3.5. Problems

3.1.

1. We consider the ground state of an electron in the hydrogen atom. Approximate the ground-state wave function by $$\left\langle{\bf r}| \psi \right\rangle= e^{-\alpha r^2}$$ and find an upper bound to the ground-state energy using variational calculus.

2. Approximate the ground state of the one-dimensional harmonic oscillator using a trial wave function $$\left\langle x | \psi \right\rangle= \begin{cases} \alpha^2 - x^2 & {\text{ for }} |x|\leq \alpha, \\ 0 & {\text{ elsewhere}}. \end{cases}$$

3.2.

The attractive potential felt by an electron in an atom is sometimes taken to be the screened potential $$V(r) = \frac{Ae^2}{r} e^{-r/\xi}.$$ Consider trial wave functions of the form $$\psi({\bf r}) = \exp(-r/\rho).$$ Minimise the variational energy for this wave function.

3.3. Sloped Well

Consider a one-dimensional potential $$V(x) = \begin{cases} \lambda x & {\text{ for }} 0<x<a; \\ \infty & {\text{ for }} x\leq 0 {\text{ and }} x\geq a. \end{cases}$$ Find the ground state for this Hamiltonian using variational calculus. Take a second order polynomial as a trial function. The polynomial should obviously satisfy the correct boundary conditions at $x=0$ and $x=a$.

3.4. Quartic potential

We consider a particle moving in one dimension in a 'quartic potential'. The Hamiltonian is given as $$H = - \frac{\hslash^2}{2m} \frac{d^2}{dx^2} + \frac{b}{4} x^4,$$ with $b$ some positive constant. Now take a trial wave function of the form $$\left\langle x | \psi\right\rangle= \frac{1}{\sqrt{\sqrt{\pi} \sigma}} e^{-x^2/(2\sigma^2)}.$$ Calculate the variational energy and minimise it to obtain the variational ground state energy.

3.5.

A particle moves in one dimension in a potential $$V(x) = V_0 \left[ \frac{b^2}{x^2} -\frac{b}{x}\right].$$

1. Show that the wave function must vanish at the origin, so that a particle starting off at the right of the origin never makes it to the left. Hint: this follows from the requirement that the expectation value for the potential energy is finite.

From now on, we restrict ourselves to $x\leq 0$.

1. Use a variational wave function of the form $$\psi(x) = \frac{x}{b} \exp(- \alpha x/b)$$ to find an estimate of the ground state energy.

2. Now consider a particle moving in $3D$, in a spherically symmetric potential of the form $$V(r) = - \frac{b}{r}.$$ Find the values of $b$ (expressed in terms of the angular momentum quantum number $l$) for which the solution of the previous part pertains to this case.

3.6.

Consider a system of two particles with equal masses $m$ and momenta ${\bf p}_1$ and ${\bf p}_2$, interacting via a potential $V(r_{12})$ where $r_{12} = \left| {\bf r}_1 - {\bf r}_2\right|$.

1. Write the Hamiltonian $\hat{H}$ of the system in terms of the momenta $$\begin{aligned} {\bf P}&= {\bf p}_1 + {\bf p}_2; \\ {\bf p}& = \frac{{\bf p}_1 - {\bf p}_2}{2} \end{aligned}$$ (all momentum vectors are operators) and of $r_{12}$. Also use the total mass $M=m_1 + m_2$ and the reduced mass $\mu = m_1 m_2 /(m_1 + m_2)$.

   Show that $\hat{H}$ can be written in the form $$\hat{H} = \frac{P^2}{2M} + \hat{H}_{12}.$$

   Also show that ${\bf p}= \hslash/(2m)\nabla_{\bf r}$ with ${\bf r}= {\bf r}_1 - {\bf r}_2$ can be written as $${\bf p}= \frac{{\bf p}_1-{\bf p}_2}{2}.$$

2. We denote by $E^{(2)}$ the ground state energy of $\hat{H}_{12}$. Give the expression for $E^{(2)}$ when $V(r) = - b^2/r$ and for $V(r) = \kappa r^2/2$.

3. Consider a system of three particles of equal mass $m$ with pairwise interactions: $$V = V\left(r_{12}\right) + V\left(r_{23}\right) + V\left(r_{31}\right).$$ Show that $$3(p_1^2 + p_2^2 + p_3^2 ) = \left( {\bf p}_1 + {\bf p}_2 + {\bf p}_3\right)^2 + \left( {\bf p}_1 - {\bf p}_2\right)^2 + \left( {\bf p}_2 - {\bf p}_3\right)^2 + \left( {\bf p}_3 - {\bf p}_1\right)^2 ,$$ and that the Hamiltonian of the three-body Hamiltonian can be written as $$H^{(3)} = \frac{{\bf P}^2}{6m} + H_{\text{rel}}^{(3)},$$ where $$H_{\text{rel}}^{(3)} = H_{12} + H_{23} + H_{31},$$ $${\bf P}= {\bf p}_1 + {\bf p}_2 + {\bf p}_3$$ and where $H_{ij}$ contains a kinetic part with a reduced mass $\mu'$. Express $\mu'$ in terms of $m$.

4. Check whether the $H_{ij}$ commute amongst each other. What can you say about the energy spectrum if this were the case?

