Question

Find the result of operating with \(d^{2} / d x^{2}-2 x^{2}\) on the function \(e^{-a x^{2}} .\) What must the value of \(a\) be to make this function an eigenfunction of the operator? What is the eigenvalue?

Short answer

The function becomes an eigenfunction when the value of \(a\) is 1. In this case, the eigenvalue \(\lambda\) is 0.

Step-by-step solution

Apply Operator

Apply the second derivative of d²/dx² and the multiplication with -2x² on the given function \(e^{-ax^2}\).

Differentiate the Function

We begin by finding the first and second derivatives of the given function \(f(x) = e^{-ax^2}\). First derivative: \(f'(x) = \frac{d}{dx} e^{-ax^2} = (-2ax) e^{-ax^2}\) Second derivative: \(f''(x) = \frac{d^2}{dx^2} e^{-ax^2} = \frac{d}{dx} (-2ax) e^{-ax^2}\) \(= (-2a)((-2ax)e^{-ax^2} + 2ae^{-ax^2})\) \(= (4a^2x^2 - 2a) e^{-ax^2}\)

Apply the Operator

Now we apply the operator to the function \(f(x) = e^{-ax^2}\): \(f''(x) - 2x^2f(x) = (4a^2x^2 - 2a)e^{-ax^2} - 2x^2 e^{-ax^2}\) Now, simplifying the expression: \(f''(x) - 2x^2f(x) = ((4a^2x^2 - 2a) - 2x^2)e^{-ax^2}\)

Determine 'a' for Eigenfunction

For an eigenfunction, the operator applied to the function results in a constant times the original function: \(((4a^2 - 2a) - 2)_{x^{2}} e^{-a {x^2}} = \lambda e^{-a {x^2}}\) Comparing the expressions for both sides, we have: \(\lambda = 4a^2x^2 - 2a - 2x^2\) Since \(x\) can take any value and \(a\) should stay constant, we can express the eigenvalue formula as follows: \(\lambda = 4a^2 - 2a - 2\) For this function to be an eigenfunction: \(\lambda = 0 \Rightarrow 4a^2 - 2a - 2 = 0\) Solving this quadratic equation to find the value of 'a'.

Calculate Eigenvalue

Now use the quadratic formula to solve for 'a': \[a = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(4)(-2)}}{2(4)}\] \[a = \frac{2 \pm \sqrt{36}}{8}\] Either \(a = \frac{2 + 6}{8} = 1\), or \(a = \frac{2 - 6}{8} = -\frac{1}{2}\). The negative solution doesn't make physical sense as exponential decay functions require a positive decay constant. So, the value of 'a' needed for the eigenfunction is 1. Now, plug the value of 'a' into the eigenvalue formula: \(\lambda = 4(1)^2 - 2(1) - 2 = 0\) The eigenvalue is 0.

Key concepts

Differential Operator

In the world of calculus and mathematical analysis, a differential operator is a crucial tool. At its core, it involves derivatives, allowing us to analyze rates of change in different functions. The differential operator used in this exercise is \( \frac{d^2}{dx^2} - 2x^2 \). It combines taking the second derivative of a function and subtracting the function multiplied by \(-2x^2\).

Understanding differential operators is vital because they help describe various physical phenomena in engineering and physics, such as heat conduction and wave propagation. In this context, applying the operator to the exponential function \( e^{-ax^2} \) involves several steps of differentiation. This paves the way to discovering deeper properties of the function, such as its eigenvalues and eigenfunctions.

Eigenvalue Problem

An eigenvalue problem typically arises when we need to find special numbers, known as eigenvalues, associated with a differential operator and a particular function, known as the eigenfunction. Here, our task is to determine whether the given function \( e^{-ax^2} \) is an eigenfunction of the operator \( \frac{d^2}{dx^2} - 2x^2 \) and if so, what is the corresponding eigenvalue.

For a function to qualify as an eigenfunction, applying the operator to it must result in the product of some constant \( \lambda \) (the eigenvalue) and the original function itself. Mathematically, this can be written as: \[\left(\frac{d^2}{dx^2} - 2x^2\right)f(x) = \lambda f(x)\]Finding \( \lambda \) involves substituting the function into this equation and simplifying. Through this process, we can reveal the inherent relationships dictated by the operator.

Exponential Function Eigenfunction

The function \( e^{-ax^2} \) is investigated as a potential eigenfunction of the operator in question. Exponential functions are popular candidates in such analyses due to their unique properties under differentiation. For this exercise, the operations of differentiation and multiplication involved in the operator lead us to check whether \( e^{-ax^2} \) adheres to the eigenfunction criteria outlined above.

In this case, through careful application of the operator and simplification of the resulting expression, we discover that for certain values of \( a \), the function indeed behaves as an eigenfunction. Specifically, this happens when \( a = 1 \), yielding an eigenvalue of \( \lambda = 0 \). This insight provides a deeper understanding of the behavior of exponential functions in relation to differential operators.

Quadratic Equation Solution

When faced with an equation like \( 4a^2 - 2a - 2 = 0 \) during the core analysis, we approach it as a quadratic equation. Solving this equation forms a key step in identifying the parameter \( a \) which ensures the function is an eigenfunction. The quadratic formula offers a systematic approach:\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \( a = 4 \), \( b = -2 \), and \( c = -2 \).

Plugging these values into the formula, two solutions emerge: \( a = 1 \) and \( a = -\frac{1}{2} \). However, given the context of the problem, \( a = 1 \) is the logical choice, leading to the only physically meaningful solution. This process illustrates the utility of quadratic solutions in resolving conditions integral to mathematical physics problems.