Question

To describe the matter wave, does the function $\mathit{A}\mathit{s}\mathit{i}\mathit{n}\left(kx\right)\mathit{c}\mathit{o}\mathit{s}\mathit{\omega }\mathit{t}$have well-defined energy? Explain

Short answer

The matter wave with the wave function $A\sin \left(kx\right)\cos \left(\omega t\right)$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$\psi \left(x,t\right)=A\sin \left(kx\right)\cos \left(\omega t\right).$

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:

$\hat{E}\psi \left(x,t\right)=i\hslash \frac{\partial }{\partial t}\psi \left(x,t\right)$

Replace all the values in the above equation.

$\begin{array}{rcl}\hat{E}\psi \left(x,t\right)& =& i\hslash \frac{\partial }{\partial t}\left(A\sin \left(kx\right)\cos \left(\omega t\right)\right)\\ & =& i\hslash \omega A\sin \left(kx\right)\sin \left(\omega t\right).\\ & & \end{array}$

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