4.2 Particle in a Box and Quantum Tunneling

Particle in a Box Problem

The particle in a box is one of the simplest exactly solvable models in quantum mechanics, yet it captures the essential physics of quantization. By confining a particle to a region with impenetrable walls, you can see directly how boundary conditions force energy to take only discrete values. This model also serves as a foundation for understanding more realistic systems like conjugated π-electron systems and quantum confinement in nanomaterials.

Solving the Particle in a Box

The model describes a particle confined to a one-dimensional potential well with infinite potential walls at $x = 0$ and $x = L$. Inside the box, the potential is zero; outside, it's infinite, so the particle has zero probability of existing beyond the walls.

Inside the box, the time-independent Schrödinger equation reduces to:

$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi$

where $\hbar$ is the reduced Planck's constant, $m$ is the particle mass, $E$ is the energy, and $\psi$ is the wave function.

The boundary conditions are $\psi(0) = \psi(L) = 0$. These arise because the wave function must be continuous, and it's zero outside the box (the particle can't penetrate infinite walls). These boundary conditions are exactly what forces quantization: only certain sinusoidal solutions "fit" inside the box with nodes at both walls.

To solve, you can follow these steps:

1. Recognize that the general solution to the differential equation is $\psi(x) = A\sin(kx) + B\cos(kx)$, where $k = \sqrt{2mE/\hbar^2}$.
2. Apply $\psi(0) = 0$: this forces $B = 0$, eliminating the cosine term.
3. Apply $\psi(L) = 0$: this requires $\sin(kL) = 0$, so $kL = n\pi$ where $n = 1, 2, 3, \ldots$
4. Solve for the allowed energies from the quantization condition on $k$.
5. Normalize the wave function to find the coefficient $A$.

Note that $n = 0$ is excluded because it would give $\psi = 0$ everywhere, meaning no particle at all.

Energy Levels and Wave Functions

The allowed energy levels are:

$E_n = \frac{n^2 h^2}{8mL^2}, \quad n = 1, 2, 3, \ldots$

Several features are worth noting:

• Energy is quantized and scales as $n^2$, so the spacing between levels grows with increasing $n$.
• The ground state ($n = 1$) has nonzero energy: $E_1 = \frac{h^2}{8mL^2}$. This zero-point energy is a purely quantum mechanical result; a classical particle at rest in a box would have zero energy.
• Energy scales inversely with $mL^2$. For a heavier particle or a larger box, the energy levels are more closely spaced and the system behaves more classically.

The normalized wave functions are:

$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$

Each wave function $\psi_n$ has exactly $n - 1$ nodes (points inside the box where $\psi = 0$, excluding the walls). More nodes correspond to higher energy, which is a general pattern across quantum systems.

The probability density $|\psi_n(x)|^2$ gives the probability of finding the particle at position $x$. It's highest at the antinodes (points of maximum amplitude) and zero at the nodes and at the walls. For the ground state ($n = 1$), the particle is most likely found near the center of the box. For higher states, the probability distributes into multiple peaks.

Quantum Tunneling and Applications

Concept of Quantum Tunneling

Quantum tunneling is the phenomenon where a particle passes through a potential energy barrier even though its kinetic energy is less than the barrier height. Classically, this is strictly forbidden: a ball rolling toward a hill it doesn't have enough energy to climb will always bounce back. Quantum mechanically, the wave function doesn't abruptly drop to zero at the barrier. Instead, it decays exponentially inside the barrier region, and if the barrier is thin enough, a nonzero amplitude emerges on the other side.

The tunneling probability depends on three factors:

• Barrier height ($V_0 - E$): a smaller difference between the barrier height and the particle's energy means less exponential decay and more tunneling.
• Barrier width ($a$): narrower barriers allow more of the wave function to survive to the other side. The probability drops exponentially with width.
• Particle mass ($m$): lighter particles tunnel more readily because the decay constant $\kappa$ depends on $\sqrt{m}$.

Applications of Quantum Tunneling

Scanning tunneling microscopy (STM) uses tunneling current to image surfaces with atomic resolution. A sharp conducting tip is brought within a few angstroms of a sample surface, and a bias voltage is applied. Electrons tunnel across the vacuum gap between tip and sample. Because the tunneling current depends exponentially on the tip-sample distance, even sub-angstrom changes in surface height produce measurable current variations. By scanning the tip across the surface and recording the current, you build up a topographic map at the atomic scale.

Electronic devices that rely on tunneling include:

• Tunnel diodes: semiconductor diodes where current flows via tunneling through a thin depletion region. Their negative differential resistance makes them useful in high-frequency oscillators and microwave circuits.
• Flash memory: data is stored by trapping charge on a floating gate. Writing and erasing involve electrons tunneling through a thin oxide layer (Fowler-Nordheim tunneling). This is the technology behind USB drives and SSDs.

