91影视

Question: In a crystal, the electric field of neighbouring ions perturbs the energy levels of an atom. As a crude model, imagine that a hydrogen atom is surrounded by three pairs of point charges, as shown in Figure 6.15. (Spin is irrelevant to this problem, so ignore it.)

(a) Assuming that $\mathit{r}\mathbf{鈮�}{\mathit{d}}_{\mathbf{1}}\mathbf{,}\mathit{r}\mathbf{鈮�}{\mathit{d}}_{\mathbf{2}}\mathbf{,}\mathit{r}\mathbf{鈮�}{\mathit{d}}_{\mathbf{3}}$show that

$\mathit{H}\mathbf{\text{'}}\mathbf{=}{\mathit{V}}_{\mathbf{0}}\mathbf{+}\mathbf{3}\mathbf{(}{\mathbf{尾}}_{\mathbf{1}}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{尾}}_{\mathbf{2}}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{尾}}_{\mathbf{3}}{\mathbf{z}}^{\mathbf{2}}\mathbf{)}\mathbf{-}\mathbf{(}{\mathbf{尾}}_{\mathbf{1}}\mathbf{+}{\mathbf{尾}}_{\mathbf{2}}\mathbf{+}{\mathbf{尾}}_{\mathbf{3}}\mathbf{)}{\mathit{r}}^{\mathbf{2}}$

where

$\mathbf{}{\mathit{尾}}_{\mathbf{i}}\mathbf{鈮�}\mathbf{-}\frac{\mathbf{e}}{\mathbf{4}{\mathbf{蟺蔚}}_{\mathbf{0}}}\frac{{\mathbf{q}}_{\mathbf{i}}}{{\mathbf{d}}_{\mathbf{i}}^{\mathbf{3}}}\mathbf{,}\mathit{a}\mathit{n}\mathit{d}\mathbf{}{\mathit{V}}_{\mathbf{0}}\mathbf{=}\mathbf{2}\mathbf{(}{\mathbf{尾}}_{\mathbf{1}}{\mathbf{d}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}{\mathbf{尾}}_{\mathbf{2}}{\mathbf{d}}_{\mathbf{2}}^{\mathbf{2}}\mathbf{+}{\mathbf{尾}}_{\mathbf{3}}{\mathbf{d}}_{\mathbf{3}}^{\mathbf{2}}\mathbf{)}$

(b) Find the lowest-order correction to the ground state energy.

(c) Calculate the first-order corrections to the energy of the first excited states Into how many levels does this four-fold degenerate system split,

(i) in the case of cubic symmetry${\mathit{尾}}_{\mathbf{1}}\mathbf{=}{\mathit{尾}}_{\mathbf{2}}\mathbf{=}{\mathit{尾}}_{\mathbf{3}}\mathbf{;}$, (ii) in the case of tetragonal symmetry${\mathit{尾}}_{\mathbf{1}}\mathbf{=}{\mathit{尾}}_{\mathbf{2}}\mathbf{鈮�}{\mathit{尾}}_{\mathbf{3}}\mathbf{;}$, (iii) in the general case of orthorhombic symmetry (all three different)?

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.

Question: Sometimes it is possible to solve Equation 6.10 directly, without having to expand in terms of the unperturbed wave functions (Equation 6.11). Here are two particularly nice examples.

(a) Stark effect in the ground state of hydrogen.

(i) Find the first-order correction to the ground state of hydrogen in the presence of a uniform external electric field${\mathit{E}}_{\mathbf{\text{ext}}}$ (the Stark effect-see Problem 6.36). Hint: Try a solution of the form

$\left(A+Br+C{r}^2\right){\mathit{e}}^{\mathbf{-}\mathbf{r}\mathbf{/}\mathbf{a}}\mathit{c}\mathit{o}\mathit{s}\mathit{胃}$

your problem is to find the constants , and C that solve Equation 6.10.

(ii) Use Equation to determine the second-order correction to the ground state energy (the first-order correction is zero, as you found in Problem 6.36(a)). Answer:$\mathbf{-}\mathit{m}{\left(3a^{2}e{E}_{\text{ext}}/2\mathrm{鈩�}\right)}^{\mathbf{2}}$ .

