91影视

$$\begin{gathered} r_1=\frac{1}{\sqrt{2}}(u+v);r_2=\frac{1}{\sqrt{2}}(u-v);r_1^2+r_2^2=\frac{1}{2}(u^2+2u鈰�v+v^2+u^2-2u鈰�v+v^2)=u^2+v^2 \\ ({鈭�}_1^2+{鈭�}_2^2)f(r_1,r_2)=\left(\frac{{鈭�}^{2}f}{鈭�x_1^2}+\frac{{鈭�}^{2}f}{鈭�y_1^2}+\frac{{鈭�}^{2}f}{鈭�z_1^2}\frac{{鈭�}^{2}f}{鈭�x_2^2}+\frac{{鈭�}^{2}f}{鈭�y_2^2}+\frac{{鈭�}^{2}f}{鈭�z_2^2}\right) \\ \frac{鈭�f}{鈭�x_1}=\frac{鈭�f}{鈭�u_x}\frac{鈭�u_x}{鈭�x_1}+\frac{鈭�f}{鈭�v_x}\frac{鈭�v_x}{鈭�x_1}=\frac{1}{\sqrt{2}}\left(\frac{鈭�f}{鈭�u_x}+\frac{鈭�f}{鈭�v_x}\right);\frac{鈭�f}{鈭�x_2}=\frac{鈭�f}{鈭�u_x}\frac{鈭�u_x}{鈭�x_2}+\frac{鈭�f}{鈭�v_x} \\ \frac{鈭�f}{鈭�x_1}=\frac{1}{\sqrt{2}}\frac{鈭�}{鈭�x_1}\left(\frac{鈭�f}{鈭�u_1}+\frac{鈭�f}{鈭�v_x}\right)=\frac{1}{\sqrt{2}}\left(\frac{{鈭�}^{2}f}{鈭�u_x^2}\frac{鈭�u_x}{鈭�x_1}+\frac{{鈭�}^{2}f}{鈭�u_x鈭�v_x}\frac{鈭�v_x}{鈭�x_1}+\frac{{鈭�}^{2}f}{鈭�u_x鈭�u_x}\right) \end{gathered}$$

$$\begin{gathered} =\frac{1}{2}\left(\frac{{鈭�}^{2}f}{鈭�u_x^2}+2\frac{{鈭�}^{2}f}{鈭�u_x鈭�v_x}+\frac{{鈭�}^{2}f}{鈭�v_x^2}\right). \\ \frac{{鈭�}^2}{鈭�x_2^2}=\frac{1}{\sqrt{2}}\frac{鈭�}{鈭�x_2}\left(\frac{鈭�f}{鈭�u_x}-\frac{鈭�f}{鈭�v_x}\right)=\frac{1}{\sqrt{2}}\left(\frac{{鈭�}^{2}f}{鈭�u_x^2}\frac{鈭�u_x}{鈭�x_2}+\frac{{鈭�}^{2}f}{鈭�u_x鈭�v_x}\frac{鈭�v_x}{鈭�x_2}-\frac{{鈭�}^{2}f}{鈭�v_2鈭�u_x}\frac{鈭�u_x}{鈭�x_2}-\frac{{鈭�}^{2}f}{鈭�{v^2}_x}\frac{鈭�v_x}{鈭�x_2}\right). \\ =\frac{1}{2}\left(\frac{{鈭�}^{2}f}{鈭�u_x^2}-2\frac{{鈭�}^{2}f}{鈭�u_x鈭�v_x}+\frac{{鈭�}^{2}f}{鈭�v_x^2}\right). \\ So\left(\frac{{鈭�}^{2}f}{鈭�x_1^2}+\frac{{鈭�}^{2}f}{鈭�x_2^2}\right)=\left(\frac{{鈭�}^{2}f}{鈭�u_x^2}+\frac{{鈭�}^{2}f}{鈭�v_x^2}\right),\mathrm{and}\mathrm{likewise}\mathrm{for}\mathrm{y}\mathrm{and}\mathrm{z}:鈭�12+鈭�22=鈭�\mathrm{u}2+鈭�\mathrm{v}2. \\ \mathrm{H}=\frac{{\mathrm{魔}}^2}{2\mathrm{m}}\left({鈭�}_{\mathrm{u}}^2+{鈭�}_{\mathrm{v}}^2\right)+\frac{1}{2}{\mathrm{m蝇}}^2\left({\mathrm{u}}^2+{\mathrm{v}}^2\right)-\frac{\mathrm{位}}{4}{\mathrm{m蝇}}^{2}2{\mathrm{v}}^2\mathrm{H} \\ =\left[-\frac{{\mathrm{魔}}^2}{2\mathrm{m}}{鈭�}_{\mathrm{u}}^2+\frac{1}{2}{\mathrm{m蝇}}^2{\mathrm{u}}^2\right]+\left[\frac{{\mathrm{魔}}^2}{2\mathrm{m}}{鈭�}_{\mathrm{v}}^2+\frac{1}{2}{\mathrm{m蝇}}^2{\mathrm{u}}^2-\frac{1}{2}{\mathrm{位m蝇}}^2{\mathrm{v}}^2\right] \end{gathered}$$

The energy is

$\frac{3}{2}魔蝇\left(\mathrm{for}\mathrm{the}\mathrm{u}\mathrm{part}\right)\mathrm{and}\frac{3}{2}魔蝇\sqrt{1-\mathrm{位}}\left(\mathrm{for}\mathrm{the}\mathrm{v}\mathrm{part}\right):E_{gs}=\frac{3}{2}魔蝇\left(1+\sqrt{1-位}\right).$

The ground state for a one-dimensional oscillator is

localid="1655995511764"

$$\begin{gathered} 3魔蝇\left(1-\frac{位}{4}\right)>\frac{3}{2}魔蝇\left(1+\sqrt{1-位}\right)鈬�2-\frac{位}{2}>1+\sqrt{1-位}鈬�1-\frac{位}{2}>\sqrt{1-位}鈬�1-位-\frac{{位}^2}{4}>1-位. \\ \mathrm{It}\mathrm{checks}.\mathrm{In}\mathrm{fact},\mathrm{expanding}\mathrm{the}\mathrm{exact}\mathrm{answer}\mathrm{in}\mathrm{powers}\mathrm{of} \\ \mathrm{位},{\mathrm{E}}_{\mathrm{gs}}鈮�\frac{3}{2}\mathrm{魔蝇}\left(1+1-\frac{1}{2}\mathrm{位}\right)=3\mathrm{魔蝇}\left(1-\frac{\mathrm{位}}{4}\right),\mathrm{we}\mathrm{recover}\mathrm{the}\mathrm{variation}\mathrm{result}. \end{gathered}$$