An electron is moving as a free particle in the −x-x-direction with momentum that has magnitude 4.50×10−244.50\$$times10^{-24} kg-m/s. $$What is the one-dimensional time-dependent wave function of the electron?

Ch 40: Quantum Mechanics I: Wave Functions

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 40: Quantum Mechanics I: Wave Functions

Problem 1

Chapter 40, Problem 1

An electron is moving as a free particle in the $- x$ -direction with momentum that has magnitude $4.50 \times 10^{- 24}$ kg-m/s. What is the one-dimensional time-dependent wave function of the electron?

Verified step by step guidance

Understand the problem: The question asks for the one-dimensional time-dependent wave function of an electron moving as a free particle. The wave function for a free particle is generally expressed as a plane wave, which depends on the particle's momentum and energy.

Recall the general form of the time-dependent wave function for a free particle: $ \psi(x, t) = A e^{i(kx - \omega t)} $, where $ A $ is the amplitude, $ k $ is the wave number, and $ \omega $ is the angular frequency. These quantities are related to the particle's momentum and energy.

Determine the wave number $ k $: The wave number is related to the momentum $ p $ by the equation $ k = \frac{p}{\hbar} $, where $ \hbar $ is the reduced Planck's constant ($ \hbar = 1.0545718 \times 10^{-34} \, \text{J·s} $). Substitute the given momentum $ p = 4.50 \times 10^{-24} \, \text{kg·m/s} $ into this equation to find $ k $.

Determine the angular frequency $ \omega $: The angular frequency is related to the energy $ E $ of the particle by $ \omega = \frac{E}{\hbar} $. For a free particle, the energy is given by $ E = \frac{p^2}{2m} $, where $ m $ is the mass of the electron ($ m = 9.10938356 \times 10^{-31} \, \text{kg} $). Use the given momentum to calculate $ E $, and then substitute into $ \omega = \frac{E}{\hbar} $.

Write the final wave function: Substitute the values of $ k $ and $ \omega $ into the general form $ \psi(x, t) = A e^{i(kx - \omega t)} $. Since the electron is moving in the -x-direction, the wave function will have a negative sign in front of $ kx $, resulting in $ \psi(x, t) = A e^{i(-kx - \omega t)} $.

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Key Concepts

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

Wave Function

In quantum mechanics, the wave function is a mathematical description of the quantum state of a particle. It contains all the information about the particle's position, momentum, and other physical properties. The wave function is typically denoted by the Greek letter psi (Ψ) and is a complex-valued function of position and time, which can be used to calculate probabilities of finding the particle in various states.

Momentum and Wave-Particle Duality

Momentum is a fundamental property of moving objects, defined as the product of mass and velocity. In quantum mechanics, particles like electrons exhibit wave-particle duality, meaning they can behave both as particles and as waves. The momentum of a particle is related to its wave properties through the de Broglie wavelength, which connects the particle's momentum to the wavelength of its associated wave function.

Time-Dependent Schrödinger Equation

The time-dependent Schrödinger equation is a key equation in quantum mechanics that describes how the wave function of a quantum system evolves over time. It incorporates both the kinetic and potential energy of the system and is essential for predicting the behavior of particles. Solving this equation allows us to determine the time-dependent wave function of a particle, which is crucial for understanding its dynamics.

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Related Practice

Textbook Question

An electron is moving as a free particle in the $- x$ -direction with momentum that has magnitude $4.50 \times 10^{- 24}$ kg*m/s. Let $k_{2} = 3 k_{1} = 3 k$ . At $t equals 0$ , the probability distribution function $∣ Ψ (x, t) ∣^{2}$ has a maximum at $x equals 0$ .

(a) What is the smallest positive value of $x$ for which the probability distribution function has a maximum at time $t = \frac{2 π}{ω}$ , where $ω = h k^{2} / 2 m$ ?

(b) From your result in part (a), what is the average speed with which the probability distribution is moving in the $+ x$ -direction?

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Textbook Question

A particle is described by a wave function $ψ (x) = A e^{- α x^{2}}$ , where $A$ and $α$ are real, positive constants. If the value of $α$ is increased, what effect does this have on (a) the particle’s uncertainty in position and (b) the particle’s uncertainty in momentum? Explain your answers.

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Textbook Question

A free particle moving in one dimension has wave function $ψ (x, t) = A [e^{i (k x - ω t)} - e^{i (2 k x - 4 ω t)}]$ where $k$ and $v$ are positive real constants.

(a) At $t equals 0$ , what are the two smallest positive values of $x$ for which the probability function $∣ ψ (x, t) ∣^{2}$ is a maximum?

(b) Repeat part (a) for time $t = \frac{2 π}{ω}$ .

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An electron is moving as a free particle in the −x-x-direction with momentum that has magnitude 4.50×10−244.50\(\times\)10^{-24} kg-m/s. What is the one-­dimensional time-­dependent wave function of the electron?

Verified video answer for a similar problem:

Key Concepts

Wave Function

Momentum and Wave-Particle Duality

Time-Dependent Schrödinger Equation

An electron is moving as a free particle in the $- x$ -direction with momentum that has magnitude $4.50 \times 10^{- 24}$ kg-m/s. What is the one-dimensional time-dependent wave function of the electron?