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.

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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:

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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

Thermodynamics

Solid State Physics

Torque and Rotational Motion

Astrophysics

Medical Physics

Scientific Method 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