91影视

The stationary state of a one-dimensional infinite square wellis:

${\mathbf{唯}}_{\mathbf{n}}^{\mathbf{0}}\mathbf{=}\sqrt{\frac{\mathbf{2}}{\mathbf{a}}}\mathbf{sin}\left(\frac{\mathrm{苍蟺}}{a}x\right)$

a)

For the infinite square well:

$\hat{\mathrm{H}}\text{'}=伪未\left(x-\frac{\mathrm{a}}{2}\right),伪=\mathrm{con}\mathrm{st}$

Solve the problem by considering the stationary state of a one-dimensional infinite square well, that is:

b)

Use the formula and substitute each value.

For n=1

Note that:$\mathrm{m}鈮�1,\left(\mathrm{n}=1,\mathrm{m}鈮�\mathrm{n}\right)$

for$\mathrm{m}=0鈬�\sin \left(0\right)=0$

for$\mathrm{m}=0鈬�\sin \left(\mathrm{尘蟺}\right)=0$

The first three non- zero terms (odd)

$$\begin{gathered} \mathrm{m}=3,5,7;\mathrm{n}=1 \\ {\mathrm{E}}_{\mathrm{n}}^0=\frac{{\mathrm{n}}^2{\mathrm{蟺}}^2\mathrm{魔}}{2{\mathrm{ma}}^2} \\ 鈬�{\mathrm{E}}_1^0=\frac{{\mathrm{蟺}}^2{\mathrm{魔}}^2}{2{\mathrm{ma}}^2} \\ 鈬�{\mathrm{唯}}_1^1=\frac{2\mathrm{a}}{\mathrm{a}}\sqrt{\frac{2}{\mathrm{a}}}\left[\frac{\sin \left({\displaystyle \frac{3\mathrm{蟺}}{2}}\right)}{{\displaystyle \frac{{\mathrm{蟺}}^2{\mathrm{魔}}^2}{2{\mathrm{ma}}^2}}\left(1-9\right)}\sin \left(\frac{3\mathrm{蟺虫}}{\mathrm{a}}\right)+\frac{\sin \left(\frac{5\mathrm{蟺}}{2}\right)\sin \left(\frac{5\mathrm{蟺虫}}{\mathrm{a}}\right)}{{\displaystyle \frac{{\mathrm{蟺}}^2{\mathrm{魔}}^2}{2{\mathrm{ma}}^2}}\left(1-25\right)}+\frac{\mathrm{sinin}\left(\frac{5\mathrm{蟺}}{2}\right)\sin \left(\frac{5\mathrm{蟺虫}}{\mathrm{a}}\right)}{\frac{{\mathrm{蟺}}^2{\mathrm{魔}}^2}{2{\mathrm{ma}}^2}\left(1-49\right)}\right] \\ =\frac{2\mathrm{a}}{\mathrm{a}}\sqrt{\frac{2}{\mathrm{a}}}\frac{2{\mathrm{ma}}^2}{{\mathrm{蟺}}^2}\frac{2{\mathrm{ma}}^2}{{\mathrm{蟺}}^2{\mathrm{魔}}^2}\left[\frac{1}{8}\sin \left(\frac{3\mathrm{蟺虫}}{\mathrm{a}}\right)-\frac{1}{24}\sin \left(\frac{5\mathrm{蟺虫}}{\mathrm{a}}\right)+\frac{1}{48}\sin \left(\frac{7\mathrm{蟺虫}}{\mathrm{a}}\right)\right] \end{gathered}$$

Proceed further and obtain the result as,

$=\sqrt{\frac{a}{2}}\frac{\mathrm{ma}}{{\mathrm{蟺}}^2{\mathrm{魔}}^2}\left[\sin \left(\frac{3\mathrm{蟺虫}}{\mathrm{a}}\right)-\frac{1}{3}\sin \left(\frac{5\mathrm{蟺虫}}{\mathrm{a}}\right)+\frac{1}{6}\sin \left(\frac{7\mathrm{蟺虫}}{\mathrm{a}}\right)\right]$