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Appendix A Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Evaluating the Value of a Functional

Substitute \(f(x) = 2x\) into the functional \(H[f] = \int_0^3 [f(x)]^2\,dx\) and find the value of \(H[f]\).

Hint

Simply substitute \([f(x)]^2 = (2x)^2 = 4x^2\) and evaluate the definite integral.

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B-2. Basic Calculation of Functional Derivatives

Find the functional derivative \(\frac{\delta F}{\delta f(x_0)}\) (\(0 \leq x_0 \leq 1\)) of the functional \(F[f] = \int_0^1 [f(x)]^4\,dx\).

Hint

This corresponds to the case \(p = 4\) and \(\varphi(y) = 1\) in Worked Example 2 of the main text. Use the same pattern as differentiating a power: "lower the \(p\)-th power to the \((p-1)\)-th power and bring out the coefficient \(p\) in front."

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B-3. Weighted Functional Derivative

Find the functional derivative \(\frac{\delta G}{\delta f(x)}\) of the functional \(G[f] = \int_{-\infty}^{\infty} [f(y)]^2\,e^{-y^2}\,dy\).

Hint

Use the formula from Worked Example 2 with \(p = 2\) and \(\varphi(y) = e^{-y^2}\). Apply the "sifting" property of the delta function at the final step.

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B-4. Functional Derivative Using the Delta Function

Write the functional \(F[f] = f(a)\) (the function value at a fixed point \(a\)) in its integral representation \(F[f] = \int f(y)\,\delta(y - a)\,dy\), and compute \(\frac{\delta F}{\delta f(x)}\) according to the definition.

Hint

Make the replacement \(f(y) \to f(y) + \epsilon\,\delta(y - x)\) and extract the terms first order in \(\epsilon\). The final result is expressed as a delta function.

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B-5. Application of the Euler-Lagrange Equation (1D Harmonic Oscillator)

For the Lagrangian \(L = \frac{1}{2}m\dot{x}^2 - \frac{1}{2}k x^2\), compute the following in order.

  1. \(\frac{\partial L}{\partial \dot{x}}\)
  2. \(\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right)\)
  3. \(\frac{\partial L}{\partial x}\)
  4. Write down the Euler-Lagrange equation and verify the resulting equation of motion.
Hint

Carry out each partial derivative carefully, such as \(\frac{\partial}{\partial \dot{x}}\left(\frac{1}{2}m\dot{x}^2\right) = m\dot{x}\). You should ultimately obtain \(m\ddot{x} = -kx\).

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B-6. Calculation of Canonical Momenta

For each of the following Lagrangians, find the canonical momentum \(p = \frac{\partial L}{\partial \dot{q}}\).

(a) \(L = \frac{1}{2}m\dot{q}^2 - mg q\) (free fall in a uniform gravitational field)

(b) For \(L = \frac{1}{2}m(\dot{r}^2 + r^2\dot{\theta}^2) - V(r)\) (2-dimensional polar coordinates), find \(p_r\) and \(p_\theta\) respectively.

Hint

For (a), simply take the partial derivative with respect to \(\dot{q}\). For (b), differentiate with respect to \(\dot{r}\) to obtain \(p_r\), and differentiate with respect to \(\dot{\theta}\) to obtain \(p_\theta\). Note that \(p_\theta = mr^2\dot{\theta}\) corresponds to the angular momentum.

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B-7. Construction of the Hamiltonian

For the Lagrangian of a 1-dimensional harmonic oscillator \(L = \frac{1}{2}m\dot{q}^2 - \frac{1}{2}m\omega^2 q^2\):

  1. Find the canonical momentum \(p\).
  2. Express \(\dot{q}\) in terms of \(p\) and \(m\).
  3. Write down the Hamiltonian \(H = p\dot{q} - L\) as a function of \(q\) and \(p\).
Hint

From \(p = m\dot{q}\), we get \(\dot{q} = p/m\). Substitute this into \(H = p\dot{q} - L\) to eliminate \(\dot{q}\). The result should be \(H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 q^2\).

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B-8. Application of the Field Euler-Lagrange Equation

For the Lagrangian density \(\mathcal{L} = \frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi\) (without mass term), apply the field Euler-Lagrange equation and derive the equation of motion.

