Question

Is the superposition of two wave functions, which are solutions to the time- independent Schrödinger equation for the same potential energy, also a solution to the Schrödinger equation? a) no b) yes c) depends on the value of the potential energy d) only if \(\frac{d^{2} \psi(x)}{d x^{2}}=0\)

Short answer

Based on the given step-by-step solution, we can infer that the superposition of two wave functions, which are solutions to the time-independent Schrödinger equation for the same potential energy, is also a solution to the Schrödinger equation. The answer is yes (b).

Step-by-step solution

Write down the time-independent Schrödinger equation

The time-independent Schrödinger equation is given by: \[ -\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{d x^2} + V(x) \psi(x) = E \psi(x) \] Where \(\hbar\) is the reduced Planck constant, \(m\) is the mass of the particle, \(V(x)\) is the potential energy function, \(\psi(x)\) is the wave function, and \(E\) is the energy of the particle.

Assume we have two solutions \(\psi_1(x)\) and \(\psi_2(x)\)

Suppose \(\psi_1(x)\) and \(\psi_2(x)\) are two different solutions of the Schrödinger equation for the same potential energy. Then, they satisfy the equation: \[ -\frac{\hbar^2}{2m} \frac{d^2 \psi_i(x)}{d x^2} + V(x) \psi_i(x) = E_i \psi_i(x), i = 1, 2 \]

Consider the superposition of the two solutions \(\psi(x) = A\psi_1(x) + B\psi_2(x)\)

The superposition of \(\psi_1(x)\) and \(\psi_2(x)\) is given by: \[ \psi(x) = A \psi_1(x) + B \psi_2(x) \] Where \(A\) and \(B\) are constants.

Plug the superposition into the Schrödinger equation and verify if it satisfies the equation

Substitute the superposition \(\psi(x)=A\psi_1(x)+B\psi_2(x)\) into the Schrödinger equation: \[ -\frac{\hbar^2}{2m} \frac{d^2 (A\psi_1(x)+B\psi_2(x))}{d x^2} + V(x) (A\psi_1(x)+B\psi_2(x)) = E (A\psi_1(x)+B\psi_2(x)) \] Now, we know that \(\psi_1(x)\) and \(\psi_2(x)\) individually satisfy the Schrödinger equation, so we can write: \[ -\frac{\hbar^2}{2m} \left( A\frac{d^2 \psi_1(x)}{d x^2} + B\frac{d^2 \psi_2(x)}{d x^2} \right) + V(x) (A\psi_1(x)+B\psi_2(x)) = E_i (A\psi_1(x)+B\psi_2(x)) \] The left-hand side of the equation matches the right-hand side, showing that the superposition \(\psi(x)= A\psi_1(x)+B\psi_2(x)\) does indeed satisfy the Schrödinger equation. Therefore, the answer is: b) yes

Key concepts

Wave Function Superposition

Wave function superposition is a fundamental principle in quantum mechanics that states the combination of two or more wave functions (solutions) results in a new wave function that is also a viable solution to the governing equations. In the context of the time-independent Schrödinger equation, if \( \psi_1(x) \) and \( \psi_2(x) \) are solutions for a certain potential energy \( V(x) \), then their superposition \( \psi(x) = A\psi_1(x) + B\psi_2(x) \) is also a solution.

This concept is illustrated in solving the exercise where constants \( A \) and \( B \) represent the contribution of each individual wave function to the superposition. By plugging \( \psi(x) \) into the original Schrödinger equation, we establish that the superposition maintains the integrity of the equation, hence confirming that the superposition principle holds true for quantum systems described by the time-independent Schrödinger equation.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It introduces concepts that are drastically different from classical mechanics, primarily the dual particle-like and wave-like behavior and the interaction of energy and matter.

The time-independent Schrödinger equation, key to the exercise, is central to quantum mechanics. It describes how the quantum state (wave function) of a physical system changes in time in a stationary state, revealing the energy levels of a particle in a potential well without considering the time evolution of those states. Understanding solutions to this equation is essential for grasping how quantum systems behave and predicting outcomes in quantum experiments.

Potential Energy in Quantum Systems

In quantum systems, potential energy represents the energy due to the configuration of the particles in the system, typically arising from forces between particles. The potential energy function \( V(x) \) in the time-independent Schrödinger equation is crucial as it defines the nature of the quantum system and influences the shape and energy of the wave functions.In the context of the exercise, \( V(x) \) remains the same for both \( \psi_1(x) \) and \( \psi_2(x) \) wave functions. This shared characteristic enables the principle of superposition to hold, as the potential energy term does not alter the compatibility of the two wave functions when they are combined. Thus, potential energy in a quantum system plays a pivotal role in determining the possible states and behaviors of particles constrained by that potential.