91影视

The anticipated value of the perturbation in the unperturbed state is the first order adjustment to the energy.

Ground state is non degenerate; Eqs. 6.9鈬�

鈥�(6.9).

localid="1658148464503" $\begin{array}{rcl}{E}^1& =& {\left(\frac{2}{a}\right)}^3{a}^3{V}_0\underset{a}{\overset{0}{鈭�}}{\sin }^2\left(\frac{蟺}{a}x\right){\sin }^2\left(\frac{蟺}{a}y\right){\sin }^2\left(\frac{蟺}{a}z\right)未\left(x-\frac{a}{4}\right)未\left(y-\frac{a}{2}\right)未\left(z-\frac{3a}{4}\right)dxdydz.\\ & =& 8V_0{\sin }^2\left(\frac{蟺}{4}\right){\sin }^2\left(\frac{蟺}{2}\right){\sin }^2\left(\frac{3蟺}{4}\right)=8V_0\left(\frac{1}{2}\right)\left(1\right)\left(\frac{1}{2}\right)=2V_0\end{array}$

First excited states:

$\begin{array}{rcl}{W}_{aa}& =& 8V_0鈭�{\sin }^2\left(\frac{蟺}{a}x\right){\sin }^2\left(\frac{蟺}{a}y\right){\sin }^2\left(\frac{2蟺}{a}z\right)未\left(x-\frac{a}{4}\right)未\left(y-\frac{a}{2}\right)未\left(z-\frac{3a}{4}\right)dxdydz.\\ & =& 8V_0\left(\frac{1}{2}\right)\left(1\right)\left(1\right)=4V_0.\end{array}$

$\begin{array}{rcl}{W}_{bb}& =& 8V_0鈭�{\sin }^2\left(\frac{蟺}{a}x\right){\sin }^2\left(\frac{2蟺}{a}y\right){\sin }^2\left(\frac{蟺}{a}z\right)未\left(x-\frac{a}{4}\right)未\left(y-\frac{a}{2}\right)未\left(z-\frac{3a}{4}\right)dxdydz.\\ & =& 8V_0\left(\frac{1}{2}\right)\left(0\right)\left(\frac{1}{2}\right)=0.\end{array}$

$\begin{array}{rcl}{W}_{cc}& =& 8V_0鈭�{\sin }^2\left(\frac{2蟺}{a}x\right){\sin }^2\left(\frac{蟺}{a}y\right){\sin }^2\left(\frac{蟺}{a}z\right)未\left(x-\frac{a}{4}\right)未\left(y-\frac{a}{2}\right)未\left(z-\frac{3a}{4}\right)dxdydz.\\ & =& 8V_0\left(1\right)\left(1\right)\left(\frac{1}{2}\right)=4V_0.\end{array}$

$\begin{array}{rcl}{W}_{ab}& =& 8V_0{\sin }^2\left(\frac{蟺}{4}\right)\sin \left(\frac{蟺}{2}\right)\sin \left(蟺\right)\sin \left(\frac{3蟺}{2}\right)\sin \left(\frac{3蟺}{4}\right)\\ & =& 0.\\ & & \end{array}$

$\begin{array}{rcl}{W}_{bc}& =& 8V_0\sin \left(\frac{蟺}{4}\right)\sin \left(\frac{蟺}{2}\right)\sin \left(蟺\right)\sin \left(\frac{蟺}{2}\right){\sin }^2\left(\frac{3蟺}{4}\right)\\ & =& 0.\\ & & \end{array}$

$\begin{array}{rcl}W& =& 4V_0\left[\begin{array}{ccc}1& 0& -1\\ 0& 0& 0\\ -1& 0& 1\end{array}\right]=4V_{0}D;\det (D-位)=\left|\begin{array}{ccc}\left(1-位\right)& 0& -1\\ 0& -位& 0\\ -1& 0& \left(1-位\right)\end{array}\right|\\ & =& -位(1-位{)}^2+位\\ & =& 0\\ & & \end{array}$

$位=0,\text{or}(1-位{)}^2=1鈬�1-位=卤1鈬�位=0.$

So the first-order corrections to the energies are $0,8V_0$.