← Part III/Finite Square Well

3. Finite Square Well

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A more realistic potential well with finite barriers: introduces bound and scattering states.

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Video Lecture

Finite Square Well - Bound States

Physics Videos by Eugene - Complete derivation of bound state energy levels in the finite square well

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

The Potential

Symmetric finite square well:

$$V(x) = \begin{cases} -V_0 & |x| < a \\ 0 & |x| > a \end{cases}$$

where $V_0 > 0$ is the well depth

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Potential Energy Diagram

Finite Square Well

Selected Energy Level: n = 1

Number of Levels: 5

Show All Energy LevelsShow Wave Function

Legend:

Blue curve - Potential energy V(x)
Cyan line - Selected energy level
Green curve - Wave function ψ(x) at that energy
Gray dashed - Other energy levels

Bound States ($E < 0$)

Inside the well ($|x| < a$):

$$\psi(x) = A\sin(kx) + B\cos(kx), \quad k = \frac{\sqrt{2m(E+V_0)}}{\hbar}$$

Outside the well ($|x| > a$):

$$\psi(x) = Ce^{-\kappa|x|}, \quad \kappa = \frac{\sqrt{-2mE}}{\hbar}$$

Wave function decays exponentially outside (evanescent wave)

Even and Odd Solutions

Even parity ($\psi(-x) = \psi(x)$):

$$k\tan(ka) = \kappa$$

Odd parity ($\psi(-x) = -\psi(x)$):

$$k\cot(ka) = -\kappa$$

These transcendental equations must be solved graphically or numerically

Finding the Ground State of a Finite Well

INTERMEDIATE

Problem: An electron is trapped in a finite square well with depth V₀ = 10 eV and width 2a = 2 nm. Estimate whether there is a bound ground state and find its approximate energy.

Given:

Well depth: V₀ = 10 eV
Well width: 2a = 2 nm → a = 1 nm = 10⁻⁹ m
Electron mass: m = 9.11 × 10⁻³¹ kg
ℏ = 1.055 × 10⁻³⁴ J·s

Find: Check if bound state exists; estimate ground state energy

Self-Check Question

A finite square well always has at least one bound state, no matter how shallow. Why is this NOT true for the infinite square well?

Number of Bound States

Define dimensionless parameter:

$$z_0 = \frac{a}{\hbar}\sqrt{2mV_0}$$

Number of bound states:

$$N \approx \frac{z_0}{\pi/2}$$

Key result: At least one bound state always exists, no matter how shallow the well

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Video Lecture

Quantum Tunneling Through Barriers

MIT OCW - Quantum tunneling phenomenon with applications to alpha decay and STM

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Self-Check Question

In the finite square well, where is the particle most likely to be found when in the ground state?

Scattering States ($E > 0$)

For $E > 0$, particle is not bound. Solutions are oscillatory everywhere:

$$k_{\text{in}} = \frac{\sqrt{2m(E+V_0)}}{\hbar}, \quad k_{\text{out}} = \frac{\sqrt{2mE}}{\hbar}$$

These states describe transmission and reflection

Transmission Coefficient

Probability of transmission through the well:

$$T = \frac{1}{1 + \frac{V_0^2\sin^2(2k_{\text{in}}a)}{4E(E+V_0)}}$$

Resonances: $T = 1$ when $k_{\text{in}}a = n\pi/2$ (perfect transmission)

Key Differences from Infinite Well

Finite number of bound states
Wave function penetrates into classically forbidden regions
Energy levels are lower than infinite well
Continuum of scattering states above $E = 0$
Spacing between levels is not uniform

Self-Check Question

What happens to the transmission coefficient T when k_in × a = nπ/2 (where n is an integer)?

Physical Applications

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Applications of the Finite Square Well Model

1. Quantum Dots and Semiconductor Nanostructures

Nanotechnology

Electrons confined in semiconductor quantum dots behave like particles in 3D finite wells. The discrete energy levels lead to size-tunable optical properties, making quantum dots ideal for displays, solar cells, and biological imaging.

