Question

Write out the total wave function$\mathit{\psi }\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}$.For an electron in the n=3 state of a 10nm wide infinite well. Other than the symbols a and t, the function should include only numerical values?

Short answer

The total wave function for n=3 state is given as,

$\psi (x,t)=\left(1.41\times {10}^4{m}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)Sin\left(9.42\times {10}^{16}{m}^{-1}x\right)e^{-1\left(5.12\times {10}^{13}{S}^{-1}\right)t}$

Step-by-step solution

Writing the wave function and energy expression for an infinite well.

The total wave function for an infinite well is

$\psi (x,t)=\sqrt{\frac{2}{L}}Sin\frac{n\mathrm{\pi x}}{L}{e}^{-1\left(\frac{E}{\sout{h}}\right)t}$

For n=3 state, the expression for energy becomes

$E_3=\frac{3^2{\mathrm{\pi }}^2{\sout{\mathrm{h}}}^2}{2m{L}^2}=\frac{9{\mathrm{\pi }}^2{\sout{\mathrm{h}}}^2}{2m{L}^2}$

Arriving at the final result.

Substituting the energy equation and given that L=10 nm & n=3

We get,

$$\begin{gathered} \Psi (x,t)=\sqrt{\frac{2}{10\times {10}^{-9}}}Sin\left(\frac{3\mathrm{\pi x}}{10\times {10}^{-9}}\right)e^{-1\left(\frac{9x^2\times 1.05\times {10}^{-34}}{2\times 9.1\times {10}^{-31}\times {10}^{16}}\right)t} \\ \Psi (x,t)=1.41\times {10}^4\sin (9.42\times {10}^{16}x)e^{-1(5.12\times {10}^3)}t \end{gathered}$$