Question

Consider the set of functions \(\\{f(x)\\}\) of the real variable \(x\), defined in the interval \(-\infty

Short answer

Operators (b) and (d) are Hermitian; (a) and (c) are not.

Step-by-step solution

Define Hermitian Operator

A linear operator \(\hat{O}\) is Hermitian if \[ \int_{-\infty}^{\infty} \psi^{*}(x) \hat{O} \varphi(x) \, dx = \int_{-\infty}^{\infty} (\hat{O} \psi(x))^{*} \varphi(x) \, dx \] for all functions \( \psi(x) \in \{ f(x) \} \) and \(\varphi(x) \in \{ f(x) \} \).

Evaluate Operator (a)

Consider \( \hat{O_{a}} = \frac{d}{d x} + x \) Compute \int_{-\infty}^{\infty} \psi^{*}(x) \left(\frac{d}{d x} + x\right)\varphi(x) \, dx : \int_{-\infty}^{\infty} \psi^{*} \frac{d \varphi}{d x} \, dx + \int_{-\infty}^{\infty} \psi^{*} x \varphi \, dx \ Integration by parts yields: \int_{-\infty}^{\infty} \left(\frac{d}{d x} \psi\right)^{*} \varphi \, dx \x and \psi(x) \varphi(x) tend to zero at infinity. Hence, \hat{O_{a}} is not Hermitian.

Evaluate Operator (b)

Consider \( \hat{O_{b}} = -i \frac{d}{d x} + x^{2} \) Compute \int_{-\infty}^{\infty} \psi^{*}(x) \left(-i \frac{d}{d x} + x^{2}\right)\varphi(x) \, dx : \int_{-\infty}^{\infty} \psi^{*}(-i \frac{d \varphi}{d x}) + \psi^{*} x^{2} \varphi \, dx \ Integration by parts yields equality of left and right sides, thus \hat{O_{b}} is Hermitian.

Evaluate Operator (c)

Consider \( \hat{O_{c}} = i x \frac{d}{d x} \) Compute \int_{-\infty}^{\infty} \psi^{*}(x) \-i x \frac{d \varphi}{d x} \, dx \ Integration by parts shows that \hat{O_{c}} is not Hermitian

Evaluate Operator (d)

Consider \( \hat{O_{d}} = i \frac{d^{3}}{d x^{3}} \) Compute \int_{-\infty}^{\infty} \psi^{*}(x) \

Key concepts

Linear Operators

In quantum mechanics, linear operators play a crucial role in describing physical systems. An operator \(\backslashhat{O}\) is said to be linear if, for any two functions \(\backslashphi(x)\) and \(\backslashpsi(x)\), and any two constants \(a\) and \(b\), it satisfies the property: \hat{O}(a\backslashphi(x) + b\backslashpsi(x)) = a\backslashhat{O}\backslashphi(x) + b\backslashhat{O}\backslashpsi(x)\. This means that when the operator acts on a sum of functions, it behaves predictably without adding any unexpected elements.

Linear operators are essential because they preserve the structure of the linear space in which quantum states exist. For example, in the problem, operators like \(\backslashfrac{d}{dx} + x\) or \(-i \backslashfrac{d}{dx} + x^2\) are linear, as they meet this property. Understanding linearity helps in simplifying the complex operations needed in quantum mechanics.

Integration by Parts

Integration by parts is a valuable technique used frequently in quantum mechanics to simplify integrals involving products of functions. The integration by parts formula is given by:
\[ \backslashint u \backslashfrac{dv}{dx} dx = uv - \backslashint \backslashfrac{du}{dx} v dx \]

In this method, you choose parts of your function to label as \(u\) and \(dv\), then apply the formula. This allows the transformation of a complicated integral into a simpler one.

For instance, in the problem's step-by-step solution, integration by parts was used to determine if the given operators are Hermitian. Specifically, in evaluating \(\backslashbackslashhat{O_{a}} = \backslashfrac{d}{dx} + x\), integration by parts helped us break down the integral into manageable parts and check if the operator equates on both sides of the Hermitian condition. This technique often leads to showing if boundary terms vanish, a necessary facet in proving Hermitian properties.

Complex Functions

Complex functions are fundamental in quantum mechanics because they can describe wave functions, probabilities, and amplitudes. A complex function can be written in the form \(f(x) = u(x) + iv(x) \), where \(u(x)\) and \(v(x)\) are real-valued functions, and \(i\) is the imaginary unit.

In quantum mechanics, the Hermitian property of operators involves complex conjugation, where a complex function \(f^*(x)\) is represented by \(f^*(x) = u(x) - iv(x)\). This conjugation plays a critical role when testing the Hermitian property of operators, as seen in the problem.

The inner product used to define Hermitian operators incorporates the complex conjugate of the first function and the original second function, as shown:
\[ \backslashint_{-\backslashinfty}^{\backslashinfty} \backslashpsi^{*}(x) \backslashhat{O}\backslashvarphi(x) dx = \backslashint_{-\backslashinfty}^{\backslashinfty} (\backslashhat{O}\backslashpsi(x))^{*} \backslashvarphi(x) dx \]

This definition helps ensure the operators used are physically meaningful, often leading to real eigenvalues for measurable quantities like energy.