Chapter 5: Q68E (page 191)

To describe the matter wave, does the function $A s i n (k x) c o s ω t$ have well-defined energy? Explain

Short Answer

Expert verified

The matter wave with the wave function $A \sin (k x) \cos (ω t)$ does not have a well-defined energy because the result still contains a wave function.

Step by step solution

Identification of given data

The wave function of the matter wave is $ψ (x, t) = A \sin (k x) \cos (ω t) .$

Concept/significance of eigenvalue of energy

Aneigenvalue means the amount of magnification of the "eigenvector" of asolution. When a transformation is applied to eigenvectors, the eigenvalues are constants that are multiplied by them.

Determination if the functionAsinkxcosωt has well-defined energy or not:

The energy eigenvalue of the function is:

$\hat{E} ψ (x, t) = i ℏ \frac{\partial}{\partial t} ψ (x, t)$

Replace all the values in the above equation.

$\begin{array}{rcl} \hat{E} ψ (x, t) & = & i ℏ \frac{\partial}{\partial t} (A \sin (k x) \cos (ω t)) \\ = & i ℏ ω A \sin (k x) \sin (ω t) . \end{array}$

Hence,the matter wave with the wave function $A \sin (k x) \cos (ω t)$ does not have a well-defined energy because the result still contains a wave function.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

To describe the matter wave, does the function $A s i n (k x) c o s ω t$ have well-defined energy? Explain

Short Answer

Step by step solution

Identification of given data

Concept/significance of eigenvalue of energy

Determination if the functionAsinkxcosωt has well-defined energy or not:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fluids

Particle Model of Matter

Conservation of Energy and Momentum

Geometrical and Physical Optics

Wave Optics

Medical Physics

Study anywhere. Anytime. Across all devices.

To describe the matter wave, does the function Asin(kx)cosωthave well-defined energy? Explain

Short Answer

Step by step solution

Identification of given data

Concept/significance of eigenvalue of energy

Determination if the functionAsinkxcosωt has well-defined energy or not:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fluids

Particle Model of Matter

Conservation of Energy and Momentum

Geometrical and Physical Optics

Wave Optics

Medical Physics

Study anywhere. Anytime. Across all devices.

To describe the matter wave, does the function $A s i n (k x) c o s ω t$ have well-defined energy? Explain