Question

Is the superposition wave function \(\psi(x)=\sqrt{2 / a}[\sin (n \pi x / a)+\sin (m \pi x / a)]\) an eigenfunction of the total energy operator for the particle in the box?

Short answer

The given superposition wave function \(\psi(x) = \sqrt{2 / a}[\sin (n \pi x / a)+\sin (m \pi x / a)]\) is not an eigenfunction of the total energy operator for the particle in the box. This is because when the total energy operator acts on ψ(x), the resulting expression does not have the form Eψ(x).

Step-by-step solution

Recall the total energy operator and the eigenfunction equation for a particle in a box

For a particle in a box, the total energy operator (Hamiltonian) is given by: \[H = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \] An eigenfunction ψ(x) satisfies the following equation: \[H \psi(x) = E \psi(x)\]

Apply the total energy operator on the given wave function

Now let's apply the total energy operator H on the given superposition wave function ψ(x): \(H \psi(x) = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \left[ \sqrt{\frac{2}{a}} \left(\sin \left(\frac{n \pi x}{a}\right) + \sin \left(\frac{m \pi x}{a}\right)\right) \right]\)

Calculate the second derivative

To compute Hψ(x), we need to find the second derivative with respect to x: \(\frac{d^2}{dx^2} \left[ \sqrt{\frac{2}{a}} \left(\sin \left(\frac{n \pi x}{a}\right) + \sin \left(\frac{m \pi x}{a}\right)\right) \right] \) Taking the second derivative, we get: \(\frac{d^2}{dx^2} \psi(x) = \frac{2}{a} \left(-\frac{n^2 \pi^2}{a^2}\sin\left(\frac{n \pi x}{a}\right) - \frac{m^2 \pi^2}{a^2}\sin\left(\frac{m \pi x}{a}\right)\right)\)

Verify if it satisfies the eigenfunction equation

Now, we can apply the total energy operator on the given wave function: \(H \psi(x) = -\frac{\hbar^2}{2m} \left(\frac{2}{a} \left(-\frac{n^2 \pi^2}{a^2}\sin\left(\frac{n \pi x}{a}\right) - \frac{m^2 \pi^2}{a^2}\sin\left(\frac{m \pi x}{a}\right)\right) \right)\) Simplify the expression: \(H \psi(x) = \frac{\hbar^2}{ma^2} \left(n^2 \pi^2 \sin\left(\frac{n \pi x}{a}\right) + m^2 \pi^2 \sin\left(\frac{m \pi x}{a}\right) \right)\) We see that the resulting expression, when the total energy operator acts on ψ(x), does not have the form Eψ(x). This implies that the given superposition wave function ψ(x) is not an eigenfunction of the total energy operator for the particle in the box.

Key concepts

Wave Function

In quantum mechanics, the wave function is a fundamental concept that describes the mathematical embodiment of a quantum system. It typically provides the probability amplitude of a particle's position in space. When you have a particle, like an electron, the wave function tells us where the particle is likely to be found. This probabilistic nature is a key departure from classical physics, where objects have defined paths. Formally, the wave function is usually denoted by the Greek letter psi (\(\psi(x)\)) and is a complex-valued function. The modulus squared of the wave function, \(|\psi(x)|^2\), gives the probability density. This means it predicts how likely it is to find a particle in a small region around a point. A wave function must satisfy certain conditions, like normalization, ensuring the total probability across all space is one. Learning how to manipulate and analyze wave functions is crucial because they allow us to calculate the likelihood of different quantum states, such as energy levels, of a system.

Particle in a Box

The concept of a 'particle in a box' is a simplified model used in quantum mechanics to demonstrate the behavior of a confined particle. Imagine a tiny particle trapped in an infinitely deep box. The walls of the box are impenetrable, and the particle has zero potential energy when inside. In this scenario, the particle's wave function must be zero at the boundaries (because the particle cannot exist beyond the walls). This boundary condition makes the system discrete, meaning the particle can only have certain specific energy levels, termed quantized energy levels. Such quantization is absent in classical physics, where energy can vary continuously. This model is key to understanding how energy quantization arises from physical constraints, providing a foundation for more complex quantum systems. The particle in a box exemplifies how boundary conditions lead to the emergence of discrete energy states.

Eigenfunctions

Eigenfunctions are special wave functions that, when acted upon by a quantum operator, yield a result that is simply a constant (the eigenvalue) times the original function. These are extremely important in quantum mechanics as they represent stable states of a system. The primary condition is given by an eigenvalue equation \(H \psi(x) = E \psi(x)\), where \(H\) is the Hamiltonian (the total energy operator), \(\psi(x)\) is the wave function, and \(E\) is the eigenvalue representing a measurable quantity like energy. For a particle in a box, eigenfunctions take the form of sinusoidal functions, specific solutions satisfying the boundary conditions. When multiple functions are expressed as a single wave function, as in the superposition, they typically do not maintain the form \(E \psi(x)\), hence they usually are not eigenfunctions. This was exactly the situation in the given exercise, where the superposition resulted in a function that was not a true eigenfunction of the energy operator. Understanding eigenfunctions allows us to predict how quantum states evolve over time, as they indicate the probabilistic nature of fundamental forces and particles.