91影视

The variational principle states that the ground-state energy is always smaller than or equal to the calculated with the trial wavefunction expectation value. We can approximate the wave function and energy of the ground-state by changing until the expectation value of is minimized.

${\mathbf{E}}_{\mathbf{gs}}\mathbf{鈮�}\mathbf{鉄�}{{\mathbf{蠄}}_{\mathbf{gs}}}^{\mathbf{o}}\mathbf{\left|}\mathbf{H}\mathbf{\right|}{{\mathbf{蠄}}_{\mathbf{gs}}}^{\mathbf{o}}\mathbf{鉄�}$

Where$E_{gs}$ is the energy in the ground state.

(a)

Solve the problem by using ${{蠄}_{gs}}^o$ as our trial wave function as:

The variational principle tells us that:

${\mathrm{E}}_{\mathrm{gs}}鈮�鉄�{{\mathrm{蠄}}_{\mathrm{gs}}}^{\mathrm{o}}\left|\mathrm{H}\right|{{\mathrm{蠄}}_{\mathrm{gs}}}^{\mathrm{o}}鉄�$

Using the perturbation theory as well;

$H=H^0+H^1$

Where ${\mathrm{H}}^1$ is the first order perturbation, and $H^{鈭�}$is the unperturbed Hamiltonian.

Thus,

$$\begin{gathered} 鉄�{{\mathrm{蠄}}_{\mathrm{gs}}}^0\left|\mathrm{H}\right|{{\mathrm{蠄}}_{\mathrm{gs}}}^0鉄�=鉄�{{\mathrm{蠄}}_{\mathrm{gs}}}^0\left|{\mathrm{H}}^{鈭�}\right|{{\mathrm{蠄}}_{\mathrm{gs}}}^0鉄�+鉄�{{\mathrm{蠄}}_{\mathrm{gs}}}^0\left|{\mathrm{H}}^1\right|{{\mathrm{蠄}}_{\mathrm{gs}}}^0鉄� \\ 鉄�{{\mathrm{蠄}}_{\mathrm{gs}}}^0\left|{\mathrm{H}}^{鈭�}\right|{{\mathrm{蠄}}_{\mathrm{gs}}}^0鉄�=E_{gs}^0 \\ 鉄�{{\mathrm{蠄}}_{\mathrm{gs}}}^0\left|{\mathrm{H}}^1\right|{{\mathrm{蠄}}_{\mathrm{gs}}}^0鉄�=E_{gs}^1 \end{gathered}$$

Then, $鉄�{{\mathrm{蠄}}_{\mathrm{gs}}}^0\left|\mathrm{H}\right|{{\mathrm{蠄}}_{\mathrm{gs}}}^0鉄�={\mathrm{E}}^0+{\mathrm{E}}^1鈮�{\mathrm{E}}_{\mathrm{gs}}$and this proves the statement.

(b)

The ground state's second order correction is denoted by $E_{gs}^2$ and from the second order perturbation theory,

${\mathrm{E}}_{\mathrm{gs}}^0=\underset{\mathrm{m}鈮�n}{鈭�}\frac{|鉄�{\mathrm{蠄}}_{\mathrm{m}}^0|{\mathrm{H}}^1|{{\mathrm{蠄}}_{\mathrm{gs}}}^0鉄�{|}^2}{{\mathrm{E}}_n^0-{\mathrm{E}}_{\mathrm{m}}^0}$

Therefore;

role="math" localid="1658317167106" ${\mathrm{E}}_{\mathrm{gs}}^0=\underset{\mathrm{m}鈮�\mathrm{gs}}{鈭�}\frac{|鉄�{\mathrm{蠄}}_{\mathrm{m}}^0|{\mathrm{H}}^1|{{\mathrm{蠄}}_{\mathrm{gs}}}^0鉄�{|}^2}{{\mathrm{E}}_{\mathrm{gs}}^0-{\mathrm{E}}_{\mathrm{m}}^0}$

Since role="math" localid="1658317017097" $E_{gs}^0$ is the ground state,

Then numerator is positive, but the denominator is negative as:

${{\mathrm{E}}_{\mathrm{gs}}}^0-{\mathrm{E}}_{\mathrm{m}}^0<0$ for all m.

Thus, role="math" localid="1658316906095" $|鉄�{\mathrm{蠄}}_{\mathrm{m}}^0|{\mathrm{H}}^1|{{\mathrm{蠄}}_{\mathrm{gs}}}^0鉄�{|}^2$, ${\mathrm{E}}_{\mathrm{gs}}^2$is definitely negative.