Skip to main content
Ch 40: One-Dimensional Quantum Mechanics
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 40: One-Dimensional Quantum MechanicsProblema 26
Capitolo 40, Problema 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.

Guida verificata passo dopo passo
1
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.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.
Video consigliato:
Percorso guidato
08:25
Kinematics Equations

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.
Video consigliato:
Percorso guidato
03:32
Superposition of Sinusoidal Wave Functions

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.
Video consigliato:
Percorso guidato
08:22
Operations with Significant Figures
Pratica correlata
Domanda del libro di testo

INT Model an atom as an electron in a rigid box of length 0.100 nm, roughly twice the Bohr radius. Calculate all the wavelengths that would be seen in the emission spectrum of this atom due to quantum jumps between these four energy levels. Give each wavelength a label λn→m to indicate the transition.

39
views
Domanda del libro di testo

Use the data from Figure 40.24 to calculate the first three vibrational energy levels of a C=O carbon-oxygen double bond.

48
views
Domanda del libro di testo

INT An electron is confined in a harmonic potential well that has a spring constant of 12.0 N/m. What is the longest wavelength of light that the electron can absorb?

57
views
Domanda del libro di testo

An electron approaches a 1.0-nm-wide potential-energy barrier of height 5.0 eV. What energy electron has a tunneling probability of (a) 10%, (b) 1.0%, and (c) 0.10%?

34
views
Domanda del libro di testo

INT Model an atom as an electron in a rigid box of length 0.100 nm, roughly twice the Bohr radius. What are the four lowest energy levels of the electron?

1294
views
Domanda del libro di testo

A 2.0-μm-diameter water droplet is moving with a speed of 1.0 μm/s in a 20-μm-long box. Estimate the particle’s quantum number.

33
views