Question

Find the result of operating with \(\left(1 / r^{2}\right)(d / d r)\left(r^{2} d / d r\right)+2 / r\) on the function \(A e^{-b r} .\) What must the values of \(A\) and \(b\) be to make this function an eigenfunction of the operator?

Short answer

In summary, to make the function \(Ae^{-br}\) an eigenfunction of the operator, the value of \(b\) must satisfy the equation \(b^2 - 2b = \lambda\), where \(\lambda\) is the eigenvalue, and the value of \(A\) can be any constant, as long as it's not equal to zero.

Step-by-step solution

Rewrite the operator

We are given the operator \(\left(\frac{1}{r^2}\right)(\frac{d}{dr})\left(r^2\frac{d}{dr}\right) + \frac{2}{r}\). Let's rewrite it using more familiar notation: \[\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{d}{dr}\right) + \frac{2}{r}\]

Apply the operator on the given function

We need to apply the operator on the function \(Ae^{-br}\): \[\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{d}{dr}(Ae^{-br})\right) + \frac{2}{r}(Ae^{-br})\] First, let's find the derivative of the function with respect to \(r\): \[\frac{d}{dr}(Ae^{-br}) = -Ab\cdot e^{-br}\] Now, we need to find the derivative of \(r^2\times(-Ab\cdot e^{-br})\), which is: \[\frac{d}{dr}\left(-r^2Ab\cdot e^{-br}\right) = -2Abr\cdot e^{-br} + r^2Ab^2\cdot e^{-br}\] Next, we need to find the derivative of the above expression divided by \(r^2\): \[\frac{1}{r^2}\frac{d}{dr}(-2Abr\cdot e^{-br} + r^2Ab^2\cdot e^{-br}) = -2Ab\cdot e^{-br} + b^2\cdot e^{-br}\] Finally, apply the second term of our operator and simplify: \[\frac{2}{r}(Ae^{-br}) + \left(-2Ab\cdot e^{-br} + b^2\cdot e^{-br}\right) = (b^2 - 2b)\cdot Ae^{-br}\]

Derive the eigenvalue equation

In order for function \(Ae^{-br}\) to be an eigenfunction, the operator acting on it must return a multiple of the function itself. In other words: \[(b^2 - 2b)Ae^{-br} = \lambda Ae^{-br}\]

Solve for \(b\) and \(A\)

From the eigenvalue equation, we have: \[(b^2 - 2b)A = \lambda A\] Since the coefficient \(A\) is not equal to zero, we can divide both sides by \(A\) to get: \[b^2 - 2b = \lambda\] As a result, we find that the value of \(b\) must satisfy the above equation in order to make the given function an eigenfunction of the operator. The value of \(A\) does not affect the eigenfunction condition, and thus it can be any constant, as long as it's not equal to zero.

Key concepts

Differential Operator

The concept of a differential operator is fundamental in various fields of mathematics and physics, serving as a key tool for describing how functions vary. In the simplest terms, a differential operator is a mathematical entity that acts on a function to produce another function, representing the rate at which the original function changes.

In our exercise, the differential operator in question, \[\begin{equation}\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{d}{dr}\right) + \frac{2}{r},\end{equation}\]is slightly more complex. It includes not just simple differentiation, but also multiplication by functions of the independent variable, which in this case is the radial coordinate, r. Such operators are ubiquitous in fields like quantum mechanics and electromagnetism, where they are used to model systems that depend on the spatial variation of some quantity.

Eigenvalue Equation

The eigenvalue equation is an equation that describes the conditions under which a particular type of solution, known as an eigenfunction, exists for a differential equation. In essence, it's a special scenario where applying a differential operator to a function (the eigenfunction) outputs the same function scaled by a constant (the eigenvalue).

Mathematical Representation

In mathematical terms, if \[\begin{equation}\mathcal{L}f = \lambda f,\end{equation}\]where \[\begin{equation}\mathcal{L}\end{equation}\]is the differential operator, \[\begin{equation}\lambda\end{equation}\]is the eigenvalue, and f is the eigenfunction, we understand that f is not simply distorted or transformed by the operator—it is merely scaled.

This concept is a cornerstone in the study of linear algebra and quantum mechanics, where solutions to problems often depend on finding eigenvalues and eigenfunctions of various operators.

Physical Chemistry

Physical chemistry is the branch of chemistry focused on understanding the physical properties of molecules, the forces acting upon them, and the physical laws governing their behavior. It is an area of science where principles of physics are applied to chemical problems, blending the macroscopic classical world and the microscopic quantum world.

Instances where differential operators and eigenvalue equations play a critical role include understanding spectroscopy, reaction kinetics, thermodynamics, and quantum chemistry. These mathematical tools help characterize the energy levels of atoms and molecules, model chemical reactions, and explain the observed behavior of matter at both the molecular and atomic scales.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. It is a domain where probabilities replace determinism, wave functions replace classical trajectories, and where eigenvalues and eigenfunctions become tools to describe the quantized nature of reality.

In quantum mechanics, operators corresponding to physical observables—like energy, momentum, and position—act on wave functions. The differential operator provided in the exercise would be analogous to the Hamiltonian operator if it corresponded to the energy of a quantum system. The search for eigenfunctions and their corresponding eigenvalues is akin to finding the allowed energy levels of a system.

The eigenvalue equation provided is a simplified representation of the more complex equations encountered in quantum systems, and learning to solve them is essential for anyone seeking to understand the quantum world. The conceptual leap from the mathematics to the physical implications is one of the defining challenges and marvels of studying quantum mechanics.