What is a partial derivative?

It is a derivative of a function with respect to one variable while all other independent variables are held constant.

What is the time-dependent Schrӧdinger equation used for?

Given the potential energy, it determines how the wave function evolves in space and time; the wave function can then be used to calculate quantum probabilities.

How is the force related to potential energy?

In one dimension, $F=-\partial U(x,t)/\partial x$. The force is the negative spatial derivative of the potential energy.

What is a free-particle plane-wave solution when $U=0$?

One solution is $\Psi(x,t)=A e^{i(kx-\omega t)}$. It is complex, but its probability density $|\Psi|^2$ is real.

What is the separated form for a state in a time-independent potential?

It is $\Psi(x,t)=\psi(x)e^{-i\omega t}=\psi(x)e^{-iEt/\hbar}$, separating the space-only and time-only factors.

Quantum mechanicsLesson 2 of 2 in the course Quantum Mechanics

Time-Dependent and Time-Independent Schrödinger Equations

5 flashcards · 5 quiz questions · 5 match pairs · 4 fill-in sentences · English

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Summary

This lesson introduces the time-dependent Schrӧdinger equation and explains how it relates to wave behavior and a particle’s energy. It also examines the equation’s free-particle solution.

A wave equation describes how a wave changes across space and time. For a light wave, the electric field $E(x,t)$ obeys

$\frac{\partial^2 E}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2 E}{\partial t^2},$

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What is a partial derivative?: It is a derivative of a function with respect to one variable while all other independent variables are held constant.
What is the time-dependent Schrӧdinger equation used for?: Given the potential energy, it determines how the wave function evolves in space and time; the wave function can then be used to calculate quantum probabilities.
How is the force related to potential energy?: In one dimension, $F=-\partial U(x,t)/\partial x$ . The force is the negative spatial derivative of the potential energy.
What is a free-particle plane-wave solution when $U=0$ ?: One solution is $\Psi(x,t)=A e^{i(kx-\omega t)}$ . It is complex, but its probability density $|\Psi|^2$ is real.
What is the separated form for a state in a time-independent potential?: It is $\Psi(x,t)=\psi(x)e^{-i\omega t}=\psi(x)e^{-iEt/\hbar}$ , separating the space-only and time-only factors.

Quiz questions

What does the symbol $\partial$ indicate in the wave equation?Answer: A partial derivative, taken with respect to one variable while other variables are held constant.
Which expression gives the nonrelativistic energy of a particle in one dimension?Answer: $E=p^2/(2m)+U(x,t)$
For a photon with momentum $p$ , which energy-momentum relationship follows from the wave equation and de Broglie relations?Answer: $E=pc$
Which property must an acceptable time-independent wave function generally have?Answer: It must be continuous in position.
What does the lowercase function $\psi(x)$ represent in a separated state?Answer: The space-dependent part of the wave function.

Based on University Physics Volume 3 by Samuel J. Ling, Jeff Sanny, William Moebs (OpenStax), licensed CC BY 4.0. Adapted into a summary, flashcards and quizzes.

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