Chapter 17 Exercises: Time-Independent Perturbation Theory — Non-Degenerate Case

Problems are organized by section and graded by difficulty: - [B] Basic — direct application of formulas and definitions - [I] Intermediate — requires multi-step reasoning or combining concepts - [A] Advanced — requires deeper analysis, proof, or synthesis

Section 17.1–17.2: Framework and Setup

Problem 17.1 [B]

A quantum system has unperturbed Hamiltonian $\hat{H}_0$ with non-degenerate eigenvalues $E_n^{(0)} = n\hbar\omega$ for $n = 1, 2, 3, \ldots$ and corresponding eigenstates $|n^{(0)}\rangle$. A perturbation $\hat{H}'$ has matrix elements $\langle m^{(0)}|\hat{H}'|n^{(0)}\rangle = V_0\delta_{m,n+1} + V_0\delta_{m,n-1}$ (i.e., $\hat{H}'$ only connects nearest neighbors).

(a) Write the perturbation expansion for $E_2$ through second order.

(b) Compute $E_2^{(1)}$ and $E_2^{(2)}$ explicitly.

(c) Write $|2^{(1)}\rangle$ explicitly in terms of unperturbed states.

Problem 17.2 [B]

Which of the following can be treated by non-degenerate perturbation theory? For each that cannot, explain why.

(a) The harmonic oscillator with perturbation $\hat{H}' = \lambda\hat{x}^3$.

(b) The hydrogen atom ($n = 2$ level) with an applied electric field.

(c) A particle in a 2D box ($L_x = L_y$) with perturbation $\hat{H}' = \lambda xy$.

(d) A particle in a 2D box ($L_x \neq L_y$, irrational ratio) with perturbation $\hat{H}' = \lambda xy$.

(e) A spin-1/2 particle in a magnetic field $B_z$ with a small additional field $B_x$.

Problem 17.3 [B]

Verify the intermediate normalization convention. Starting from $|n\rangle = |n^{(0)}\rangle + \lambda|n^{(1)}\rangle + \lambda^2|n^{(2)}\rangle + \cdots$ and requiring $\langle n^{(0)}|n\rangle = 1$, show that $\langle n^{(0)}|n^{(k)}\rangle = 0$ for all $k \ge 1$.

Problem 17.4 [I]

Suppose we had instead chosen the normalization $\langle n|n\rangle = 1$ (unit norm for the perturbed state). Show that this leads to the condition:

$$\text{Re}\,\langle n^{(0)}|n^{(1)}\rangle = -\frac{1}{2}\langle n^{(1)}|n^{(1)}\rangle \lambda + O(\lambda^2)$$

and explain why the intermediate normalization $\langle n^{(0)}|n\rangle = 1$ is simpler for practical calculations.

Section 17.3: First-Order Energy Correction

Problem 17.5 [B]

A particle is in a one-dimensional harmonic oscillator potential with a perturbation $\hat{H}' = \epsilon m\omega^2 x$ (a constant force).

(a) Compute the first-order energy correction $E_n^{(1)}$ for all $n$.

(b) This problem can be solved exactly by completing the square. Find the exact energies and show that $E_n^{(1)}$ agrees with the first-order term of the Taylor expansion of the exact result.

Problem 17.6 [B]

A particle in an infinite square well ($0 < x < L$) is subjected to the perturbation:

$$\hat{H}' = V_0\sin\left(\frac{\pi x}{L}\right)$$

Compute $E_n^{(1)}$ for the first three states ($n = 1, 2, 3$).

(Hint: Use the identity $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$ and $\sin^2 A = \frac{1}{2}(1 - \cos 2A)$.)

Problem 17.7 [I]

A particle in the ground state of an infinite square well ($0 < x < L$) is subjected to a perturbation that raises the bottom of the left half:

$$\hat{H}' = \begin{cases} V_0 & 0 < x < L/2 \\ 0 & L/2 < x < L \end{cases}$$

(a) Compute $E_n^{(1)}$ for general $n$.

(b) For $n = 1$, show that $E_1^{(1)} = V_0/2 + V_0/\pi$. Interpret this result physically.

(c) For $n = 2$, show that $E_2^{(1)} = V_0/2$. Explain why the correction is independent of whether the step is in the left or right half.

Problem 17.8 [I]

A one-dimensional harmonic oscillator has the perturbation $\hat{H}' = \lambda\hat{x}^2$.

