Question

Verify that $\mathbf{\text{Asin(kx) + Bcos(kx)}}$is a solution of equation $\mathbf{\text{(5 - 12)}}$.

Short answer

The wave function equation satisfies the Schrodinger Equation.

Step-by-step solution

Wave function equation and Schrodinger Equation.

Schrodinger Equation,

$\frac{d^2\psi }{{\displaystyle {\text{dx}}^{\text{2}}}}{\text{= -k}}^{\text{2}}\text{ψ(x),k =}\sqrt{\frac{{\displaystyle \text{2mE}}}{{\displaystyle {\text{h}}^{\text{2}}}}}\text{...................}\left(1\right)$

Wave function equation,

$\psi =\text{= A sin(kx)+B cos(kx)........}\left(1\right)$

Substituting (2) in to (1).

Substitute the values

$$\begin{gathered} \frac{d^2\psi {\displaystyle \text{(x)}}}{{\displaystyle {\text{dx}}^{\text{2}}}}{\text{= -k}}^{\text{2}}{\text{A sin(kx) -k}}^{\text{2}}\text{B cos(kx)} \\ \frac{{\displaystyle d\psi \text{(x)}}}{{\displaystyle \text{dx}}}\text{= k A cos(kx) - k B sin(kx)} \\ \frac{{\displaystyle d^2\psi \text{(x)}}}{{\displaystyle {\text{dx}}^{\text{2}}}}{\text{= -k}}^{\text{2}}{\text{A sin(kx) -k}}^{\text{2}} \\ =\frac{{\displaystyle {\text{k}}^{\text{2}}}}{{\psi }_{\left(x\right)}} \end{gathered}$$So, thatwave function equation satisfies the Schrodinger Equation.