Phase Difference in Delta Potential Scattering

A particle of energy $E$ is scattered off a one-dimensional potential $\lambda \delta(x)$, where $\lambda$ is a real positive constant, with a transition amplitude $t_x$. In a different experiment, the same particle is scattered off another one dimensional potential $-\lambda \delta(x)$ with a transmission amplitude $t_{-}$ In the limit $E \to 0$, the phase difference between $t_{+}$ and $t_{-}$ is

1. $\pi / 2$

2. $\pi$

3. 0

4. $3\pi / 2$

This problem explores the scattering phenomena of particles in a one-dimensional quantum system, specifically through a delta potential. The delta potential is a common idealized model in quantum mechanics due to its simplicity and ease of theoretical treatment. It helps in understanding the fundamentals of scattering theory, a key part of quantum mechanics that describes the behavior of particles interacting with potential barriers or wells. Here, we are considering a pair of delta potentials, one positive and one negative, which presents an intriguing problem of symmetry and invariance in the context of quantum scattering.

The critical aspect of this problem is to find the phase difference between the transition amplitudes as the energy of the particle approaches zero. This phase difference is essential in determining the symmetry properties and conservation laws at play in quantum scattering. The transition amplitude in quantum mechanics describes the probability amplitude for a particle to tunnel through a potential barrier; thus, the phase difference provides insight into how these probabilities vary between the two modeled potential types.

It's crucial to note the underlying principles of quantum superposition and interference, which govern how phase differences can lead to observable differences in scattering outcomes. This problem not only illustrates fundamental concepts in one-dimensional quantum systems but also highlights the role of mathematical methods in the analysis of quantum transport properties.