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Appendix D Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Calculation of Canonical Momentum from the Lagrangian

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

(a)

\[ L = \frac{1}{2}m\dot{q}^2 - \frac{1}{2}k q^2 \]

(b)

\[ L = \frac{1}{2}m(\dot{r}^2 + r^2\dot{\theta}^2) - V(r) \]

Find \(p_r = \frac{\partial L}{\partial \dot{r}}\) and \(p_\theta = \frac{\partial L}{\partial \dot{\theta}}\) respectively.

(c)

\[ L = \frac{1}{2}m\dot{q}^2 + e\dot{q}A(q) - e\phi(q) \]

where \(A(q)\) is the vector potential, \(\phi(q)\) is the scalar potential, and \(e\) is the electric charge.

Hint

Apply the definition of canonical momentum (D.12) \(p_j = \frac{\partial L}{\partial \dot{q}_j}\) directly. When taking the partial derivative with respect to \(\dot{q}\), treat \(q\) as a constant. In (c), note that \(A(q)\) does not depend on \(\dot{q}\).

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B-2. Construction of the Hamiltonian via Legendre Transform

For each Lagrangian from Drill D1, find the Hamiltonian \(H(q, p) = p\dot{q} - L\) as a function of \(q\) and \(p\). Here, \(\dot{q}\) should be eliminated by solving the definition of \(p\) inversely.

(a) For the case of D1(a)

(b) For the case of D1(b) (find \(H(r, \theta, p_r, p_\theta)\))

(c) For the case of D1(c)

Hint

Solve the expression \(p = \frac{\partial L}{\partial \dot{q}}\) obtained in D1 for \(\dot{q}\), and substitute into \(H = p\dot{q} - L\). For (b), note that \(H = p_r\dot{r} + p_\theta\dot{\theta} - L\).

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B-3. Application of Hamilton's Canonical Equations

For the Hamiltonian of a one-dimensional harmonic oscillator

\[ H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 q^2 \]

write down Hamilton's canonical equations (D.21) and find \(\dot{q}\) and \(\dot{p}\). Furthermore, by differentiating the equation for \(\dot{q}\) with respect to time and combining it with the equation for \(\dot{p}\), derive a second-order differential equation for \(q\) and confirm that it agrees with the equation of motion for a harmonic oscillator \(m\ddot{q} = -m\omega^2 q\).

Hint

Compute \(\dot{q} = \frac{\partial H}{\partial p}\) and \(\dot{p} = -\frac{\partial H}{\partial q}\) respectively. Differentiating \(\dot{q} = p/m\) with respect to time gives \(\ddot{q} = \dot{p}/m\), so substitute the expression for \(\dot{p}\).

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B-4. Direct Application of the Euler-Lagrange Equation

For the following Lagrangians, write down the Euler-Lagrange equation (D.9) and derive the equations of motion.

(a) Free particle: \(L = \frac{1}{2}m\dot{q}^2\)

(b) Particle in a gravitational field: \(L = \frac{1}{2}m\dot{q}^2 - mgq\) (where \(g\) is the gravitational acceleration)

(c) General one-dimensional potential: \(L = \frac{1}{2}m\dot{q}^2 - V(q)\)

Hint

Compute \(\frac{\partial L}{\partial \dot{q}}\) and \(\frac{\partial L}{\partial q}\) respectively, then substitute into \(\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0\). For (b), note that \(\frac{\partial}{\partial q}(mgq) = mg\).

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B-5. Direct Verification of Energy Conservation

For the Hamiltonian of the 1-dimensional harmonic oscillator (same as in D3), expand \(\frac{dH}{dt}\) using the chain rule and substitute Hamilton's canonical equations to show that \(\frac{dH}{dt} = 0\). Reproduce the calculation of equation (D.25) in the main text with your own hands.

Hint

Write \(\frac{dH}{dt} = \frac{\partial H}{\partial q}\dot{q} + \frac{\partial H}{\partial p}\dot{p}\), and substitute \(\dot{q}\), \(\dot{p}\) and \(\frac{\partial H}{\partial q}\), \(\frac{\partial H}{\partial p}\) obtained in D3.

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B-6. Concrete Calculation of the Action

Consider a free particle (\(V = 0\)) that is at position \(q_A = 0\) at time \(t_1 = 0\) and reaches position \(q_B = d\) at time \(t_2 = T\). Assuming the path is uniform linear motion \(q(t) = \frac{d}{T}t\), calculate the action

\[ S[q] = \int_0^T \frac{1}{2}m\dot{q}^2\, dt \]
Hint

Use the fact that for uniform linear motion, \(\dot{q} = d/T\) is a constant. The integral of a constant can be evaluated straightforwardly.