Hint

\(\frac{\partial\mathcal{L}}{\partial\phi} = 0\) (there is no term containing \(\phi\) itself), \(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)} = \partial^\mu\phi\). What is the equation \(\partial_\mu(\partial^\mu\phi) = 0\) called?

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B-9. Equation of Motion for \(\phi^3\) Theory

Apply the Euler-Lagrange equation for fields to the Lagrangian density \(\mathcal{L} = \frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - \frac{m^2}{2}\phi^2 - \frac{g}{3!}\phi^3\), and derive the equation of motion.

Hint

\(\frac{\partial}{\partial\phi}\left(\frac{g}{3!}\phi^3\right) = \frac{g}{3!}\times 3\phi^2 = \frac{g}{2}\phi^2\). The rest follows the same procedure as the \(\phi^4\) theory example in the main text.

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B-10. Chain Rule for Functional Derivatives

Compute the functional derivative \(\frac{\delta S}{\delta q(t')}\) (\(t_1 < t' < t_2\)) of the functional \(S[q] = \int_{t_1}^{t_2}\frac{1}{2}m[\dot{q}(t)]^2\,dt\). You may use the boundary conditions \(\delta q(t_1) = \delta q(t_2) = 0\).

Hint

Substituting \(q(t) \to q(t) + \epsilon\,\delta(t - t')\) gives \(\dot{q}(t) \to \dot{q}(t) + \epsilon\,\frac{d}{dt}\delta(t - t')\). Extract the terms first order in \(\epsilon\) and use integration by parts to find the coefficient in front of \(\delta(t - t')\). The result is \(-m\ddot{q}(t')\).

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Medium

M-1. Deriving Newton's Gravitational Equation of Motion from the Action Principle

A particle of mass \(m\) moves vertically in a uniform gravitational field. The Lagrangian is

\[ L = \frac{1}{2}m\dot{z}^2 - mgz \]

Show the following.

  1. Compute the variation \(\delta S\) of the action \(S[z] = \int_{t_1}^{t_2} L\,dt\) and arrange it into an integral containing \(\delta z(t)\) (explicitly showing the integration by parts).
  2. Derive the Euler-Lagrange equation from \(\delta S = 0\) and obtain \(m\ddot{z} = -mg\).
  3. Confirm that the resulting equation is consistent with Newton's equation of motion \(F = ma\).
Hint

Follow exactly the same procedure as the derivation in Section A.4.3 of the text. Use \(\frac{\partial L}{\partial \dot{z}} = m\dot{z}\) and \(\frac{\partial L}{\partial z} = -mg\). Show from the boundary conditions that the surface term vanishes upon integration by parts.

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M-2. Canonical Momentum and Hamiltonian Density of a Field

Given the Lagrangian density of the Klein-Gordon field

\[ \mathcal{L} = \frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - \frac{m^2}{2}\phi^2 = \frac{1}{2}\dot{\phi}^2 - \frac{1}{2}(\nabla\phi)^2 - \frac{m^2}{2}\phi^2 \]

perform the following:

  1. Find the canonical momentum density \(\pi(x) = \frac{\partial\mathcal{L}}{\partial\dot{\phi}}\).
  2. Write down the Hamiltonian density \(\mathcal{H} = \pi\dot{\phi} - \mathcal{L}\) in terms of \(\phi\), \(\pi\), and \(\nabla\phi\).
  3. Verify that the resulting \(\mathcal{H}\) is an energy density (positive definite).
Hint

\(\pi = \dot{\phi}\). Substitute \(\dot{\phi} = \pi\) into \(\mathcal{H} = \pi\dot{\phi} - \mathcal{L}\) and simplify. The result is \(\mathcal{H} = \frac{1}{2}\pi^2 + \frac{1}{2}(\nabla\phi)^2 + \frac{m^2}{2}\phi^2\), and since all terms are squared, it is positive definite.

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M-3. Poisson Brackets and Hamilton's Equations of Motion

Consider a one-dimensional particle with Hamiltonian given by \(H(q, p) = \frac{p^2}{2m} + V(q)\). The Poisson bracket is defined as

\[ \{A, B\}_{\mathrm{PB}} = \frac{\partial A}{\partial q}\frac{\partial B}{\partial p} - \frac{\partial A}{\partial p}\frac{\partial B}{\partial q} \]

Show the following.

