Question

For the wave function in Example 2.2, find the expectation value of H, at time $\mathbf{t}\mathbf{=}\mathbf{0}$ ,the “old fashioned way:

$\mathbf{⟨}\mathbf{H}\mathbf{⟩}\mathbf{=}\int \Psi {(x,0)}^{\ast }\widehat{H}\Psi (x,0)\mathrm{dx}\mathbf{.}$

Compare the result obtained in Example 2.3. Note: Because is independent of time, there is no loss of generality in using$\mathbf{t}\mathbf{=}\mathbf{0}$

Short answer

The expectation value of$H$isrole="math" localid="1655393475453" $\frac{5{\mathrm{\hslash }}^2}{{\mathrm{ma}}^2}$ which is same as Example .

Step-by-step solution

Definition of wave function

A wave function is a function that describes the probability of a particle's quantum state as a function of position, momentum, time or spin. The variable Ψ is widely used to represent wave functions.

Finding the value of  H^Ψ(x,0)

The expectation value of $\mathrm{H}$at the time $\mathrm{t}=0$will be evaluated.

The value of$\widehat{\mathrm{H}}\mathrm{\Psi }(\mathrm{x},0)$ have to be found for the calculation of expectation value of H:

Finding the value of H

The expectation value of H can be calculated as,

Thus, the expectation value of $H$is $\frac{5{\mathrm{\hslash }}^2}{{\mathrm{ma}}^2}$which is same as Example $2.3$.