91影视

Two identical spin-zero bosons are placed in an infinite square well (Equation 2.19). They interact weakly with one another, via the potential

$\mathit{V}\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}\mathbf{,}}{\mathbf{x}}_{\mathbf{2}}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathit{a}{\mathbf{V}}_{\mathbf{0}}\mathbf{未}\mathbf{(}{\mathbf{x}}_{\mathbf{1}}\mathbf{-}{\mathbf{x}}_{\mathbf{2}}\mathbf{)}\mathbf{.}$ (2.19).

(where ${\mathit{V}}_{\mathbf{0}}$is a constant with the dimensions of energy, and a is the width of the well).

(a)First, ignoring the interaction between the particles, find the ground state and the first excited state鈥攂oth the wave functions and the associated energies.

(b) Use first-order perturbation theory to estimate the effect of the particle鈥� particle interaction on the energies of the ground state and the first excited state.

(a) Find the second-order correction to the energies$\left(E_n^2\right)$for the potential in Problem 6.1. Comment: You can sum the series explicitly, obtaining -for odd n.

(b) Calculate the second-order correction to the ground state energy$\mathbf{\left(}{\mathbf{E}}_{\mathbf{0}}^{\mathbf{2}}\mathbf{\right)}$for the potential in Problem 6.2. Check that your result is consistent with the exact solution.

Consider a charged particle in the one-dimensional harmonic oscillator potential. Suppose we turn on a weak electric field (E), so that the potential energy is shifted by an amount${\mathbf{H}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{-}\mathbf{qEx}$.(a) Show that there is no first-order change in the energy levels, and calculate the second-order correction. Hint: See Problem 3.33.

(b) The Schr枚dinger equation can be solved directly in this case, by a change of variables${\mathit{x}}^{\mathbf{\text{'}}}\mathbf{鈮�}\mathit{x}\mathbf{-}\left(\mathrm{qE}/{\mathrm{尘蝇}}^2\right)$. Find the exact energies, and show that they are consistent with the perturbation theory approximation.

Problem 6.6 Let the two "good" unperturbed states be

${\mathbf{蠄}}_{\mathbf{卤}}^{\mathbf{0}}\mathbf{=}\mathbf{伪}\mathbf{卤}{\mathbf{蠄}}_{\mathbf{a}}^{\mathbf{0}}\mathbf{+}{\mathbf{尾}}_{\mathbf{卤}}{\mathbf{蠄}}_{\mathbf{b}}^{\mathbf{0}}$

where${\mathbf{伪}}_{\mathbf{卤}}$and${\mathbf{尾}}_{\mathbf{卤}}$are determined (up to normalization) by Equation 6.22(orEquation6.24). Show explicitly that

(a)are orthogonal;role="math" localid="1655966589608" $\mathbf{(}\mathbf{鉄�}{\mathbf{蠄}}_{\mathbf{+}}^{\mathbf{0}}\mathbf{鈭�}{\mathbf{蠄}}_{\mathbf{-}}^{\mathbf{0}}\mathbf{鉄�}\mathbf{=}\mathbf{0}\mathbf{)}\mathbf{;}$

(b) $\mathbf{鉄�}{\mathbf{蠄}}_{\mathbf{+}}^{\mathbf{0}}\mathbf{\left|}{\mathbf{H}}^{\mathbf{\text{'}}}\mathbf{\right|}{\mathbf{蠄}}_{\mathbf{-}}^{\mathbf{0}}\mathbf{鉄�}\mathbf{=}\mathbf{0}\mathbf{;}$

(c)$\mathbf{鉄�}{\mathbf{蠄}}_{\mathbf{卤}}^{\mathbf{0}}\mathbf{\left|}{\mathbf{H}}^{\mathbf{\text{'}}}\mathbf{\right|}{\mathbf{蠄}}_{\mathbf{卤}}^{\mathbf{0}}\mathbf{鉄�}\mathbf{=}{\mathbf{E}}_{\mathbf{卤}}^{\mathbf{1}}\mathbf{,}$with${\mathbf{E}}_{\mathbf{卤}}^{\mathbf{1}}$given by Equation 6.27.

Suppose we perturb the infinite cubical well (Equation 6.30) by putting a delta function 鈥渂ump鈥� at the point$\mathbf{(}\mathbf{a}\mathbf{/}\mathbf{4}\mathbf{}\mathbf{,}\mathbf{}\mathbf{a}\mathbf{/}\mathbf{2}\mathbf{}\mathbf{,}\mathbf{}\mathbf{}\mathbf{3}\mathbf{a}\mathbf{/}\mathbf{4}\mathbf{)}$$\mathbf{:}{\mathbf{H}}^{\mathbf{\text{'}}}\mathbf{=}{\mathbf{a}}^{\mathbf{3}}{\mathbf{V}}_{\mathbf{0}}\mathbf{未}\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{a}\mathbf{/}\mathbf{4}\mathbf{)}\mathbf{未}\mathbf{(}\mathbf{y}\mathbf{-}\mathbf{a}\mathbf{/}\mathbf{2}\mathbf{)}\mathbf{未}\mathbf{(}\mathbf{z}\mathbf{-}\mathbf{3}\mathbf{a}\mathbf{/}\mathbf{4}\mathbf{)}\mathbf{.}$

Find the first-order corrections to the energy of the ground state and the (triply degenerate) first excited states.