Chapter 19 Exercises: The Variational Principle

Contributors to Quantum Mechanics: From Wavefunctions to Qubits

Chapter 19 Exercises: The Variational Principle

Part A: Conceptual Questions (⭐)

These questions test your understanding of the core ideas. No (or minimal) calculations required.

A.1 State the variational theorem in your own words. Why is the result an inequality rather than an equality? Under what condition does equality hold?

A.2 A student claims: "The variational method always underestimates the ground state energy, because you're using a guess instead of the exact wavefunction." Explain precisely why this student is wrong. What does the variational method always do — overestimate or underestimate?

A.3 Suppose you apply the variational method with two different trial functions to the same Hamiltonian. Trial function A gives an energy of $-3.2$ eV, and trial function B gives an energy of $-3.5$ eV. What can you conclude about the true ground state energy $E_0$? Which trial function is the better approximation?

A.4 Can the variational method give you a lower bound on $E_0$? Explain why or why not. (Hint: think about what would be needed to establish $E_0 \geq E_{\text{lower}}$.)

A.5 Why is a Gaussian trial function $\psi \propto e^{-\alpha r^2}$ a poor choice for the hydrogen atom, even though the variational theorem still gives a valid upper bound? What specific physical features does the Gaussian get wrong?

A.6 Explain the physical meaning of $Z_{\text{eff}} = 27/16$ for the helium ground state. Why is $Z_{\text{eff}} < 2$ rather than $Z_{\text{eff}} > 2$?

A.7 In the LCAO treatment of H₂⁺, why does the bonding orbital have lower energy than the antibonding orbital? Give both a mathematical reason (in terms of the exchange integral $K$) and a physical reason (in terms of where the electron probability density is concentrated).

A.8 The zero-variance principle in VMC states that the variance of the local energy vanishes when the trial function is exact. Why is this true? What is $E_L(\mathbf{r})$ when $\psi$ is an exact eigenstate?

Part B: Applied Problems (⭐⭐)

These problems require direct application of the variational method.

B.1: Variational Bound for the Infinite Square Well

Consider a particle in an infinite square well of width $L$ (walls at $x = 0$ and $x = L$). Use the trial function:

$$\psi(x; \alpha) = x^{\alpha}(L - x)^{\alpha}$$

(a) Verify that this trial function satisfies the boundary conditions for any $\alpha > 0$.

(b) Compute $\langle \psi | \psi \rangle$, $\langle T \rangle$, and $\langle V \rangle$ (where $V = 0$ inside the well) as functions of $\alpha$. You will need the Beta function $B(a, b) = \int_0^1 t^{a-1}(1-t)^{b-1} dt = \Gamma(a)\Gamma(b)/\Gamma(a+b)$.

(c) Show that $E(\alpha) = \frac{\hbar^2}{2m} \cdot \frac{\alpha(2\alpha - 1)}{L^2(2\alpha + 1)}$ (after some algebra with the Beta function ratios).

(d) Minimize over $\alpha$ and compare with the exact ground state energy $E_1 = \pi^2\hbar^2/(2mL^2)$.

(e) What value of $\alpha$ gives the best bound? How close is it to the exact energy?

B.2: Helium Ground State — The Full Calculation

Reproduce the helium variational calculation from Section 19.3 in full detail.

(a) Starting from the trial function $\psi(\mathbf{r}_1, \mathbf{r}_2; Z') = (Z'^3/\pi) e^{-Z'(r_1 + r_2)}$, verify that $\langle \hat{H}_0(Z') \rangle = -(Z')^2$.

(b) Using the hydrogen-like expectation value $\langle 1/r \rangle_{1s} = Z'/a_0$ (in atomic units, $a_0 = 1$), compute $\langle (Z' - Z)(1/r_1 + 1/r_2) \rangle$.

(c) Using Unsöld's result $\langle 1/r_{12} \rangle = 5Z'/8$ for the product of two 1s wavefunctions with effective charge $Z'$, write the total energy $E(Z')$.

(d) Minimize $E(Z')$ and verify that $Z'_{\text{opt}} = 27/16$.

(e) Calculate $E_{\min}$ numerically and compare with the first-order perturbation theory result ($-2.750$ hartree) and experiment ($-2.9037$ hartree).

B.3: Gaussian Trial Function for the Hydrogen Atom

(a) Starting from $\psi(r; \alpha) = (2\alpha/\pi)^{3/4} e^{-\alpha r^2}$, compute $\langle T \rangle = 3\alpha\hbar^2/(2m)$ and $\langle V \rangle = -e^2\sqrt{2\alpha/\pi}$ (in Gaussian units) or equivalently $\langle T \rangle = 3\alpha/2$ and $\langle V \rangle = -2\sqrt{2\alpha/\pi}$ in atomic units.