  1. Verify that \(\{q, p\}_{\mathrm{PB}} = 1\).
  2. Compute Hamilton's equations of motion \(\dot{q} = \{q, H\}_{\mathrm{PB}}\), \(\dot{p} = \{p, H\}_{\mathrm{PB}}\), and obtain \(\dot{q} = p/m\) and \(\dot{p} = -\frac{dV}{dq}\), respectively.
  3. Using the canonical quantization prescription "\(\{A, B\}_{\mathrm{PB}} \to \frac{1}{i\hbar}[\hat{A}, \hat{B}]\)", confirm that \([\hat{q}, \hat{p}] = i\hbar\) is obtained.
Hint

Substitute \(A = q\), \(B = p\) into the definition of the Poisson bracket. Use \(\frac{\partial q}{\partial q} = 1\), \(\frac{\partial p}{\partial p} = 1\), etc. For Hamilton's equations of motion, compute \(\{q, H\}_{\mathrm{PB}} = \frac{\partial q}{\partial q}\frac{\partial H}{\partial p} - \frac{\partial q}{\partial p}\frac{\partial H}{\partial q}\).

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M-4. Legendre Transform for Systems with Multiple Degrees of Freedom

For a system with \(N\) generalized coordinates \(q_1, \ldots, q_N\) and Lagrangian \(L(q_i, \dot{q}_i)\):

  1. Define the canonical momenta \(p_i = \frac{\partial L}{\partial \dot{q}_i}\) and construct the Hamiltonian as
\[ H = \sum_{i=1}^N p_i \dot{q}_i - L \]

Show that \(H\) does not contain \(\dot{q}_i\) and is a function only of \((q_i, p_i)\), by computing the total differential \(dH\).

  1. From the expression for \(dH\), derive Hamilton's canonical equations
\[ \dot{q}_i = \frac{\partial H}{\partial p_i}, \qquad \dot{p}_i = -\frac{\partial H}{\partial q_i} \]
Hint

Compute \(dH = \sum_i(\dot{q}_i\,dp_i + p_i\,d\dot{q}_i) - \frac{\partial L}{\partial q_i}dq_i - \frac{\partial L}{\partial \dot{q}_i}d\dot{q}_i\). Using the definition \(p_i = \frac{\partial L}{\partial \dot{q}_i}\) and the Euler-Lagrange equations \(\dot{p}_i = \frac{\partial L}{\partial q_i}\), the terms in \(d\dot{q}_i\) cancel.

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M-5. Relationship Between Functional Derivatives and the Euler-Lagrange Equation

For the action

\[ S[q] = \int_{t_1}^{t_2} L(q(t), \dot{q}(t))\,dt \]

compute the functional derivative \(\frac{\delta S}{\delta q(t')}\) and show that the result is

\[ \frac{\delta S}{\delta q(t')} = \frac{\partial L}{\partial q}\bigg|_{t=t'} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right)\bigg|_{t=t'} \]

Using this, reconfirm in the language of functional derivatives that \(\delta S = 0\) is equivalent to the Euler-Lagrange equation.

Hint

Make the replacement \(q(t) \to q(t) + \epsilon\,\delta(t - t')\), and expand \(L\) to first order in \(\epsilon\). Note that \(\dot{q}(t) \to \dot{q}(t) + \epsilon\,\frac{d}{dt}\delta(t - t')\), and handle the term containing \(\frac{d}{dt}\delta(t - t')\) by integration by parts.

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Advanced

A-1. Charged Particle in an Electromagnetic Field and Gauge Dependence of Canonical Momentum

The Lagrangian of a charged particle (charge \(e\), mass \(m\)) in an electromagnetic field \((V, \mathbf{A})\) is given by

\[ L = \frac{1}{2}m\dot{\mathbf{r}}^2 - eV(\mathbf{r}, t) + e\dot{\mathbf{r}}\cdot\mathbf{A}(\mathbf{r}, t) \]

Carry out the following.

