Question

Let \(|f\rangle,|g\rangle \in \mathbb{C}(a, b)\) with the additional property that $$ f(a)=g(a)=f(b)=g(b)=0 . $$ Show that for such functions, the derivative operator \(\mathbf{D}\) is anti- hermitian. The inner product is defined as usual: $$ \langle f \mid g\rangle \equiv \int_{a}^{b} f^{*}(t) g(t) d t . $$

Short answer

Using the boundary condition that at \(a\) and \(b\) both \(f\) and \(g\) are zero, and by applying integration by parts and the definition of an inner product, we have shown that the derivative operator in this case is anti-hermitian.

Step-by-step solution

Express \(\langle f|D|g \rangle\) and \(\langle Df|g \rangle\) using definition of inner product and derivative operator

The first step involves rewriting the expressions involving the derivative operator as inner products. This results in the following expressions: \[ \langle f | D | g \rangle = \int_a^b f^*(t) g'(t) dt \] \[ \langle Df | g \rangle = \int_a^b (f'(t))^* g(t) dt \] Here, the derivative operator \( D \) is acting on the function that it is next to in each inner product, creating a derivative. The integrals run from \( a \) to \( b \).

Apply integration by parts

Apply integration by parts on \(\langle f | D | g \rangle\). With integration by parts, we obtain: \[ \int_a^b f^*(t) g'(t) dt = [f^*(t)g(t)]_a^b - \int_a^b (f^{'*}(t)) g(t) dt \] Here, note that \(g'(t)\) is equivalent to \(d/dt(g(t))\). Now we need to apply the given boundary conditions.

Apply the boundary conditions

The exercise states that \(f(a)=g(a)=f(b)=g(b)=0\). So \(f^*(t)g(t)\) final term, evaluated from \(a\) to \(b\), vanishes. We get: \[ \langle f | D | g \rangle = - \int_a^b (f^{'*}(t)) g(t) dt \] Here, note that the integrand in this expression is just the complex conjugate of the integrand in the expression for \(\langle Df | g \rangle\) from step 1.

Conclude that the derivative operator is anti-hermitian

With an hermitian operator, we expect that \(\langle f|D|g \rangle = \langle Df|g \rangle^*\). However, we have seen in this case that \(\langle f|D|g \rangle = - \langle Df|g \rangle^*\). Hence the derivative operator is not hermitian, but anti-hermitian for our given boundary conditions. This is because the two expressions are equal up to a sign change, with the one in \(\langle f|D|g \rangle\) being negative.

Key concepts

Derivative Operator

In mathematics, particularly in calculus, the derivative operator is an essential concept. It represents the rate at which a function changes at any point. For instance, in physics, it can represent how velocity changes over time, essentially acceleration.

When dealing with functions that represent quantum states—like in our exercise—the derivative operator acts on these functions to provide the derivative of the quantum state with respect to an independent variable, usually time or position. In our case, the derivative operator symbolized as \( \mathbf{D} \) generates the derivative of quantum state functions, effectively providing a way to study their rates of change within the context of quantum mechanics.

Inner Product

The inner product is a pivotal part of vector space theory, offering a way to quantify the 'closeness' of two vectors. In quantum mechanics, which often uses complex vectors, the inner product is especially important as it provides a means to determine probabilities and expectation values—crucial aspects of the theory.

In the context of our textbook problem, the inner product is defined between two functions, \( f \) and \( g \), as an integral which involves the complex conjugate of \( f \) and the function \( g \), represented as \( \langle f | g \rangle \) over a certain interval. This value can tell us about the overlap between the states described by \( f \) and \( g \) in a physical sense, which is fundamental in determining the likelihood of transitions between states in quantum systems.

Integration by Parts

Integration by parts is a mathematical technique that is derived from the product rule for differentiation. It is a way to simplify the integration of products of functions. Generally, it comes in handy when integrating the product of two functions where one can be easily differentiated, and the other can be easily integrated.

In the exercise, integration by parts is used to transform the inner product involving the derivative operator. By applying this technique, terms that vanish due to the boundary conditions are identified, ultimately simplifying the integration task and bringing us closer to proving the anti-hermitian nature of the derivative operator.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics describing the physical properties of nature at the scale of atoms and subatomic particles. It is a complex and often non-intuitive branch of physics that introduces a wide array of new concepts, like quantization of energy, wave-particle duality, and uncertainty principle, among others.

In quantum mechanics, operators, including the derivative operator, play a crucial role as they operate on wavefunctions to yield observables—measureable quantities. Our scenario deals specifically with how certain operators behave with respect to an inner product space, which is essential for understanding the physical implications and behaviors of quantum systems.

Boundary Conditions

Boundary conditions are constraints necessary to solve differential equations and are usually determined by the physics of the problem. In quantum mechanics, boundary conditions can dictate the probabilistic nature of particles and the allowed quantum states.

In our anti-hermitian operator problem, the prescribed boundary conditions tell us that the functions vanish at the boundaries of the interval \( [a, b] \). This information is leveraged when applying integration by parts to show that certain terms drop out, thus simplifying our integral expressions and enabling us to conclude the anti-hermitian character of the derivative operator.