(b) Minimize $E(\alpha)$ and find $\alpha_{\text{opt}}$ and $E_{\min}$.

(c) Show that $E_{\min}/E_{\text{exact}} = 8/(3\pi) \approx 0.849$, so the Gaussian overestimates the ground state energy by about 15%.

(d) Now try the trial function $\psi(r; \alpha) = N e^{-\alpha r}$. Show that the variational method recovers the exact ground state energy $E_0 = -1/2$ hartree with $\alpha_{\text{opt}} = 1$.

B.4: Anharmonic Oscillator

The anharmonic oscillator has Hamiltonian $\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + \lambda x^4$ (a quartic potential, no quadratic term).

(a) Use the Gaussian trial function $\psi(x; \alpha) = (2\alpha/\pi)^{1/4} e^{-\alpha x^2}$ to compute $E(\alpha)$.

(b) Find $\alpha_{\text{opt}}$ and $E_{\min}$ in terms of $\hbar$, $m$, and $\lambda$.

(c) Express $E_{\min}$ in the form $E_{\min} = C (\hbar^2\lambda/m)^{1/3}$ and find the numerical constant $C$.

(d) The exact ground state energy is $E_0 \approx 0.668 (\hbar^2\lambda/m)^{1/3}$. How close is your variational bound?

B.5: Half-Harmonic Oscillator

Consider a particle in the "half-harmonic oscillator" potential:

$$V(x) = \begin{cases} \frac{1}{2}m\omega^2 x^2 & x > 0 \\ \infty & x \leq 0 \end{cases}$$

(a) What are the exact energy eigenvalues? (Hint: which eigenstates of the full harmonic oscillator satisfy $\psi(0) = 0$?)

(b) Use the trial function $\psi(x; \alpha) = N x e^{-\alpha x^2}$ (for $x > 0$) to find a variational upper bound on the ground state energy. (This trial function automatically satisfies $\psi(0) = 0$.)

(c) Compare your variational bound with the exact ground state energy.

B.6: H₂⁺ Potential Energy Curve

Using the LCAO treatment of H₂⁺ from Section 19.4:

(a) Show that the overlap integral is $S(R) = e^{-R}(1 + R + R^2/3)$.

(b) Verify that $S(0) = 1$ (complete overlap when protons coincide) and $S(\infty) = 0$ (no overlap when infinitely separated). Sketch $S(R)$.

(c) Write expressions for $E_+(R)$ and $E_-(R)$ in terms of $S$, $J$, $K$, $E_{1s}$, and $1/R$.

(d) Sketch $E_+(R)$ and $E_-(R)$ as functions of $R$. Which has a minimum? What is the physical significance?

(e) Why does the antibonding state $E_-(R)$ have higher energy than the separated atom limit for all $R$?

Part C: Analytical Challenges (⭐⭐⭐)

These problems require more advanced mathematical techniques and deeper physical reasoning.

C.1: The Schwarz Inequality and the Variational Principle

(b) Let $|\phi\rangle = (\hat{H} - E_0)|0\rangle = 0$ since $|0\rangle$ is the ground state. For any state $|\psi\rangle$, use the Cauchy-Schwarz inequality with $|\phi\rangle = (\hat{H} - E_0)|\psi\rangle$ and the spectral decomposition to provide an alternative proof of the variational theorem.

(c) Derive a lower bound on $E_0$ given $\langle \psi | \hat{H} | \psi \rangle$, $\langle \psi | \hat{H}^2 | \psi \rangle$, and $E_1$ (the first excited state energy). This is the Weinstein lower bound:

$$E_0 \geq \langle H \rangle - \sqrt{\langle H^2 \rangle - \langle H \rangle^2} \cdot \frac{\langle H \rangle - E_0}{E_1 - E_0}$$

Simplify for the case where you know that $E_1 - E_0 = \Delta$ (the spectral gap).

C.2: Two-Parameter Helium Wavefunction

Extend the one-parameter helium calculation by using the trial function:

$$\psi(\mathbf{r}_1, \mathbf{r}_2; a, b) = N\left(e^{-ar_1 - br_2} + e^{-br_1 - ar_2}\right)$$

This function allows asymmetric screening — one electron can be closer to the nucleus than the other.

(a) Why must the function be symmetrized (the sum of both terms) rather than just $e^{-ar_1 - br_2}$? (Hint: what symmetry must the spatial wavefunction of two electrons in a spin-singlet state have?)

(b) Write the energy functional $E(a, b)$ in terms of hydrogen-like integrals.