(b) If the proton had an electric dipole moment p the potential energy of the electron in hydrogen would be perturbed in the amount

$\mathit{H}\mathbf{\text{'}}\mathbf{=}\mathbf{-}\frac{\mathbf{e}\mathbf{p}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{胃}}{\mathbf{4}\mathbf{蟺}\mathbf{虁}{\mathbf{o}}_{\mathbf{0}}{\mathbf{r}}^{\mathbf{2}}}$

(i) Solve Equation 6.10 for the first-order correction to the ground state wave function.

(ii) Show that the total electric dipole moment of the atom is (surprisingly) zero, to this order.

(iii) Use Equation 6.14 to determine the second-order correction to the ground state energy. What is the first-order correction?

Sometimes it is possible to solve Equation 6.10 directly, without having to expand ${{\mathbf{蠄}}^{\mathbf{1}}}_{\mathbf{n}}$in terms of the unperturbed wave functions (Equation 6.11). Here are two particularly nice examples.

(a) Stark effect in the ground state of hydrogen.

(i) Find the first-order correction to the ground state of hydrogen in the presence of a uniform external electric field (the Stark effect-see Problem 6.36). Hint: Try a solution of the form

$\mathbf{(}\mathit{A}\mathbf{+}\mathit{B}\mathit{r}\mathbf{+}\mathit{C}{\mathit{r}}^{\mathbf{2}}\mathbf{)}{\mathit{e}}^{\mathbf{-}\mathbf{r}\mathbf{/}\mathbf{a}}\mathit{c}\mathit{o}\mathit{s}\mathit{胃}$

your problem is to find the constants , and C that solve Equation 6.10.

(ii) Use Equation to determine the second-order correction to the ground state energy (the first-order correction is zero, as you found in Problem 6.36(a)).

Answer:$\mathbf{-}\mathit{m}{\mathbf{(}\mathbf{3}{\mathit{a}}^{\mathbf{2}}\mathit{e}{\mathit{E}}_{\mathbf{ext}}\mathbf{/}\mathbf{2}\mathit{h}\mathbf{)}}^{\mathbf{2}}$

(b) If the proton had an electric dipole moment p the potential energy of the electron in hydrogen would be perturbed in the amount

$\mathit{H}\mathbf{\text{'}}\mathbf{=}\mathbf{-}\frac{\mathbf{别辫肠辞蝉胃}}{\mathbf{4}\mathbf{蟺}\stackrel{\mathbf{~}}{{\mathbf{0}}_{\mathbf{0}}{\mathbf{r}}^{\mathbf{2}}}}$

(i) Solve Equation 6.10 for the first-order correction to the ground state wave function.

(ii) Show that the total electric dipole moment of the atom is (surprisingly) zero, to this order.

(iii) Use Equation 6.14 to determine the second-order correction to the ground state energy. What is the first-order correction?

(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.

Question: Consider a quantum system with just three linearly independent states. Suppose the Hamiltonian, in matrix form, is

$$\begin{gathered} \mathbf{H}\mathbf{=}{\mathbf{V}}_{\mathbf{0}}\left(\left(1-\stackrel{藱}{o}\right)00 \\ 00\stackrel{藱}{o} \\ 0\stackrel{藱}{o}2\right) \end{gathered}$$

Where${\mathbf{V}}_{\mathbf{0}}$is a constant, and$\stackrel{藱}{\mathrm{o}}$is some small number$\left(鈭�鈮�1\right)$.

(a) Write down the eigenvectors and eigenvalues of the unperturbed Hamiltonian$\mathbf{(}\stackrel{\mathbf{藱}}{\mathbf{o}}\mathbf{=}\mathbf{0}\mathbf{)}$.

(b) Solve for the exact eigen values of H. Expand each of them as a power series in$\stackrel{\mathbf{藱}}{\mathbf{o}}$, up to second order.

(c) Use first- and second-order non degenerate perturbation theory to find the approximate eigen value for the state that grows out of the non-degenerate eigenvector of${\mathit{H}}^{\mathbf{0}}$. Compare the exact result, from (a).

(d) Use degenerate perturbation theory to find the first-order correction to the two initially degenerate eigen values. Compare the exact results.