Solve ihbarpartialPsi/partialt=-hbar^2/2mpartial^2Psi/partialx^2+VPsi

$$i \hbar \frac { \partial \Psi } { \partial t } = - \frac { \hbar ^ { 2 } } { 2 m } \frac { \partial ^ { 2 } \Psi } { \partial x ^ { 2 } } + V \Psi$$

Solve for V

$$\left\{\begin{matrix}V=0\text{, }&m\neq 0\\V\in \mathrm{C}\text{, }&\Psi =0\text{ and }m\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2m.

$$iℏ\frac{\mathrm{d}(\Psi )}{\mathrm{d}t}\times 2m=\left(-\frac{ℏ^{2}}{2m}\right)\frac{\mathrm{d}(\Psi )}{\mathrm{d}x^{2}}\times 2m+V\Psi \times 2m$$

Multiply i and 2 to get 2i.

$$2iℏ\frac{\mathrm{d}(\Psi )}{\mathrm{d}t}m=\left(-\frac{ℏ^{2}}{2m}\right)\frac{\mathrm{d}(\Psi )}{\mathrm{d}x^{2}}\times 2m+V\Psi \times 2m$$

Express \left(-\frac{ℏ^{2}}{2m}\right)\frac{\mathrm{d}(\Psi )}{\mathrm{d}x^{2}} as a single fraction.

$$2iℏ\frac{\mathrm{d}(\Psi )}{\mathrm{d}t}m=\frac{-ℏ^{2}\frac{\mathrm{d}(\Psi )}{\mathrm{d}x^{2}}}{2m}\times 2m+V\Psi \times 2m$$

Express \frac{-ℏ^{2}\frac{\mathrm{d}(\Psi )}{\mathrm{d}x^{2}}}{2m}\times 2 as a single fraction.

$$2iℏ\frac{\mathrm{d}(\Psi )}{\mathrm{d}t}m=\frac{-ℏ^{2}\frac{\mathrm{d}(\Psi )}{\mathrm{d}x^{2}}\times 2}{2m}m+V\Psi \times 2m$$

Cancel out 2 in both numerator and denominator.

$$2iℏ\frac{\mathrm{d}(\Psi )}{\mathrm{d}t}m=\frac{-ℏ^{2}\frac{\mathrm{d}(\Psi )}{\mathrm{d}x^{2}}}{m}m+V\Psi \times 2m$$

Express \frac{-ℏ^{2}\frac{\mathrm{d}(\Psi )}{\mathrm{d}x^{2}}}{m}m as a single fraction.

$$2iℏ\frac{\mathrm{d}(\Psi )}{\mathrm{d}t}m=\frac{-ℏ^{2}\frac{\mathrm{d}(\Psi )}{\mathrm{d}x^{2}}m}{m}+V\Psi \times 2m$$

Cancel out m in both numerator and denominator.

$$2iℏ\frac{\mathrm{d}(\Psi )}{\mathrm{d}t}m=-ℏ^{2}\frac{\mathrm{d}(\Psi )}{\mathrm{d}x^{2}}+V\Psi \times 2m$$

Swap sides so that all variable terms are on the left hand side.

$$-ℏ^{2}\frac{\mathrm{d}(\Psi )}{\mathrm{d}x^{2}}+V\Psi \times 2m=2iℏ\frac{\mathrm{d}(\Psi )}{\mathrm{d}t}m$$

Add ℏ^{2}\frac{\mathrm{d}(\Psi )}{\mathrm{d}x^{2}} to both sides.

$$V\Psi \times 2m=2iℏ\frac{\mathrm{d}(\Psi )}{\mathrm{d}t}m+ℏ^{2}\frac{\mathrm{d}(\Psi )}{\mathrm{d}x^{2}}$$

The equation is in standard form.

$$2m\Psi V=0$$

Divide 0 by 2\Psi m.

$$V=0$$

Solve for m

$$m\neq 0$$

$$\Psi =0\text{ or }V=0$$