# 16.2: Exercises - Perturbations

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

An asterisk denotes a harder problem, which you are nevertheless encouraged to try!

The following trigonometric identities may prove useful in the first two questions:

$\cos^2 A \equiv 1 − \sin^2 A \nonumber$

$\sin 2A \equiv 2 \sin A \cos A \nonumber$

$\sin A \sin B \equiv \frac{1}{2} [\cos (A − B) − \cos (A + B)] \nonumber$

$\cos A \cos B \equiv \frac{1}{2} [\cos (A − B) + \cos (A + B)] \nonumber$

1. A quantum dot is a self assembled nanoparticle in which a single electron state can be confined. A model for such an object is a particle moving in one dimension in the potential

$V (x) = \infty , \quad |x| > a, \quad V (x) = V_0 \cos (\pi x/2a), \quad |x| \leq a \nonumber$

Identify an appropriate unperturbed system and perturbation term.

Calculate the energies of the two lowest states to first order in perturbation theory.

What is the sign of $$V_0$$?

State two ways in which the colour of a material containing dots can be shifted towards the red.

2. A particle moves in one dimension in the potential

$V (x) = \infty , \quad |x| > a, \quad V (x) = V_0 \sin(\pi x/a), |x| \leq a \nonumber$

• show that the first order energy shift is zero;
• *obtain the expression for the second order correction to the energy of the ground state,

$\boxed{\Delta E_{1}^{(2)}=-\left(\frac{32 V_{0}}{15 \pi}\right)^{2} \frac{8 m a^{2}}{3 \pi^{2} \hbar^{2}}-\left(\frac{64 V_{0}}{105 \pi}\right)^{2} \frac{8 m a^{2}}{15 \pi^{2} \hbar^{2}}-\ldots} \nonumber$

3. The 1-d anharmonic oscillator: a particle of mass m is described by the Hamiltonian

$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2 \hat{x}^2 + \gamma \hat{x}^4 \nonumber$

• Assuming that $$\gamma$$ is small, use first-order perturbation theory to calculate the ground state energy;

$E_n \simeq (n + \frac{1}{2} )\hbar \omega + 3\gamma \left( \frac{\hbar}{2m\omega} \right)^2 (2n^2 + 2n + 1) \nonumber$

Hint: to evaluate matrix elements of powers of $$\hat{x}$$, write $$\hat{x}$$ in terms of the harmonic oscillator raising and lowering operators $$\hat{a}$$ and $$\hat{a}^{\dagger}$$. Recall that the raising and lowering operators are defined by

$\hat{a} \equiv \sqrt{\frac{m\omega}{2\hbar}} \hat{x} + \frac{i}{\sqrt{2m\omega \hbar}} \hat{p} \quad \text{ and } \quad \hat{a}^{\dagger} \equiv \sqrt{\frac{m\omega}{2\hbar}} \hat{x} − \frac{i}{\sqrt{2m\omega \hbar}} \hat{p} \nonumber$

with the properties that

$\hat{a}|n \rangle = \sqrt{n} |n − 1\rangle \quad \text{ and } \quad \hat{a}^{\dagger} |n \rangle = \sqrt{n + 1} |n + 1 \rangle \nonumber$

4. A 1-dimensional harmonic oscillator of mass $$m$$ carries an electric charge, $$q$$. A weak, uniform, static electric field of magnitude $$\mathcal{E}$$ is applied in the $$x$$-direction. Write down an expression for the classical electrostatic potential energy for a point particle at x.

The quantum operator is given by the same expression with $$x \rightarrow \hat{x}$$. By considering the symmetry of the integrals, or otherwise, how that, to first order in perturbation theory, the oscillator energy levels are unchanged, and calculate the second-order shift. Can you show that the second-order result is in fact exact?

Hint: to evaluate matrix elements of $$\hat{x}$$, write $$\hat{x}$$ in terms of the harmonic oscillator raising and lowering operators $$\hat{a}$$ and $$\hat{a}^{\dagger}$$ and use the results $$\hat{a}|n \rangle = \sqrt{ n}|n − 1 \rangle$$ and $$\hat{a}^{\dagger} |n \rangle = \sqrt{n + 1}|n + 1 \rangle$$. To obtain an exact solution, change variables to complete the square in the potential and show it remains a harmonic oscillator.

5. Starting from the relativistic expression for the total energy of a single particle, $$E = (m^2 c^4 + p^2 c^2 )^{1/2}$$, and expanding in powers of $$p^2$$, obtain the leading relativistic correction to the kinetic energy, for a plane wavefunction $$\Phi (x) = A \cos(kx)$$, and determine whether $$\Phi (x)$$ is an eigenstate for a relativistic free particle. Hint For normalisation of the wavefunction, it helps to keep the integral form for $$|A|^{−2} = \int \cos^2 kxdx$$.

This page titled 16.2: Exercises - Perturbations is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform.