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Ch 40: One-Dimensional Quantum Mechanics
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 40: One-Dimensional Quantum MechanicsProblem 26
Chapter 40, Problem 26

CALC Suppose that ψ1(x) and ψ2(x) are both solutions to the Schrödinger equation for the same potential energy U(x). Prove that the superposition ψ(x)=Aψ1(x) + Bψ2(x) is also a solution to the Schrödinger equation.

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Start with the time-independent Schrödinger equation, which is written as: -\(\frac{\hbar^2}{2m}\)\(\frac{d^2\psi(x)}{dx^2}\) + U(x)\(\psi\)(x) = E\(\psi\)(x). Here, \(\psi\)(x) is the wavefunction, U(x) is the potential energy, E is the energy, and \(\hbar\) is the reduced Planck's constant.
Substitute the superposition wavefunction \(\psi\)(x) = A\(\psi\)_1(x) + B\(\psi\)_2(x) into the Schrödinger equation. This gives: -\(\frac{\hbar^2}{2m}\)\(\frac{d^2}{dx^2}\)[A\(\psi\)_1(x) + B\(\psi\)_2(x)] + U(x)[A\(\psi\)_1(x) + B\(\psi\)_2(x)] = E[A\(\psi\)_1(x) + B\(\psi\)_2(x)].
Use the linearity of differentiation to separate the terms: -\(\frac{\hbar^2}{2m}\)[A\(\frac{d^2\psi_1(x)}{dx^2}\) + B\(\frac{d^2\psi_2(x)}{dx^2}\)] + U(x)[A\(\psi\)_1(x) + B\(\psi\)_2(x)] = E[A\(\psi\)_1(x) + B\(\psi\)_2(x)].
Since \(\psi\)_1(x) and \(\psi\)_2(x) are both solutions to the Schrödinger equation, they individually satisfy: -\(\frac{\hbar^2}{2m}\[\frac{d^2\psi_1(x)}{dx^2}\) + U(x)\(\psi\)_1(x) = E\(\psi\)_1(x) and -\(\frac{\hbar^2}{2m}\]\frac{d^2\psi_2(x)}{dx^2}\) + U(x)\(\psi\)_2(x) = E\(\psi\)_2(x). Substitute these into the equation for the superposition.
After substitution, the terms combine to show that the superposition \(\psi\)(x) = A\(\psi\)_1(x) + B\(\psi\)_2(x) also satisfies the Schrödinger equation. This proves that the superposition of solutions is itself a solution, as the Schrödinger equation is linear.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Schrödinger Equation

The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is a key component in understanding wave functions, which represent the probabilities of finding a particle in various states. The equation can be time-dependent or time-independent, depending on the context, and is central to predicting the behavior of quantum systems.
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Superposition Principle

The superposition principle in quantum mechanics states that if two or more solutions to a linear equation exist, any linear combination of these solutions is also a solution. This principle is crucial for understanding quantum states, as it allows for the combination of different wave functions to form new states. In the context of the Schrödinger equation, it implies that multiple wave functions can coexist and contribute to the overall state of a system.
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Linear Operators

In quantum mechanics, observables are represented by linear operators acting on wave functions. The Schrödinger equation is a linear differential equation, meaning that the operators involved satisfy the properties of linearity. This linearity is essential for the superposition principle to hold, allowing for the addition of wave functions and ensuring that the resulting function remains a valid solution to the equation.
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