Nuclear and radioactive processes also depend on tunneling:

• Nuclear fusion in stars: protons must overcome the Coulomb repulsion barrier to fuse. At stellar core temperatures, their thermal energies are far too low classically, but quantum tunneling makes fusion possible at rates sufficient to power stars.
• Alpha decay: an alpha particle ($^4\text{He}$ nucleus) is bound inside a heavy nucleus by the nuclear potential. It tunnels through the Coulomb barrier to escape, and the decay rate depends exponentially on the barrier parameters, which is why half-lives vary enormously across isotopes.

Transmission Probability for a Barrier

Calculating Transmission Probability

The transmission coefficient $T$ quantifies the probability that a particle incident on a barrier will tunnel through it. For a rectangular barrier of height $V_0$ and width $a$, with a particle of energy $E < V_0$:

$T = \frac{1}{1 + \frac{V_0^2}{4E(V_0 - E)}\sinh^2(\kappa a)}$

where the decay constant inside the barrier is:

$\kappa = \sqrt{\frac{2m(V_0 - E)}{\hbar^2}}$

Note the $\hbar^2$ in the denominator under the square root. The quantity $\kappa$ has units of inverse length and characterizes how rapidly the wave function decays inside the barrier.

For a thick, high barrier ($\kappa a \gg 1$), $\sinh(\kappa a) \approx \frac{1}{2}e^{\kappa a}$, and the expression simplifies to:

$T \approx 16\frac{E}{V_0}\left(1 - \frac{E}{V_0}\right)e^{-2\kappa a}$

The exponential factor $e^{-2\kappa a}$ dominates, so the transmission probability drops off extremely rapidly with increasing barrier width or increasing $\kappa$ (which grows with particle mass and barrier height).

Factors Influencing Transmission Probability

• As $E \to V_0$, $\kappa \to 0$ and the barrier becomes effectively transparent. At $E = V_0$, the transmission coefficient approaches unity, though the exact behavior requires taking the limit carefully in the formula above ($\sinh$ approaches its argument for small values).
• For $E > V_0$, the particle is above the barrier. Classically you'd expect $T = 1$, but quantum mechanically there can still be partial reflection due to the abrupt potential steps. $T$ oscillates and equals 1 only at specific resonance energies where an integer number of half-wavelengths fit inside the barrier region.
• The exponential sensitivity to barrier width is what makes STM possible: a change in tip-sample distance of just 1 Å can change the tunneling current by roughly an order of magnitude.

Quantum Confinement in Nanomaterials

Quantum Confinement Effects

Quantum confinement occurs when a material's dimensions shrink to the point where they become comparable to the de Broglie wavelength of its electrons:

$\lambda = \frac{h}{p}$

For a typical conduction electron in a semiconductor, this wavelength is on the order of a few nanometers. Once the material's size approaches this scale, electrons can no longer be treated as free carriers in a bulk band structure. Instead, their motion is restricted, and the energy spectrum becomes discrete, much like the particle-in-a-box levels.

The key consequences of confinement:

• Discrete energy levels replace the continuous bands of the bulk material. The spacing between levels increases as the confining dimension shrinks, following the $1/L^2$ dependence from the particle-in-a-box model.
• The band gap increases relative to the bulk value. A smaller nanocrystal has a larger effective band gap because the lowest allowed energy level sits higher above the valence band edge.
• Optical properties become size-tunable. Because the band gap determines which wavelengths of light are absorbed and emitted, you can shift the color of a nanomaterial simply by changing its size.

Applications of Quantum-Confined Nanomaterials

Quantum dots are semiconductor nanocrystals (typically 2-10 nm in diameter) that are the most prominent example of quantum confinement in practice.

• Their emission wavelength is directly controlled by size: smaller dots emit bluer (higher energy) light, larger dots emit redder (lower energy) light. CdSe quantum dots, for instance, can be tuned across the entire visible spectrum.
• Applications include color-tunable LEDs (quantum dot displays are now commercially available), fluorescent labels for biological imaging (offering superior brightness and photostability compared to organic dyes), and light absorbers in next-generation solar cells.
• Quantum dots can also serve as single-photon emitters for quantum information and cryptography applications.

Beyond quantum dots, confinement effects are exploited in:

• Optoelectronics: quantum wells (2D confinement) are the active layers in most semiconductor lasers and LEDs. Quantum wires (1D confinement) and dots (0D) offer further tunability.
• Photovoltaics: quantum dots can absorb multiple photons or generate multiple excitons per photon, potentially exceeding the Shockley-Queisser efficiency limit for single-junction solar cells.
• Catalysis and sensing: nanomaterials have a high surface-to-volume ratio, providing more active sites for chemical reactions. Surface functionalization with specific ligands enables targeted drug delivery and selective biosensing.