Question

Prove that the transitional-state wave function (5.33) does not have a well-defined energy.

Short answer

The obtained solution for the energy still hasa wave function, so the original wave function does not have a defined energy.

Step-by-step solution

Identification of given data 

The given data from equation 5.33 is:

• The sum of the two different solutions is $A{\psi }_n\left(x\right)e^{-i\left(E_n/\hslash \right)t}+B{\psi }_m{e}^{-i\left(E_m/\hslash \right)t}$

Concept/Significance of wave function

A wave function is a function that plots the probability of a particle's existence in a quantum system as a function of position, momentum, duration, and/or rotation.

Proof of the transitional state wave function does not have defined energy 

The transitional state wave function is:

$\psi \left(x,t\right)={\psi }_n\left(x\right)e^{-i\left(E_n/\hslash \right)t}+{\psi }_m{e}^{-i\left(E_m/\hslash \right)t}$

Apply the energy operator on the above equation.

$\begin{array}{rcl}\hat{E}\psi \left(x,t\right)& =& \left(i\hslash \frac{\partial }{\partial x}\right)\left({\psi }_n\left(x\right)e^{-i\left(E_n/\hslash \right)t}+{\psi }_m{e}^{-i\left(E_m/\hslash \right)t}\right)\\ & =& \left(i\hslash \frac{\partial {\psi }_n\left(x\right)e^{-i\left(E_n/\hslash \right)t}}{\partial x}\right)+\left(i\hslash \frac{\partial {\psi }_m{e}^{-i\left(E_m/\hslash \right)t}}{\partial x}\right)\\ & =& E_n{\psi }_n\left(x\right)e^{-i\left(E_n/\hslash \right)t}+E_m{\psi }_m{e}^{-i\left(E_m/\hslash \right)t}.\\ & & \end{array}$

The obtained solution for the energy still hasa wave function;therefore, the original wave function does not have defined energy.