(c) Show that the minimum of $E(a, b)$ occurs at $a = b = 27/16$ — the symmetric solution. This means the one-parameter calculation was already optimal within this family. (The asymmetric improvement requires including $r_{12}$ dependence, as Hylleraas showed.)

C.3: Variational Proof that Bound States Exist

Consider a one-dimensional particle in the potential $V(x)$ that is negative somewhere: $V(x_0) < 0$ for some $x_0$. Assume $V(x) \to 0$ as $|x| \to \infty$.

(a) Use a Gaussian trial function $\psi(x; \alpha) = (\pi/\alpha)^{-1/4} e^{-\alpha(x - x_0)^2/2}$ centered at $x_0$ and show that for sufficiently small $\alpha$ (wide Gaussian), $E(\alpha) < 0$.

(b) Conclude that at least one bound state exists (with $E_0 < 0$) whenever the potential has any attractive region in one dimension.

(c) Does this argument work in three dimensions? (Hint: how does $\langle T \rangle$ scale with $\alpha$ in $d$ dimensions? You will need to think about whether $\langle V \rangle$ can always dominate $\langle T \rangle$ for small $\alpha$.)

C.4: The Hylleraas Variational Calculation

Hylleraas improved the helium ground state energy dramatically by using coordinates $s = r_1 + r_2$, $t = r_1 - r_2$, and $u = r_{12}$, and the trial function:

$$\psi = e^{-Z's}(1 + c_1 u + c_2 t^2)$$

(a) Explain why the function depends on $t^2$ rather than $t$ (hint: exchange symmetry).

(b) Explain why including $u = r_{12}$ explicitly is physically important. What correlation does it capture that the simple product wavefunction $e^{-Z'(r_1 + r_2)}$ misses?

(c) With three parameters ($Z'$, $c_1$, $c_2$), Hylleraas obtained $E = -2.9033$ hartree, compared to the exact $-2.9037$ hartree. Calculate the percentage error.

(d) Why is explicit $r_{12}$ dependence so much more effective than simply adding more single-particle orbitals?

C.5: The Ritz Method and the Harmonic Oscillator

Apply the Ritz method to the harmonic oscillator using the basis set $\{x^{2n} e^{-x^2/2}\}$ for $n = 0, 1, 2, \ldots, N-1$ (in dimensionless units where $\hbar = m = \omega = 1$).

(a) Compute the $3 \times 3$ Hamiltonian and overlap matrices for $N = 3$.

(b) Solve the generalized eigenvalue problem and find the three energy eigenvalues.

(c) Compare these with the exact eigenvalues $E_n = (2n + 1/2)$ for $n = 0, 1, 2$ (only even-parity states, since the basis functions are all even). How accurate is the $N = 3$ Ritz approximation?

(d) Why does this basis set give exact results? (Hint: what are the exact even-parity eigenstates of the harmonic oscillator?)

C.6: Variational Method for a Delta-Function Potential

The one-dimensional delta-function potential is $V(x) = -\alpha \delta(x)$ (with $\alpha > 0$).

(a) The exact ground state energy is $E_0 = -m\alpha^2/(2\hbar^2)$ and the exact wavefunction is $\psi_0(x) = (\kappa) e^{-\kappa|x|}$ with $\kappa = m\alpha/\hbar^2$. Verify this by substitution.

(b) Use the Gaussian trial function $\psi(x; b) = (2b/\pi)^{1/4} e^{-bx^2}$. Show that $\langle V \rangle = -\alpha\sqrt{2b/\pi}$ and $\langle T \rangle = \hbar^2 b/(2m)$.

(c) Find $b_{\text{opt}}$ and $E_{\min}$. Express $E_{\min}/E_0$ as a simple fraction. How close is the Gaussian bound to the exact answer?

(d) Now use the Lorentzian trial function $\psi(x; \gamma) = N/(x^2 + \gamma^2)$. Compute the variational energy (you will need $\langle T \rangle$ via integration by parts). Is the Lorentzian better or worse than the Gaussian?

Part D: Computational Problems (⭐⭐⭐)

These problems require Python implementation. Use the toolkit functions from code/example-01-variational.py or write your own.

D.1: Variational Energy Landscape

(a) For the hydrogen atom with Gaussian trial function $\psi = (2\alpha/\pi)^{3/4} e^{-\alpha r^2}$, plot $E(\alpha)$ vs. $\alpha$ for $\alpha \in [0.01, 2.0]$. Mark the minimum. Verify that the minimum gives $E_{\min} = -4/(3\pi)$ hartree.

