Show that the correctly normalised wavefunction of a...

Show that the correctly normalised wavefunction of a particle trapped by the potential $V(x)=-V_{\delta} \delta(x)$ is $\psi(x)=\sqrt{K} e^{-K|x|}$, where $K=$ $m V_{\delta} / \hbar^{2}$. Show that although this wavefunction makes it certain that a measurement of $x$ will find the particle outside the well where its kinetic energy is negative, the expectation value of its kinetic energy $\left\langle E_{K}\right\rangle=\frac{1}{2} m V_{\delta}^{2} / \hbar^{2}$ is in fact positive. Reconcile this apparent paradox as follows: (i) Show that for a narrow, deep potential well of depth $V_{0}$ and half-width $a$, with $2 V_{0} a=V_{\delta}, k a \simeq W \equiv\left(2 m V_{0} a^{2} / \hbar^{2}\right)^{1 / 2}$, while $K a \simeq W^{2}$. (ii) Hence show that the contribution from inside the well to $\left\langle E_{K}\right\rangle$ is $|\psi(0)|^{2} V_{\delta}$ regardless of the value of $a .$ Explain physically what is happening as we send $a \rightarrow 0$.

Key Concepts

Regularization of Singular Potentials

Regularization involves replacing an idealized singular potential (like the delta function) with a physically more realistic, smooth potential, and then carefully taking a limit to recover the idealized case. This procedure clarifies how the singular behavior of the delta potential at a point can yield finite, physically meaningful quantities, such as the kinetic energy expectation value. In this problem, it explains how the contribution from inside the well, even as the potential becomes infinitely narrow, remains finite and crucial for the overall energy balance.

Narrow Deep Potential Wells and the Delta-Function Limit

A narrow, deep potential well offers an instructive approximation to the delta function potential. By taking the limits of the well width to zero and the well depth to infinity while keeping the product (which corresponds to the strength of the delta potential) constant, one can derive how the finite contributions inside the well lead to finite expectation values for measurable quantities. Specifically, as the width of the well approaches zero, the potential's contribution becomes concentrated yet finite, ensuring that the kinetic energy expectation remains positive.

Kinetic Energy Expectation Value

The expectation value of the kinetic energy is obtained by applying the kinetic energy operator (involving the second spatial derivative of the wavefunction) and then integrating the resulting expression weighted by the probability density. Although measurements indicate that the particle is found outside the well where one might naively think the kinetic energy is negative, the proper quantum mechanical treatment shows that the overall expectation value of the kinetic energy remains positive. This reconciles the apparent paradox by considering contributions from regions where the potential affects the wavefunction's curvature.

Normalisation of Wavefunctions

Normalization is a core concept in quantum mechanics that ensures the total probability of finding a particle in all space is one. For singular potentials like the delta potential, the exact form of the wavefunction must be normalized carefully. In this context, the exponential wavefunction ?(x) = ?K e^(–K|x|) is normalized such that its integral over all space equals one, ensuring a valid probability interpretation.

Delta Potential

A delta potential is an idealized, infinitely short ranged potential represented mathematically by a Dirac delta function. It captures the essence of a localized interaction that is strong and confined to a single point in space. In quantum mechanics, it is used to simplify problems involving bound states and tunneling, while still retaining significant physical insights despite its mathematical singularity.

Bound States in a Singular Potential

Bound states are solutions of the Schrödinger equation where the particle is confined to a finite region of space with energy less than the potential at infinity. In the case of a delta potential, even though the potential is nonzero only at a point, it supports exactly one bound state. The corresponding normalized wavefunction decays exponentially away from the location of the delta function, indicating confinement despite the potential's singularity.

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