5. Show that the three-body ground state energy $E^{(3)}$ is related to the ground state energy $E^{(2)}$ of the two-body problem described by $H_{ij}$ by the inequality $$E^{(3)} \geq 3 E^{(2)}.$$ Note that the latter depends on $\mu'$ calculated in (c).\ Hint: write $E^{(3)}$ as the expectation value of $H^{(3)}$ for the ground state $\left| \Omega \right\rangle$. Then use the fact that $\left\langle\Omega \left| H^{(2)} \right| \Omega \right\rangle\geq E^{(2)}$.

6. Give the lower bounds for the case where $V(r) = - b^2/r$ and where $V(r) = \kappa r^2/2$. How does this lower bound for the first case compare with the numerical result $$E^{(3)} \approx -1.067 m b ^4/\hslash^2?$$

3.7.

In this problem, we start from the Hamiltonian $$H = E_{\text{C}} p^2 - E_{\text{J}} \cos\delta$$ which describes a Josephson junction. Here, $\delta$ is a phase which runs between $0$ and $2\pi$.\ In the quantized form, this Hamiltonian leads to a Schrödinger equation of the form: $$\left[ -E_{\text{C}} \frac{d^2}{d\delta^2} - E_{\text{J}} \cos\delta\right] \psi(\delta) = E \psi(\delta).$$

We try to solve for the ground state of this Hamiltonian using variational calculus. We use basis functions $\psi(\delta) = \frac{1}{\sqrt{2\pi}} e^{{\rm i}n \delta}$ for integer $n$.

1. Show that these functions form an orthonormal basis.

   Formulate the Hamiltonian in this basis and give this matrix as for the three basis vectors $n=1, 0, -1$ (preferably in that order).

2. Diagonalize the matrix in the last part. Hint: use vectors reflecting the symmetry between the $n=1$ and the $n=-1$ components; the ground state wave vector has all positive elements.

3. Write a computer code which solves the problem for arbitrary basis set sizes. You can use any computer language for this.\ As a check: Taking $E_{\text{C}} = 0.2 e$V and $E_{\text{J}} = 0.6 e$V, the ground state converges for large basis sizes towards $-0.36844154 e$V.

   What is the first, simple check for the result you have obtained in (b)?

3.8. Linear variational calculus for the Cooper-pair box.

The Cooper-pair box consists of a small piece of superconductor (called the island) coupled to a larger piece (called the reservoir) via a Josephson junction: a thin, insulating layer. Superconductivity will be addressed later in this course, and Josephson junctions not at all. Detailed knowledge however is not needed to do this exercise. In a superconductor, Cooper pairs, pairs of electrons with opposite spin and momentum, form a condensate which requires a finite energy to create excitations. The Josephson junction is thin enough for allowing Cooper pairs to tunnel through it, and this tunnelling generates a particular coupling between the two superconducting volumes connected by the junction.

We use the charge basis $\left| n \right\rangle$, where $n$ denotes the number of Cooper pairs that have tunneled through the junction, and therefore the number of charges that have moved to the small superconductor.

This island has an electrostatic capacity $C$, which means that every electron contributes an amount $E_C = e^2/{2C}$ to the energy. The Hamiltonian reads $$H=4 E_C \sum_{n}(n-n_g)^2 \left| n \right\rangle\left\langle n \right| - \frac{E_J}{2}\sum_{n}\left(\left| n+1 \right\rangle\left\langle n \right|+\left| n \right\rangle\left\langle n+1 \right|\right),$$ with $n={...-1,0,1,...}$ and $n_g$ is some charge offset. The first term reflects the electrostatic charging, and the second reflects tunnelling (also called hopping). The parameter $E_J$ is the so-called Josephson energy. Crucially, note that $n_g$ is a constant (not an operator) under the control of the experimentalist. Note that the charge basis is orthonormal, $\langle n|m\rangle=\delta_{n,m}$.

We will now use linear variational calculus to estimate the ground and first-excited state energies and wave functions. Caution: please do not attempt to solve this problem analytically. Rather, use your favourite mathematical software: Mathematica, Matlab, anything you like! Please print out your code.

1. Write the matrix $H_{qp}$ in the charge basis. The matrix is of course infinite dimensional, show just a subset of it, revealing its basic structure.

2. Write the matrix $S_{qp}$ also in the charge basis.

3. Consider $E_C=10 E_J$. Restrict your trial functions to the subspace $n=-N,...,0,...+N$. Plot the ground and first-excited state energies as a function of $n_g$ in the range $n_g \in \left[-1,1\right]$. Do this for $N=1$, $N=2$ and $N=3$.

4. Repeat for $E_J=10 E_C$.

5. For $E_J=10 E_C$ and $n_g=0.5$, plot the ground and first-excited state energies as a function of $N$ in the range $N \in\left[1,5\right]$. Confirm McDonald's theorem: show that the energies decrease monotonically as you increase $N$.

6. For $E_J=10 E_C$, plot the ground and first-excited state wave functions in the charge basis. That is, plot the coefficient $c_n$ in the expansion $$\left| \Psi \right\rangle=\sum_{n=-N}^{N} c_n \left| n \right\rangle$$ Do this for $n_g=0$ and $n_g=0.5$. What choice of $N$ would you say is accurate enough?