Question

Question: Show that Eq.$\mathbf{38}\mathbf{-}\mathbf{24}$ is indeed a solution of Eq.$\mathbf{38}\mathbf{-}\mathbf{22}$ by substituting $\mathbf{\psi }\left(x\right)$ and its second derivative into Eq.$\mathbf{38}\mathbf{-}\mathbf{22}$ and noting

that an identity results.

Short answer

It is shown that Eq. $38-24$is indeed a solution of Eq. $38-22$by substituting $\psi \left(x\right)$and its second derivative into Eq.$38-22$.

Step-by-step solution

 Step 1: Identifying the data given in the question

Eq. $38-24$is $\frac{d^2\psi }{d{x}^2}+k^2\psi =0$

Eq.$38-22$is$\mathrm{\psi }\left(\mathrm{x}\right)={\mathrm{Ae}}^{\mathrm{ikx}}+{\mathrm{Be}}^{-\mathrm{ikx}}$

 Step 2: Concept used to solve the question.

The solution of the equation always satisfies that equation.

 Step 3: Finding the solution

The given function is

$\mathrm{\psi }\left(\mathrm{x}\right)={\mathrm{Ae}}^{\mathrm{ikx}}+{\mathrm{Be}}^{-\mathrm{ikx}}$

Differentiating the function with respect to x

$$\begin{gathered} \frac{d\left(\psi \left(x\right)\right)}{dx}=\frac{d}{dx}\left(A{e}^{ikx}+B{e}^{ikx}\right) \\ \frac{d\psi }{dx}=ikA{e}^{ikx}-ikB{e}^{ikx} \end{gathered}$$

Differentiating again with respect to x

$$\begin{gathered} \frac{d^2\psi }{d{x}^2}=\frac{d}{dx}\left(ikA{e}^{ikx}-ikB{e}^{ikx}\right) \\ \frac{d^2\psi }{d{x}^2}={\left(ik\right)}^{2}A{e}^{ikx}+{\left(-ik\right)}^{2}B{e}^{ikx} \\ \frac{d^2\psi }{d{x}^2}=-k^{2}A{e}^{ikx}-k^{2}B{e}^{ikx} \end{gathered}$$

Now substituting the value of $\frac{d^2\psi }{d{x}^2}$and $\psi \left(x\right)$in the equation Eq.$38-24$

$$\begin{gathered} \frac{d^2\psi }{d{x}^2}+k^2\psi =-k^{2}A{e}^{ikx}-k^{2}B{e}^{ikx}+k^2\left(A{e}^{ikx}+B{e}^{ikx}\right) \\ \frac{d^2\psi }{d{x}^2}+k^2\psi =-k^{2}A{e}^{ikx}-k^{2}B{e}^{ikx}+k^{2}A{e}^{ikx}+k^{2}B{e}^{ikx} \\ \frac{d^2\psi }{d{x}^2}+k^2\psi =0 \end{gathered}$$

Since$\frac{d^2\psi }{d{x}^2}$and $\psi \left(x\right)$ satisfying the equation $\frac{d^2\psi }{d{x}^2}+k^2\psi$

Therefore, they are the solution of the equation.

Hence, it is shown that Eq. $38-24$is indeed a solution of Eq.$38-22$ by substituting $\psi \left(x\right)$ and its second derivative into Eq.$38-22$ .