6.4 Delta function potential

The Dirac delta function is a powerful tool in quantum mechanics, representing an infinitely sharp peak with unit area. It's used to model localized interactions and approximate potentials that act over very small distances, like those in atomic nuclei.

Delta potentials come in two flavors: attractive and repulsive. They help us understand quantum phenomena like bound states, scattering, and tunneling. Attractive potentials can trap particles, while repulsive ones only allow scattering. These simplified models are key to grasping quantum behavior.

Dirac Delta Function and Potential Types

Understanding the Dirac Delta Function

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• Dirac delta function represents an infinitely sharp peak with unit area
• Mathematical representation: $\delta(x) = \begin{cases} \infty, & x = 0 \\ 0, & x \neq 0 \end{cases}$
• Integral property: $\int_{-\infty}^{\infty} \delta(x) dx = 1$
• Serves as a useful tool for modeling localized interactions in quantum mechanics
• Physically interpreted as an extremely short-range potential
• Used to approximate potentials that act over very small distances (atomic nuclei)

Attractive and Repulsive Delta Potentials

• Attractive delta potential defined as $V(x) = -\alpha \delta(x)$, where $\alpha > 0$
• Repulsive delta potential defined as $V(x) = \alpha \delta(x)$, where $\alpha > 0$
• Attractive potentials can support bound states
• Repulsive potentials only allow scattering states
• Strength of the potential determined by the magnitude of $\alpha$
• Delta potentials provide simplified models for studying quantum tunneling and scattering

Bound and Scattering States

Characteristics of Bound States

• Bound states occur when particles are confined to a specific region
• Energy of bound states discrete and negative
• Wavefunctions of bound states normalized and square-integrable
• Decay exponentially as $x \rightarrow \pm \infty$
• Exist only for attractive delta potentials
• Number of bound states depends on the strength of the potential
• For attractive delta potential, only one bound state exists with energy $E = -\frac{m\alpha^2}{2\hbar^2}$

Properties of Scattering States

• Scattering states represent particles with positive energy
• Occur when particles interact with a potential but remain unbound
• Energy spectrum continuous
• Wavefunctions not square-integrable, extend to infinity
• Described by plane waves with modifications due to the potential
• Exist for both attractive and repulsive delta potentials
• Scattering states analyzed using transmission and reflection coefficients

Resonances in Delta Potential Systems

• Resonances represent quasi-bound states in scattering systems
• Occur at specific energies where transmission probability peaks
• Associated with temporary trapping of particles near the potential
• Characterized by sharp peaks in scattering cross-sections
• Resonances in delta potentials manifest as rapid phase shifts in transmitted waves
• Resonance energies can be complex, with imaginary part related to state lifetime
• Fano resonances possible in systems with multiple delta potentials

Transmission and Reflection

Calculating Transmission and Reflection Coefficients

• Transmission coefficient (T) represents probability of particle passing through potential
• Reflection coefficient (R) represents probability of particle bouncing back
• For delta potential, transmission coefficient given by $T = \frac{4E}{4E + \alpha^2}$
• Reflection coefficient calculated as $R = 1 - T = \frac{\alpha^2}{4E + \alpha^2}$
• Sum of T and R always equals 1 due to conservation of probability
• Coefficients depend on particle energy (E) and potential strength ($\alpha$)
• At low energies, reflection dominates; at high energies, transmission dominates

Analyzing Transmission and Reflection Behavior

• Transmission increases monotonically with energy
• Reflection decreases monotonically with energy
• For repulsive potentials, transmission always less than 1
• For attractive potentials, perfect transmission possible at specific energies
• Tunneling occurs when particles transmit through classically forbidden regions
• Group velocity and phase velocity of transmitted waves differ from incident waves
• Transmission and reflection coefficients used to study quantum transport in nanostructures (quantum dots)