Examples:

QLED TV displays (Samsung, LG)
Quantum dot solar cells (45% theoretical efficiency)
Fluorescent biomarkers for cancer detection
Single-photon sources for quantum cryptography

2. Nuclear Physics - Alpha Decay

Nuclear Physics

Alpha particles escape nuclei via quantum tunneling through the finite Coulomb barrier. The finite square well approximation helps model the nuclear potential and predict decay rates.

Examples:

Radioactive dating (C-14, U-238)
Nuclear medicine (Am-241 in smoke detectors)
Understanding stellar nucleosynthesis
Nuclear reactor fuel decay calculations

3. Molecular Binding and Chemical Bonds

Chemistry

Electrons in molecules experience finite potential wells from nuclear attraction. The finite well model explains bonding energies, ionization potentials, and molecular stability.

Examples:

Hydrogen molecular ion (H₂⁺)
Conjugated π-electron systems in organic molecules
Metal-ligand bonding in coordination chemistry
Electron affinities of atoms and molecules

4. Scanning Tunneling Microscope (STM)

Surface Science

STM exploits quantum tunneling through the finite barrier between a sharp tip and sample surface. Tunneling current depends exponentially on tip-sample distance, enabling atomic-resolution imaging.

Examples:

Imaging individual atoms on surfaces
Manipulation of single atoms (IBM 'quantum corrals')
Measuring local density of states
Studying surface chemical reactions

5. Quantum Wells in Laser Diodes

Optoelectronics

Semiconductor lasers use quantum wells (finite potential wells) to confine electrons and holes, creating sharp energy levels that produce specific wavelengths of light with high efficiency.

Examples:

Blu-ray laser diodes (405 nm wavelength)
Fiber optic telecommunications
LIDAR systems for autonomous vehicles
Medical laser surgery devices

6. Electron Capture and Exotic Atoms

Particle Physics

Exotic atoms like muonic hydrogen have particles bound in finite wells. The finite well model helps calculate binding energies and spectral lines for precision tests of QED.

Examples:

Muonic hydrogen spectroscopy
Positronium (electron-positron bound state)
Antihydrogen production at CERN
Tests of fundamental constants

💡 These applications demonstrate how fundamental quantum concepts translate into modern technology.

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Video Lecture

Quantum Wells in Semiconductor Devices

Applications of finite quantum wells in modern technology including LEDs, lasers, and transistors

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

📝 Summary: Finite Square Well

Key Equations

Even states:

$$k\tan(ka) = \kappa$$

Odd states:

$$k\cot(ka) = -\kappa$$

Number of states:

$$N \approx \frac{a}{\hbar\pi/2}\sqrt{2mV_0}$$

Transmission:

$$T = \frac{1}{1 + \frac{V_0^2\sin^2(2k_{\text{in}}a)}{4E(E+V_0)}}$$

Key Concepts

Wave function penetrates classically forbidden regions (tunneling)
Finite number of bound states (unlike infinite well)
Energy levels lower than infinite well of same width
Always at least one bound state for symmetric well
Transcendental equations require numerical/graphical solutions
Scattering states (E > 0) show resonant transmission
Even and odd parity solutions alternate in energy
Essential model for quantum dots, STM, molecular bonds

Why It Matters: The finite well bridges idealized infinite wells and realistic potentials, introducing key phenomena like tunneling and resonances that are crucial in modern nanotechnology and quantum devices.

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Quantum Mechanics Course

3. Finite Square Well

Your Progress

Video Lecture

The Potential

Potential Energy Diagram

Bound States ($E < 0$)

Even and Odd Solutions

Finding the Ground State of a Finite Well

Self-Check Question

Number of Bound States

Video Lecture

Self-Check Question

Scattering States ($E > 0$)

Transmission Coefficient

Key Differences from Infinite Well

Self-Check Question

Physical Applications

Applications of the Finite Square Well Model

1. Quantum Dots and Semiconductor Nanostructures

2. Nuclear Physics - Alpha Decay

3. Molecular Binding and Chemical Bonds

4. Scanning Tunneling Microscope (STM)

5. Quantum Wells in Laser Diodes

6. Electron Capture and Exotic Atoms

Video Lecture

📝 Summary: Finite Square Well

Key Equations

Key Concepts

Related Topics

Infinite Square Well

Quantum Tunneling

Harmonic Oscillator

Scattering Theory

WKB Approximation