(a) Compute the first-order energy correction $E_n^{(1)}$. (Express your answer in terms of $n$, $\hbar$, $m$, and $\omega$.)

(b) This problem can be solved exactly by noting that $\hat{H}_0 + \lambda\hat{x}^2 = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega'^2\hat{x}^2$ with $\omega' = \omega\sqrt{1 + 2\lambda/(m\omega^2)}$. Find the exact energies and verify that your first-order correction is the correct leading-order approximation.

(c) At what value of $\lambda/(m\omega^2)$ does the first-order approximation begin to deviate from the exact answer by more than 10% for the ground state?

Problem 17.9 [B]

A hydrogen atom is subjected to a small perturbation $\hat{H}' = Ar^2$ (where $A$ is a constant and $r$ is the electron-nucleus distance).

(a) Compute the first-order correction to the ground-state energy. You will need $\langle r^2\rangle_{100} = 3a_0^2$.

(b) Does this perturbation raise or lower the ground-state energy? Is this consistent with physical intuition?

Problem 17.10 [I]

The gravitational Stark effect. A hydrogen atom sits in a uniform gravitational field $g$. The perturbation is $\hat{H}' = m_e g z$ (where $m_e$ is the electron mass and $z$ is measured from the nucleus).

(a) Explain, without computation, why $E_1^{(1)} = 0$ for the ground state.

(b) Is the gravitational field strong enough to produce a measurable energy shift? Estimate $E_1^{(2)}$ using the bound $|E_1^{(2)}| \le (m_e g)^2 a_0^2 / (E_2^{(0)} - E_1^{(0)})$ and compare with $E_1^{(0)}$.

Section 17.4: First-Order Wavefunction Correction

Problem 17.11 [B]

For the system in Problem 17.1, write $|2^{(1)}\rangle$ as a linear combination of unperturbed states.

Problem 17.12 [I]

A harmonic oscillator has perturbation $\hat{H}' = \lambda\hat{x}^3$.

(a) Show that $E_n^{(1)} = 0$ for all $n$ by a parity argument.

(b) Despite the vanishing first-order energy, the wavefunction is corrected at first order. Compute $|0^{(1)}\rangle$ for the ground state. Use $\hat{x} = \sqrt{\hbar/(2m\omega)}(\hat{a}+\hat{a}^\dagger)$ and the fact that $\hat{x}^3$ connects $|0\rangle$ only to $|1\rangle$ and $|3\rangle$.

(c) What is the physical significance of the first-order wavefunction correction when the first-order energy correction vanishes?

Problem 17.13 [I]

For a particle in an infinite square well with perturbation $\hat{H}' = V_0 x/L$:

(a) Calculate $|1^{(1)}\rangle$ keeping terms through $m = 5$ in the sum.

(b) Which unperturbed state contributes the most to the correction? Explain why in terms of matrix elements and energy denominators.

(c) Plot the first-order corrected ground-state wavefunction (qualitatively) and explain how it differs from the unperturbed wavefunction. Which way does the probability shift, and why?

Problem 17.14 [A]

Prove the following identity for the first-order wavefunction correction:

$$\langle n^{(1)}|n^{(1)}\rangle = \sum_{m \neq n} \frac{|\langle m^{(0)}|\hat{H}'|n^{(0)}\rangle|^2}{(E_n^{(0)} - E_m^{(0)})^2}$$

Then show that the condition for perturbation theory to be valid ($|n^{(1)}\rangle$ should be small compared to $|n^{(0)}\rangle$) requires:

$$|\langle m^{(0)}|\hat{H}'|n^{(0)}\rangle| \ll |E_n^{(0)} - E_m^{(0)}| \quad \text{for all } m \neq n$$

State this condition in words.

Section 17.5: Second-Order Energy Correction

Problem 17.15 [B]

For the system in Problem 17.1 (nearest-neighbor coupling $V_0$), compute $E_n^{(2)}$ for general $n \ge 2$.

Problem 17.16 [I]

Show that the second-order energy correction for the ground state is always negative (or zero): $E_0^{(2)} \le 0$. Under what condition is $E_0^{(2)} = 0$?

Problem 17.17 [I]

For the harmonic oscillator with perturbation $\hat{H}' = \lambda\hat{x}^3$:

(a) Compute $E_0^{(2)}$ using the result of Problem 17.12(b) and the formula $E_n^{(2)} = \langle n^{(0)}|\hat{H}'|n^{(1)}\rangle$.

