91影视

Two orthogonal wave functions $\mathit{蠁}\left(x\right)$and$\mathit{蠄}\left(x\right)$ represent mutually exclusive physical states: if one is true, in the sense that it is a valid description of the quantum system, the other is false, in the sense that it is an incorrect description of the quantum system.

(a)

For construction the value of ${蠄}_2$

localid="1656345370561" $$\begin{gathered} a_{+}{蠄}_0=\frac{1}{\sqrt{2魔m蝇}}\left(-魔\frac{d}{dx}+m蝇x\right){\left(\frac{m蝇}{\mathrm{蟺魔}}\right)}^{1/4}{e}^{-\frac{m蝇}{2魔}} \\ =\frac{1}{\sqrt{2魔m蝇}}{\left(\frac{m蝇}{\mathrm{蟺魔}}\right)}^{1/4}\left[-魔\left(\frac{m蝇}{2魔}\right)2x+m蝇x\right]e^{-\frac{m蝇}{2魔}} \\ =\frac{1}{\sqrt{2魔m蝇}}{\left(\frac{m蝇}{\mathrm{蟺魔}}\right)}^{1/4}2m蝇x{e}^{-\frac{m蝇}{2魔}} \\ {\left(a_{+}\right)}^2{蠄}_0=\frac{1}{\sqrt{2魔m蝇}}{\left(\frac{m蝇}{\mathrm{蟺魔}}\right)}^{1/4}2m蝇\left(-魔\frac{d}{dx}+m蝇x\right)+e^{-\frac{m蝇}{2魔}{x}^2} \\ =\frac{1}{魔}{\left(\frac{m蝇}{\mathrm{蟺魔}}\right)}^{1/4}2m蝇\left[-魔\left(1-x\frac{m蝇}{2魔}2x\right)+m蝇x^2\right]e^{-\frac{m蝇}{2魔}{x}^2} \\ ={\left(\frac{m蝇}{\mathrm{蟺魔}}\right)}^{1/4}\left(\frac{2m蝇}{魔}{x}^2-1\right)e^{-\frac{m蝇}{2魔}{x}^2} \\ {蠄}_2=\frac{1}{\sqrt{2}}{\left(a_x\right)}^2{蠄}_0 \\ =\frac{1}{\sqrt{2}}{\left(\frac{m蝇}{\mathrm{蟺魔}}\right)}^{1/4}\left(\frac{2m蝇}{魔}{x}^2-1\right)e^{-\frac{m蝇}{2魔}{x}^2} \\ \mathrm{The}\mathrm{value}\mathrm{of}{\mathrm{蠄}}_2\mathrm{is}\frac{1}{\sqrt{2}}{\left(\frac{\mathrm{m蝇}}{\mathrm{蟺魔}}\right)}^{1/4}\left(\frac{2\mathrm{m蝇}}{\mathrm{魔}}{\mathrm{x}}^2-1\right){\mathrm{e}}^{-\frac{\mathrm{m蝇}}{2\mathrm{魔}}{\mathrm{x}}^2} \end{gathered}$$

(b)

The above diagrams are of three functions.

(c)

Since ${蠄}_0$and ${蠄}_2$are even, whereas ${蠄}_1$is odd. $鈭�{{蠄}_0}^{*}{蠄}_{1}dxand鈭�{{蠄}_2}^{*}{蠄}_{1}dx$vanish automatically. The only one we need to check is $鈭�{{蠄}_2}^{*}{蠄}_{1}dx$:

localid="1656345463405" $$\begin{gathered} 鈭�{{蠄}_2}^{*}{蠄}_{1}dx=\frac{1}{\sqrt{2}}\sqrt{\frac{m蝇}{\mathrm{蟺魔}}}{鈭�}_{-鈭�}^{鈭�}(\frac{2m蝇}{\mathrm{魔}}{x}^2-1e^{-\frac{m蝇}{2\mathrm{魔}}} \\ =\sqrt{\frac{m蝇}{2\mathrm{蟺魔}}}\left({鈭�}_{-鈭�}^{鈭�}{e}^{-\frac{m蝇}{2\mathrm{魔}}}dx-\frac{2m蝇}{\mathrm{魔}}{鈭�}_{-鈭�}^{鈭�}{x}^2{e}^{-\frac{m蝇}{2\mathrm{魔}}}dx\right) \\ =\sqrt{\frac{m蝇}{2\mathrm{蟺魔}}}\left(\sqrt{\frac{\mathrm{蟺魔}}{m蝇}-}\frac{2m蝇}{\mathrm{魔}}\frac{\mathrm{魔}}{2m蝇}\sqrt{\frac{\mathrm{蟺魔}}{m蝇}-}\right) \\ =0 \\ so,{蠄}_0{蠄}_1{蠄}_2\mathrm{is}\mathrm{orthogonal} \end{gathered}$$