(b) Now use the two-parameter trial function $\psi = N(e^{-\alpha r} + \beta r e^{-\alpha r})$ for the hydrogen atom. Plot $E(\alpha, \beta)$ as a contour plot and find the minimum numerically using scipy.optimize.minimize.

(c) Add a third parameter: $\psi = N(e^{-\alpha r} + \beta r e^{-\gamma r})$. Does the three-parameter function significantly improve the energy?

D.2: Helium with Multiple Parameters

(a) Implement the one-parameter helium calculation ($Z'$ parameter) and verify $E_{\min} = -2.8477$ hartree.

(b) Add a second variational parameter by using $\psi = N(e^{-ar_1 - br_2} + e^{-br_1 - ar_2})$ and optimize over both $a$ and $b$. Does the energy improve beyond the symmetric $a = b$ solution?

(c) Include correlation by multiplying by a Jastrow factor: $\psi = N e^{-Z'(r_1 + r_2)} e^{c \cdot r_{12}/(1 + d \cdot r_{12})}$. Optimize over $Z'$, $c$, and $d$ using VMC with 10,000 samples. How close do you get to the exact $-2.9037$ hartree?

D.3: H₂⁺ Potential Energy Curves

(a) Implement the LCAO calculation for H₂⁺ and plot $E_+(R)$ and $E_-(R)$ for $R \in [0.5, 10.0]$ $a_0$.

(b) Find the equilibrium bond length $R_e$ and binding energy $D_e$ numerically.

(c) Add the variational parameter $Z'$ to the atomic orbitals and repeat. How much do $R_e$ and $D_e$ improve?

(d) Compare your curves with the exact (numerically computed) H₂⁺ potential energy curve.

D.4: VMC for the Harmonic Oscillator

(a) Implement the VMC algorithm for the 1D harmonic oscillator using the trial function $\psi(x; \alpha) = e^{-\alpha x^2}$.

(b) Run VMC with 10,000 Metropolis samples for various values of $\alpha$. Plot $\bar{E}(\alpha)$ with error bars.

(c) Verify that the variance of the local energy is minimized at $\alpha_{\text{opt}} = m\omega/(2\hbar)$ (the zero-variance principle).

(d) Extend to a trial function with two parameters: $\psi(x; \alpha, \beta) = e^{-\alpha x^2}(1 + \beta x^4)$. Does the energy improve?

D.5: Ritz Method — Anharmonic Oscillator

(a) Implement the Ritz method for the anharmonic oscillator $\hat{H} = \hat{p}^2/(2m) + \lambda x^4$ using harmonic oscillator eigenstates $\{|0\rangle, |2\rangle, |4\rangle, \ldots, |2N\rangle\}$ as a basis (only even-parity states, since the ground state has even parity).

(b) Compute the ground state energy for $N = 1, 2, 3, 5, 10, 20$ and plot $E_0(N)$ vs. $N$. How fast does the Ritz energy converge to the exact value?

(c) How does the convergence rate depend on $\lambda$? Try $\lambda = 0.1, 1.0, 10.0$. For which value is convergence fastest? Explain why.

D.6: VMC for the Helium Atom

(a) Implement a VMC solver for the helium atom using the trial function $\psi = e^{-Z'(r_1 + r_2)}$ with $Z'$ as a variational parameter.

(b) Use Metropolis sampling in the 6-dimensional configuration space $(\mathbf{r}_1, \mathbf{r}_2)$. Tune the step size for ~50% acceptance.

(c) Compute the local energy at each sample point and estimate $E(Z')$ for $Z' = 1.5, 1.6, 1.7, 1.8, 1.9$.

(d) Find the optimal $Z'$ and compare with the analytical result $27/16 = 1.6875$.

(e) Add a Jastrow factor $e^{r_{12}/2(1 + \beta r_{12})}$ and optimize $\beta$ along with $Z'$. How much does the energy improve?

Part E: Synthesis and Reflection (⭐⭐⭐)

E.1 You need to estimate the ground state energy of a lithium atom (3 electrons, $Z = 3$). Which approach would you use — perturbation theory, nonlinear variational, Ritz method, or VMC? Justify your choice and describe your trial function.

E.2 The variational method always overestimates $E_0$, while the experimental measurement is exact (in principle). Is there any situation where the variational method gives a result closer to experiment than the exact $E_0$ of the Hamiltonian? (Hint: what if the Hamiltonian is not perfectly accurate — e.g., it neglects relativistic effects?)

E.3 Explain why the linear variational (Ritz) method is the foundation of modern computational quantum chemistry. What are the main practical challenges in applying it to molecules with 100+ electrons?

E.4 Compare the roles of physical intuition in perturbation theory vs. the variational method. In which method does the physicist's judgment matter more, and why?