We consider the scattering of free particles of mass m that...

We consider the scattering of free particles of mass $m$ that move in one dimension in the potential $V(x)=-W \delta(x)$, with $W>0$. (a) For a well of finite depth $V_{0}$ and width $2 a$ the condition on the phases $\phi$ and $\phi^{\prime}$ of the even- and odd-parity wavefunctions $\psi \propto \sin (k x+\phi)$, etc., for free particles are $$ \tan (k a+\phi)=-\frac{k}{K} \cot (K a), \quad \tan \left(k a+\phi^{\prime}\right)=-\frac{k}{K} \tan (K a) $$ Show that in the limit $a \rightarrow 0, V_{0}=W / 2 a \rightarrow \infty$ we have $\tan \phi \rightarrow$ $-\hbar^{2} k / m W$ and $\phi^{\prime} \rightarrow 0$. Hence obtain the scattering cross-section by the $\delta$-function potential $$ \sigma=\frac{2}{1+(\hbar k / m W)^{2}} $$ (b) Re-derive the equation above for $\phi$ by requiring that $\psi=\sin (k|x|+\phi)$ satisfy the TISE. Convince yourself that $\psi=\sin (k x)$ is also consistent with the TISE.

We consider the scattering of free particles of mass $m$ that move in one dimension in the potential $V(x)=-W \delta(x)$, with $W>0$. (a) For a well of finite depth $V_{0}$ and width $2 a$ the condition on the phases $\phi$ and
$\phi^{\prime}$ of the even- and odd-parity wavefunctions $\psi \propto \sin (k x+\phi)$, etc., for free particles are
$$
\tan (k a+\phi)=-\frac{k}{K} \cot (K a), \quad \tan \left(k a+\phi^{\prime}\right)=-\frac{k}{K} \tan (K a)
$$
Show that in the limit $a \rightarrow 0, V_{0}=W / 2 a \rightarrow \infty$ we have $\tan \phi \rightarrow$ $-\hbar^{2} k / m W$ and $\phi^{\prime} \rightarrow 0$. Hence obtain the scattering cross-section by the $\delta$-function potential
$$
\sigma=\frac{2}{1+(\hbar k / m W)^{2}}
$$
(b) Re-derive the equation above for $\phi$ by requiring that $\psi=\sin (k|x|+\phi)$ satisfy the TISE. Convince yourself that $\psi=\sin (k x)$ is also consistent with the TISE.

Key Concepts

Boundary Conditions and the Time-Independent Schrödinger Equation

The Time-Independent Schrödinger Equation (TISE) governs the spatial behavior of quantum mechanical wave functions. When a potential has discontinuities or singularities, such as in the case of a delta function potential, precise boundary conditions must be imposed. These conditions, including the continuity of the wave function and the specific jump in its derivative, are essential for obtaining valid solutions and connecting physical observables like phase shifts and cross-sections.

Scattering Theory in Quantum Mechanics

Scattering theory investigates how particles behave when they encounter potentials, particularly focusing on how incident waves split into reflected and transmitted components upon interacting with a localized potential. This theory quantifies the effects of the potential via phase shifts and cross-sections, which reveal the nature and strength of the interaction between the particle and the potential.

Delta Function Potential

The delta function potential is an idealized, singular potential that models interactions concentrated at a single point in space. It is obtained as the limit of a finite potential well with vanishing width and diverging depth while keeping the area (or integrated strength) constant. This simplification allows for analytical solutions that capture essential features of point-like interactions in quantum systems.

Phase Shifts in Scattering

Phase shifts represent the change in the phase of a particle’s wave function due to its interaction with a potential. They are determined by matching the wave functions and their derivatives across regions with different potentials. These shifts are critical for deriving scattering observables and understanding the impact of the potential on the wave propagation and interference.

Parity and Symmetry in Wave Functions

Parity refers to the symmetry properties of wave functions under spatial inversion. In one-dimensional scattering problems, exploiting parity simplifies the analysis by allowing the separation of solutions into even and odd components. This separation leads to distinct boundary conditions and phase relations for each symmetry sector, facilitating the derivation of scattering characteristics.

Limit Process from a Finite Well to a Delta Function Potential

Taking the limit of a finite-width and finite-depth potential well to obtain a delta function potential involves simultaneously letting the width shrink to zero and the depth grow such that their product remains constant. This mathematical limit preserves the overall 'strength' of the potential and provides a simplified model that captures the essential physics of a highly localized interaction.

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