Solve d^2psi/dx^2=-2mE/hbar^2psi

$$\frac { d ^ { 2 } \psi } { d x ^ { 2 } } = - \frac { 2 m E } { \hbar ^ { 2 } } \psi$$

Solve for E (complex solution)

$$\left\{\begin{matrix}E=0\text{, }&ℏ\neq 0\\E\in \mathrm{C}\text{, }&\left(\psi =0\text{ or }m=0\right)\text{ and }ℏ\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by ℏ^{2}.

$$ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}=\left(-\frac{2mE}{ℏ^{2}}\right)\psi ℏ^{2}$$

Express \left(-\frac{2mE}{ℏ^{2}}\right)\psi as a single fraction.

$$ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}=\frac{-2mE\psi }{ℏ^{2}}ℏ^{2}$$

Express \frac{-2mE\psi }{ℏ^{2}}ℏ^{2} as a single fraction.

$$ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}=\frac{-2mE\psi ℏ^{2}}{ℏ^{2}}$$

Cancel out ℏ^{2} in both numerator and denominator.

$$ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}=-2Em\psi$$

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

$$-2Em\psi =ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}$$

The equation is in standard form.

$$\left(-2m\psi \right)E=0$$

Divide 0 by -2m\psi .

$$E=0$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=0\text{, }&ℏ\neq 0\\m\in \mathrm{C}\text{, }&\left(\psi =0\text{ or }E=0\right)\text{ and }ℏ\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by ℏ^{2}.

$$ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}=\left(-\frac{2mE}{ℏ^{2}}\right)\psi ℏ^{2}$$

Express \left(-\frac{2mE}{ℏ^{2}}\right)\psi as a single fraction.

$$ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}=\frac{-2mE\psi }{ℏ^{2}}ℏ^{2}$$

Express \frac{-2mE\psi }{ℏ^{2}}ℏ^{2} as a single fraction.

$$ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}=\frac{-2mE\psi ℏ^{2}}{ℏ^{2}}$$

Cancel out ℏ^{2} in both numerator and denominator.

$$ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}=-2Em\psi$$

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

$$-2Em\psi =ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}$$

The equation is in standard form.

$$\left(-2E\psi \right)m=0$$

Divide 0 by -2E\psi .

$$m=0$$

Solve for E

$$\left\{\begin{matrix}E=0\text{, }&ℏ\neq 0\\E\in \mathrm{R}\text{, }&\left(\psi =0\text{ or }m=0\right)\text{ and }ℏ\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by ℏ^{2}.

$$ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}=\left(-\frac{2mE}{ℏ^{2}}\right)\psi ℏ^{2}$$

Express \left(-\frac{2mE}{ℏ^{2}}\right)\psi as a single fraction.

$$ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}=\frac{-2mE\psi }{ℏ^{2}}ℏ^{2}$$

Express \frac{-2mE\psi }{ℏ^{2}}ℏ^{2} as a single fraction.

$$ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}=\frac{-2mE\psi ℏ^{2}}{ℏ^{2}}$$

Cancel out ℏ^{2} in both numerator and denominator.

$$ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}=-2Em\psi$$

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

$$-2Em\psi =ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}$$

The equation is in standard form.

$$\left(-2m\psi \right)E=0$$

Divide 0 by -2m\psi .

$$E=0$$

Solve for m

$$\left\{\begin{matrix}m=0\text{, }&ℏ\neq 0\\m\in \mathrm{R}\text{, }&\left(\psi =0\text{ or }E=0\right)\text{ and }ℏ\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by ℏ^{2}.

$$ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}=\left(-\frac{2mE}{ℏ^{2}}\right)\psi ℏ^{2}$$

Express \left(-\frac{2mE}{ℏ^{2}}\right)\psi as a single fraction.

$$ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}=\frac{-2mE\psi }{ℏ^{2}}ℏ^{2}$$

Express \frac{-2mE\psi }{ℏ^{2}}ℏ^{2} as a single fraction.

$$ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}=\frac{-2mE\psi ℏ^{2}}{ℏ^{2}}$$

Cancel out ℏ^{2} in both numerator and denominator.

$$ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}=-2Em\psi$$

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

$$-2Em\psi =ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}$$

The equation is in standard form.

$$\left(-2E\psi \right)m=0$$

Divide 0 by -2E\psi .

$$m=0$$