(b) Alternatively, compute $E_0^{(2)}$ directly from equation (17.17). Identify all states $|m^{(0)}\rangle$ for which $\langle m^{(0)}|\hat{x}^3|0^{(0)}\rangle \neq 0$ and evaluate the matrix elements using ladder operators.

(c) Verify that your two answers agree.

Problem 17.18 [I]

Level repulsion. Consider a two-state system with $\hat{H}_0 = E_a|a\rangle\langle a| + E_b|b\rangle\langle b|$ and perturbation $\hat{H}' = V(|a\rangle\langle b| + |b\rangle\langle a|)$ with $V$ real.

(a) Compute the second-order energy corrections $E_a^{(2)}$ and $E_b^{(2)}$.

(b) Show that $E_a^{(2)} + E_b^{(2)} = 0$: the levels repel symmetrically.

(c) This problem can be solved exactly by diagonalizing the $2\times 2$ matrix. Find the exact eigenvalues and expand them to second order in $V/(E_a - E_b)$. Verify agreement with perturbation theory.

(d) Plot the exact eigenvalues as a function of $V/(E_a - E_b)$ from 0 to 2. At what point does second-order perturbation theory begin to fail?

Problem 17.19 [A]

Sum rule for second-order corrections. Prove that:

$$\sum_n E_n^{(2)} = \sum_n \sum_{m \neq n} \frac{|\langle m^{(0)}|\hat{H}'|n^{(0)}\rangle|^2}{E_n^{(0)} - E_m^{(0)}} = 0$$

when the sum runs over all states. Interpret this result physically: what does it say about the "total" second-order shift?

Section 17.6: Anharmonic Oscillator

Problem 17.20 [B]

Verify equation (17.23) by computing $\langle 1|(\hat{a} + \hat{a}^\dagger)^4|1\rangle$ step by step using ladder operators. Show that the result is $6(1)^2 + 6(1) + 3 = 15$.

Problem 17.21 [I]

For the anharmonic oscillator $\hat{H} = \hat{H}_0 + \lambda\hat{x}^4$:

(a) Compute the second-order energy correction $E_0^{(2)}$ for the ground state. You will need the matrix elements $\langle m|\hat{x}^4|0\rangle$ for all $m$ (only $m = 0, 2, 4$ contribute).

(b) Express $E_0^{(2)}$ in terms of $\lambda$, $\hbar$, $m$, and $\omega$.

(c) For what value of $\lambda$ (in natural units $m = \omega = \hbar = 1$) does the second-order correction equal 10% of the first-order correction?

Problem 17.22 [I]

Consider the cubic-plus-quartic perturbation: $\hat{H}' = \alpha\hat{x}^3 + \beta\hat{x}^4$.

(a) What is the first-order energy correction for state $|n\rangle$? (Only one term contributes at first order.)

(b) For the ground state, compute the second-order correction due to the cubic term only ($\alpha\hat{x}^3$).

(c) Under what conditions on $\alpha$ and $\beta$ can the cubic contribution to $E_0^{(2)}$ cancel the quartic contribution to $E_0^{(1)}$? Does this cancellation have physical significance?

Problem 17.23 [A]

Numerical verification. Use the Python code from code/example-01-perturbation.py (or write your own) to numerically diagonalize the anharmonic oscillator Hamiltonian in a truncated basis of $N$ harmonic oscillator states.

(a) For $\lambda = 0.1$ (in natural units), compute the ground-state energy using $N = 10, 20, 50, 100$ basis states. Show that the result converges as $N$ increases.

(b) Compare the converged numerical result with the first-order perturbation theory prediction and the first-plus-second-order prediction. Compute the percentage error of each.

(c) Repeat for $\lambda = 1.0$. At what point does perturbation theory fail qualitatively?

Section 17.7: Stark Effect

Problem 17.24 [B]

For the hydrogen ground state in an electric field:

(a) Verify that $\langle 1,0,0|r\cos\theta|1,0,0\rangle = 0$ by performing the angular integral explicitly.

(b) Use the selection rule $\Delta l = \pm 1$ to identify which excited states contribute to $E_1^{(2)}$.

Problem 17.25 [I]

Compute the matrix element $\langle 2,1,0|r\cos\theta|1,0,0\rangle$ for hydrogen.

