/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 What are the complex conjugates ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 25

What are the complex conjugates of the following wavefunctions? (a) \(\Psi=3 x\) (b) \(\Psi-4-3 i\) (c) \(\Psi=\cos 4 x\) (d) \(\Psi=-1 / \hbar \sin 4 x\) (e) \(\Psi=e^{3 t}\) (f) \(\Psi=e^{-2 \pi i 4 n}\)

Short Answer

Expert verified
(a) 3x (b) -4+3i (c) \cos 4x (d) -1/\hbar \sin 4x (e) e^{3t} (f) e^{2\pi i 4n}

Step by step solution

01

Understand Complex Conjugate

The complex conjugate of a number or expression involves reversing the sign of any imaginary component. For example, if the wavefunction is of the form \(a + bi\), its complex conjugate is \(a - bi\).
02

Solve Part (a)

For the wavefunction \(\Psi = 3x\), there is no imaginary part, so the complex conjugate is the same: \(\overline{\Psi} = 3x\).
03

Solve Part (b)

The wavefunction \(\Psi = -4 - 3i\) has an imaginary part \(-3i\). To find the complex conjugate, change the sign of the imaginary part: \(\overline{\Psi} = -4 + 3i\).
04

Solve Part (c)

For \(\Psi = \cos 4x\), there is again no imaginary part, so the complex conjugate is the same: \(\overline{\Psi} = \cos 4x\).
05

Solve Part (d)

The wavefunction \(\Psi = -\frac{1}{\hbar} \sin 4x\) has no imaginary part, thus the complex conjugate is the same: \(\overline{\Psi} = -\frac{1}{\hbar} \sin 4x\).
06

Solve Part (e)

For \(\Psi = e^{3t}\), there is no imaginary component in the expression, thus the complex conjugate is the same: \(\overline{\Psi} = e^{3t}\).
07

Solve Part (f)

The expression \(\Psi = e^{-2\pi i 4n}\) is a complex exponential. To find the complex conjugate, change the sign of the imaginary exponent: \(\overline{\Psi} = e^{2\pi i 4n}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wavefunctions
Wavefunctions are a fundamental concept in quantum mechanics that describe the quantum state of a system. They can often be represented by complex numbers or mathematical expressions to capture all the necessary physical information about a particle or a system.
  • Wavefunctions can tell us the probability of finding a particle in a particular state or location.
  • They are key to understanding phenomena such as interference and diffraction in quantum systems.
Wavefunctions can take various forms, such as trigonometric functions, exponential functions, or polynomials. Sometimes, they may appear as simple expressions, such as a sine or cosine function, but they can also include complex numbers involving imaginary parts. Understanding the nature of these wavefunctions helps physicists and engineers to predict how systems behave under different conditions.
Imaginary Part
The imaginary part of a complex number plays a crucial role in wavefunctions and mathematical expressions. It is typically represented by a number multiplied by the imaginary unit `i`, where `i` is defined as the square root of -1.
  • In a complex wavefunction, the imaginary part provides additional dimensions and information, often related to the wave's phase or time evolution.
  • The process of obtaining a complex conjugate involves reversing the sign of the imaginary part.
By manipulating the imaginary part of wavefunctions, we can explore various quantum states and understand more about superposition and quantum interference. This concept helps us to describe the full scope of wave characteristics beyond mere amplitude.
Complex Numbers
Complex numbers, composed of a real part and an imaginary part, are essential in mathematics and physics.They are written in the form of \(a + bi\), where \(a\) is the real component and \(bi\) is the imaginary component.Complex numbers can represent phenomena that are otherwise difficult to describe using just real numbers.
  • In wavefunctions, complex numbers enable a richer description of waves, allowing for expressions of both amplitude and phase.
  • The complex conjugate of a number \(a + bi\) is \(a - bi\), which reverses the imaginary portion.
Handling complex numbers is a fundamental skill for solving problems in quantum physics and engineering, where they help in analyzing and understanding complex systems.Applications include electrical engineering, signal processing, and quantum computing.
Mathematical Expressions
Mathematical expressions are used to succinctly communicate relationships and operations between numbers and variables. They form the language of mathematics, providing a structured way to represent and solve problems across various fields.
  • Wavefunctions use mathematical expressions to concisely describe quantum states.
  • Trigonometric functions, exponential expressions, and polynomials commonly appear in wavefunction representations.
Complex conjugates come into play often, especially when dealing with wave functions that involve imaginary numbers. Mathematical rigor in manipulating these expressions is essential for precisely modeling and predicting outcomes in quantum mechanics and other scientific inquiries. Being adept at interpreting and forming these expressions allows students to grasp complex concepts with greater ease, fostering a deeper understanding of both theoretical and applied mathematics.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Verify that the following wavefunctions are indeed eigenfunctions of the Schrodinger equation, and determine their energy eigenvalues. (a) \(\Psi=e^{a x r}\) where \(V=0\) and \(K\) is a constant (b) \(\Psi=e^{k x}\) where \(V=k, k\) is some constant potential energy, and \(K\) is a constant (c) \(\Psi=\sqrt{\frac{2}{a}} \sin \frac{\pi x}{a}\) where \(V=0\)

Find the first three energies of a proton confined to a box having a width of \(1.00 \mathrm{~nm}\).

Determine the classical velocity of an electron in a \(0.50-n m\) box having \(n=1\). Do this by assuming that the quantized energy is equal to its kinetic energy, \(1 / 2 m v^{2}\). Comment on its velocity.

Normalize the following wavefunctions over the range indicated. You may have to use the integral table in Appendix \(1 .\) (a) \(\Psi=x^{2}, x-0\) to 1 (b) \(\Psi=1 / x, x-5\) to 6 (c) \(\Psi=\cos x, x=-\pi / 2\) to \(\pi / 2\) (d) \(\Psi=e^{-n l}, f=0\) to \(x, a\) is a constant, \(d x=4 r^{2} d t\) (e) \(\Psi=e^{-i / a}, r=-\infty

What are \(\left(x^{2}\right),\left(y^{2}\right)\), and \(\left(z^{2}\right)\) for \(\Psi_{111}\) of a 3-D particlein-a-box? Assume that the operator \(x^{2}\) is simply multiplication by \(x^{2}\) and that the other operators are defined similarly. Check the integral table in Appendix 1 for needed integrals.
See all solutions

Recommended explanations on Chemistry Textbooks

Nuclear Chemistry

Read Explanation

The Earths Atmosphere

Read Explanation

Chemistry Branches

Read Explanation

Physical Chemistry

Read Explanation

Chemical Reactions

Read Explanation

Making Measurements

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.