Potential wells and obstacles

In quantum mechanics, it is fundamental to understand how particles behave under different potential energies. The central tool in such analysis is the Schrödinger equation, which provides a way to describe how the quantum state of a physical system changes over time. Two highly influential concepts that arise from such analysis are potential wells and potential barriers. These concepts help us understand phenomena such as quantum tunneling and bound states.

Schrödinger Equation

Before taking a deeper look at potential wells and barriers, it is necessary to understand the Schrödinger equation, which is given as:

iħ ∂ψ/∂t = Ĥψ

Here, ψ is the wave function, which contains all the information about the particle; ħ is the reduced Planck constant; ∂ψ/∂t represents the partial derivative of the wave function with respect to time, and Ĥ is the Hamiltonian operator representing the total energy of the system.

Potential wells

A potential well is a region where the potential energy V(x) is less than the surrounding region. Imagine a bowl-shaped valley that a particle can enter but exit only if it gains enough energy.

Finite potential well

Consider a one-dimensional finite well, where the potential V(x) is given by:

V(x) = { 0, if |x| < a V₀, if |x| ≥ a }

The potential well has regions of different potential energy: zero inside the well and V₀ outside.

Within the well, the time-independent Schrödinger equation is:

-ħ²/2m ∂²ψ/∂x² = Eψ

Solving this produces a sinusoidal solution inside the well:

ψ(x) = A sin(kx) + B cos(kx)

Where k = √(2mE)/ħ.

Outside the well, where the potential energy is V₀, the solution takes the form of exponential decay or growth because the energy inside the well is generally less than the potential energy outside. Thus:

ψ(x) = F e^(αx) + G e^(-αx)

Where α = √(2m(V₀ - E))/ħ.

Graphical representation of a finite potential well

Quantum tunneling and barriers

Quantum tunneling is a phenomenon in which particles can pass through potential barriers even if their energy is less than the height of the barrier. This is in contrast to classical physics where it is impossible to do so.

Potential barrier

Consider the following constraint:

V(x) = { 0, if x < 0 or x > L V₀, if 0 ≤ x ≤ L }

Inside the barrier, the wave function can be written using the equality of potential wells:

ψ(x) = C e^(κx) + D e^(-κx)

Where κ = √(2m(V₀ - E))/ħ. This represents the exponential decay inside the barrier.

Graphical representation of potential barrier

Solution examples

Let us take the example of a particle approaching a barrier. Consider a particle whose energy is less than E V₀. The probability of the particle being on the other side of the barrier is non-zero, defined as the tunneling probability.

For a potential barrier of height V₀ and width L, the tunneling probability T is approximately:

T ≈ exp(-2κL)

This formula shows that tunneling depends sharply on the difference between the width of the barrier and the height of the barrier and the energy of the particle.

Application

Potential wells and barriers are not just theoretical constructs; they have applications in the real world. Quantum tunneling is important in nuclear physics, for example, in the process of nuclear fusion in stars. Semiconductor devices such as tunnel diodes depend on quantum tunneling to operate. The phenomena discovered in potential wells are also important in quantum dots, nanoscale devices that confine electrons in a potential well, affecting their quantum mechanical properties.

Closing thoughts

The study of potential wells and barriers in quantum mechanics provides insight into the behavior of particles at the quantum scale, which is very different from classical predictions. Through the Schrödinger equation, we learn that energy levels are quantized, and phenomena such as tunneling become possible, leading to technological advancements in various fields.

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