(a) Separate the integral into radial and angular parts.

(b) Evaluate the angular integral using the fact that $\cos\theta = \sqrt{4\pi/3}\,Y_1^0$.

(c) Evaluate the radial integral $\int_0^\infty r^3 R_{21}(r) R_{10}(r) r^2\,dr$ using the explicit hydrogen radial wavefunctions.

(d) Use your result to estimate the contribution of the $n=2$ states alone to the hydrogen ground-state polarizability. How does this compare to the exact result $\alpha/(4\pi\epsilon_0) = 4.5a_0^3$?

Problem 17.26 [I]

Stark effect for the harmonic oscillator ground state. A charged particle ($q$) in a 1D harmonic oscillator is placed in a constant electric field $\mathcal{E}$, so the perturbation is $\hat{H}' = -q\mathcal{E}\hat{x}$.

(a) Compute $E_0^{(1)}$ and $E_0^{(2)}$.

(b) Solve this problem exactly (hint: complete the square in the potential) and verify your perturbative results.

(c) Show that the exact answer happens to be captured exactly by the second-order perturbation theory — i.e., all higher-order corrections vanish. Why does this happen?

Problem 17.27 [A]

The quadratic Stark effect in alkali atoms. In an alkali atom (Li, Na, K, ...), the valence electron sees an effective potential that is not purely Coulombic — the inner electrons partially screen the nucleus. As a result, states with different $l$ but the same $n$ are not degenerate (this is the "quantum defect" effect).

(a) Explain qualitatively why the Stark effect for the ground state of sodium is still quadratic (proportional to $\mathcal{E}^2$), just like hydrogen.

(b) The polarizability of sodium is $\alpha/(4\pi\epsilon_0) = 24.1\,\text{\AA}^3$, compared to $0.667\,\text{\AA}^3$ for hydrogen. Why is sodium so much more polarizable? (Hint: think about the energy denominator and the size of the atom.)

Section 17.8: Convergence

Problem 17.28 [I]

Consider a particle in an infinite square well with a delta-function perturbation at the center: $\hat{H}' = \alpha\delta(x - L/2)$.

(a) Compute $E_n^{(1)}$ for all $n$.

(b) For which values of $n$ does $E_n^{(1)} = 0$? Explain by symmetry.

(c) Compute $|1^{(1)}\rangle$, keeping enough terms to assess convergence. Does the sum converge? (Hint: examine the behavior of $|\langle m^{(0)}|\hat{H}'|1^{(0)}\rangle|^2 / (E_1^{(0)} - E_m^{(0)})$ as $m \to \infty$.)

Problem 17.29 [A]

Divergence of the perturbation series. For the anharmonic oscillator $\hat{H} = \frac{1}{2}\hat{p}^2 + \frac{1}{2}\hat{x}^2 + \lambda\hat{x}^4$ (in natural units), the perturbation series for the ground-state energy is known to have the large-order behavior:

$$E_0^{(k)} \sim (-1)^{k+1} \frac{3^k}{\pi^{3/2}} \Gamma\left(k + \frac{1}{2}\right) \quad \text{as } k \to \infty$$

(a) Show that this implies the series $\sum_k E_0^{(k)} \lambda^k$ has zero radius of convergence.

(b) For $\lambda = 0.1$, estimate the optimal truncation order $N^*$ (the order at which adding one more term begins to increase the error rather than decrease it).

(c) Estimate the best accuracy achievable by truncation at order $N^*$.

Problem 17.30 [A]

The double-well problem. Consider the symmetric double-well potential $V(x) = \lambda(x^2 - a^2)^2$.

(a) By expanding around one of the minima (say $x = +a$), show that the effective harmonic frequency is $\omega = 2a\sqrt{2\lambda/m}$.

(b) Explain why perturbation theory around the single-well minimum cannot reproduce the tunnel splitting $\Delta E$ between the symmetric and antisymmetric ground states.

(c) The tunnel splitting has the WKB estimate $\Delta E \propto e^{-S/\hbar}$ where $S = \int_{-a}^{a}\sqrt{2mV(x)}\,dx$. Evaluate $S$ for the potential $V(x) = \lambda(x^2-a^2)^2$ and show that $\Delta E \propto e^{-c/\lambda}$ for some constant $c$ (i.e., the splitting is non-perturbative in $\lambda$).