  1. Find the canonical momentum \(\mathbf{p} = \frac{\partial L}{\partial \dot{\mathbf{r}}}\) and show that it differs from the kinetic momentum \(m\dot{\mathbf{r}}\).
  2. Write down the Hamiltonian \(H = \mathbf{p}\cdot\dot{\mathbf{r}} - L\) in terms of \((\mathbf{r}, \mathbf{p})\).
  3. Show how the canonical momentum transforms under the gauge transformation \(\mathbf{A} \to \mathbf{A} + \nabla\chi\), \(V \to V - \frac{\partial\chi}{\partial t}\), and verify that the Hamiltonian (and hence the equations of motion) is gauge invariant.
  4. Discuss how this result serves as the classical origin of the prescription (minimal coupling) of "replacing \(\hat{\mathbf{p}}\) with \(\hat{\mathbf{p}} - e\hat{\mathbf{A}}\)" in Ch. 6 (quantization of QED) in the main text.
Hint

We obtain \(\mathbf{p} = m\dot{\mathbf{r}} + e\mathbf{A}\). Substitute \(\dot{\mathbf{r}} = (\mathbf{p} - e\mathbf{A})/m\) to construct \(H\). Under the gauge transformation \(\mathbf{p} \to \mathbf{p} + e\nabla\chi\), but the combination \((\mathbf{p} - e\mathbf{A})\) appearing in \(H\) is gauge invariant.

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A-2. From Field Poisson Brackets to Canonical Quantization

For a scalar field \(\phi(\mathbf{x}, t)\) and the canonical momentum density \(\pi(\mathbf{x}, t) = \dot{\phi}(\mathbf{x}, t)\), the field Poisson bracket is defined as

\[ \{A, B\}_{\mathrm{PB}} = \int d^3x\left(\frac{\delta A}{\delta\phi(\mathbf{x})}\frac{\delta B}{\delta\pi(\mathbf{x})} - \frac{\delta A}{\delta\pi(\mathbf{x})}\frac{\delta B}{\delta\phi(\mathbf{x})}\right) \]

Show the following.

  1. Verify that \(\{\phi(\mathbf{x}), \pi(\mathbf{y})\}_{\mathrm{PB}} = \delta^3(\mathbf{x} - \mathbf{y})\).
  2. Verify that \(\{\phi(\mathbf{x}), \phi(\mathbf{y})\}_{\mathrm{PB}} = 0\) and \(\{\pi(\mathbf{x}), \pi(\mathbf{y})\}_{\mathrm{PB}} = 0\).
  3. For the Hamiltonian \(H = \int d^3x\,\mathcal{H}\) (where \(\mathcal{H} = \frac{1}{2}\pi^2 + \frac{1}{2}(\nabla\phi)^2 + \frac{m^2}{2}\phi^2\)), compute Hamilton's equation of motion \(\dot{\phi}(\mathbf{x}) = \{\phi(\mathbf{x}), H\}_{\mathrm{PB}}\) and obtain \(\dot{\phi} = \pi\).
  4. Similarly, compute \(\dot{\pi}(\mathbf{x}) = \{\pi(\mathbf{x}), H\}_{\mathrm{PB}}\) and obtain \(\dot{\pi} = \nabla^2\phi - m^2\phi\). Confirm that combining these reproduces the Klein-Gordon equation.
  5. Apply the canonical quantization prescription \(\{\cdot, \cdot\}_{\mathrm{PB}} \to \frac{1}{i\hbar}[\cdot, \cdot]\) and verify that the equal-time commutation relation \([\hat{\phi}(\mathbf{x}), \hat{\pi}(\mathbf{y})] = i\hbar\,\delta^3(\mathbf{x} - \mathbf{y})\) from Ch. 4 is obtained.
Hint

For 1, set \(A = \phi(\mathbf{x})\), \(B = \pi(\mathbf{y})\) and compute the functional derivatives. Use \(\frac{\delta\phi(\mathbf{x})}{\delta\phi(\mathbf{z})} = \delta^3(\mathbf{x} - \mathbf{z})\). For 4, when computing \(\frac{\delta H}{\delta\phi(\mathbf{x})}\), pay attention to the \((\nabla\phi)^2\) term: using integration by parts yields \(-\nabla^2\phi\).

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