Solve -hbar^2/2mpartial^2psi/partialx^2+Upsi=Epsi

$$- \frac { \hbar ^ { 2 } } { 2 m } \frac { \partial ^ { 2 } \psi } { \partial x ^ { 2 } } + U \psi = E \psi$$

Solve for E

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

Steps for Solving Linear Equation

Multiply both sides of the equation by 2m.

$$\left(-\frac{ℏ^{2}}{2m}\right)\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}\times 2m+U\psi \times 2m=E\psi \times 2m$$

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

$$\frac{-ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}}{2m}\times 2m+U\psi \times 2m=E\psi \times 2m$$

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

$$\frac{-ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}\times 2}{2m}m+U\psi \times 2m=E\psi \times 2m$$

Cancel out 2 in both numerator and denominator.

$$\frac{-ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}}{m}m+U\psi \times 2m=E\psi \times 2m$$

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

$$\frac{-ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}m}{m}+U\psi \times 2m=E\psi \times 2m$$

Cancel out m in both numerator and denominator.

$$-ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}+U\psi \times 2m=E\psi \times 2m$$

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

$$E\psi \times 2m=-ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}+U\psi \times 2m$$

The equation is in standard form.

$$2m\psi E=2Um\psi$$

Divide both sides by 2\psi m.

$$\frac{2m\psi E}{2m\psi }=\frac{2Um\psi }{2m\psi }$$

Dividing by 2\psi m undoes the multiplication by 2\psi m.

$$E=\frac{2Um\psi }{2m\psi }$$

Divide 2U\psi m by 2\psi m.

$$E=U$$

Solve for U

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

Steps for Solving Linear Equation

Multiply both sides of the equation by 2m.

$$\left(-\frac{ℏ^{2}}{2m}\right)\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}\times 2m+U\psi \times 2m=E\psi \times 2m$$

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

$$\frac{-ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}}{2m}\times 2m+U\psi \times 2m=E\psi \times 2m$$

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

$$\frac{-ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}\times 2}{2m}m+U\psi \times 2m=E\psi \times 2m$$

Cancel out 2 in both numerator and denominator.

$$\frac{-ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}}{m}m+U\psi \times 2m=E\psi \times 2m$$

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

$$\frac{-ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}m}{m}+U\psi \times 2m=E\psi \times 2m$$

Cancel out m in both numerator and denominator.

$$-ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}+U\psi \times 2m=E\psi \times 2m$$

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

$$U\psi \times 2m=E\psi \times 2m+ℏ^{2}\frac{\mathrm{d}(\psi )}{\mathrm{d}x^{2}}$$

The equation is in standard form.

$$2m\psi U=2Em\psi$$

Divide both sides by 2\psi m.

$$\frac{2m\psi U}{2m\psi }=\frac{2Em\psi }{2m\psi }$$

Dividing by 2\psi m undoes the multiplication by 2\psi m.

$$U=\frac{2Em\psi }{2m\psi }$$

Divide 2E\psi m by 2\psi m.

$$U=E$$