oscillation frequency, or tunneling rate, is directly related to the probability of the particle penetrating the barrier.
Understanding Energy States and Symmetry
When a quantum particle is confined to a symmetric double well, tunneling fundamentally changes the resulting energy states, or eigenstates. If the barrier were infinitely high, the particle could be in either the left or right well, and both states would share the same energy, a condition known as degeneracy. The possibility of tunneling removes this degeneracy.
The particle’s true state is a quantum superposition of being in both wells simultaneously. Tunneling splits the single energy level into two distinct levels very close in value, known as doublet splitting or tunnel splitting.
The two resulting eigenstates are characterized by their symmetry. The lower-energy state is the symmetric state, where the wave function has the same sign in both wells. The higher-energy state is the antisymmetric state, where the wave function has opposite signs in the two wells.
The energy difference between these two split levels determines the tunneling frequency of the particle. A small energy splitting corresponds to a low tunneling rate. Conversely, a larger energy splitting indicates a faster oscillation between the wells.
Practical Uses of the Double Well Model
The double well potential serves as a powerful theoretical tool modeling numerous physical systems across different scientific disciplines.
One historically significant example is the inversion of the ammonia (\(NH_3\)) molecule. The nitrogen atom can exist on either side of the plane formed by the three hydrogen atoms, creating a double well potential for its position. The nitrogen atom tunnels through this plane, causing the molecule to rapidly invert its shape. This tunneling is responsible for the precise energy splitting measured spectroscopically, which was utilized in the first maser device.
In quantum computing, the double well potential models a qubit, the basic unit of quantum information. The two potential minima represent the two distinct quantum states, such as 0 and 1. The particle’s ability to tunnel between these states is the mechanism used to manipulate the qubit and perform quantum logic operations.
The model is also applied in condensed matter physics to understand phase transitions in certain materials. For example, it describes the collective motion of atoms in a crystal lattice where two stable positions are possible. The double well physics captures the behavior of systems where a thermodynamic state transitions from a symmetric phase to one with broken symmetry.