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B-7. Practice Calculating Variations

Consider a free particle action (same setup as D6), where a small deviation \(\delta q(t) = \epsilon \sin\!\left(\frac{\pi t}{T}\right)\) (with \(\epsilon\) being a small parameter) is added to the uniform linear motion path \(q_0(t) = \frac{d}{T}t\), giving the path \(q(t) = q_0(t) + \delta q(t)\).

(a) Verify that \(\delta q(t)\) satisfies the endpoint conditions \(\delta q(0) = \delta q(T) = 0\).

(b) Calculate \(S[q_0 + \delta q]\) to second order in \(\epsilon\), and confirm that \(S[q_0 + \delta q] - S[q_0]\) has no first-order term in \(\epsilon\).

Hint

Substitute \(\dot{q} = \dot{q}_0 + \epsilon\frac{\pi}{T}\cos\!\left(\frac{\pi t}{T}\right)\) into \(\frac{1}{2}m\dot{q}^2\) and expand. It is useful to apply \(\int_0^T \cos^2\!\left(\frac{\pi t}{T}\right)dt = T/2\). The fact that the first-order term in \(\epsilon\) vanishes is the meaning of "stationary."

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B-8. Calculation of Poisson Brackets

The Poisson bracket introduced in Section D.6 of the main text (outside the scope of the main text excerpt, but the definition is given below) is defined as follows:

\[ \{A, B\}_{\mathrm{PB}} = \sum_j \left(\frac{\partial A}{\partial q_j}\frac{\partial B}{\partial p_j} - \frac{\partial A}{\partial p_j}\frac{\partial B}{\partial q_j}\right) \]

For the case of one degree of freedom, calculate the following.

(a) \(\{q, p\}_{\mathrm{PB}}\)

(b) \(\{q, q\}_{\mathrm{PB}}\) and \(\{p, p\}_{\mathrm{PB}}\)

(c) \(\{q^2, p\}_{\mathrm{PB}}\)

(d) \(\{q, p^2\}_{\mathrm{PB}}\)

Hint

Substitute \(A\) and \(B\) into the definition and compute the partial derivatives. For example, in (a), set \(A = q\), \(B = p\) and use \(\frac{\partial q}{\partial q} = 1\), \(\frac{\partial q}{\partial p} = 0\), etc.

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Medium

M-1. Derivation of Euler-Lagrange Equations in 2D Polar Coordinates

Consider a particle of mass \(m\) moving in a two-dimensional plane under a central force potential \(V(r)\). The Lagrangian in polar coordinates \((r, \theta)\) is

\[ L = \frac{1}{2}m(\dot{r}^2 + r^2\dot{\theta}^2) - V(r) \]

(a) Derive the Euler-Lagrange equation with respect to \(r\) and explain its physical meaning (the radial equation of motion).

(b) Derive the Euler-Lagrange equation with respect to \(\theta\) and explain why \(\frac{d}{dt}(mr^2\dot{\theta}) = 0\) implies conservation of angular momentum.

(c) Compare these results with Newton's equations of motion expressed in polar coordinates and discuss the advantages of the Lagrangian formalism.

Hint

(a) Compute \(\frac{\partial L}{\partial \dot{r}} = m\dot{r}\) and \(\frac{\partial L}{\partial r} = mr\dot{\theta}^2 - V'(r)\). (b) Note that \(L\) does not depend explicitly on \(\theta\)—what does the Euler-Lagrange equation imply when \(\frac{\partial L}{\partial \theta} = 0\)? (c) Compare with the Newtonian formalism, where one must separately introduce the acceleration components in polar coordinates (centrifugal and Coriolis force terms).

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M-2. Inverse of the Legendre Transform

Verify that the Legendre transform "preserves information" by following the steps below.

(a) Starting from a one-degree-of-freedom Hamiltonian \(H(q, p)\), describe the procedure for obtaining \(p = p(q, \dot{q})\) by solving \(\dot{q} = \frac{\partial H}{\partial p}\) inversely for \(p\).

(b) Define the inverse Legendre transform \(L(q, \dot{q}) = p\dot{q} - H(q, p)\) (where \(p\) is expressed in terms of \(q, \dot{q}\)), and starting from the one-dimensional harmonic oscillator Hamiltonian \(H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 q^2\), recover \(L = \frac{1}{2}m\dot{q}^2 - \frac{1}{2}m\omega^2 q^2\).

(c) Show that, in general, applying the Legendre transform twice returns the original function (proof of involutivity).

Hint

(a) \(\dot{q} = \partial H/\partial p\) is an implicit equation for \(p\). (b) From \(H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 q^2\), we get \(\dot{q} = p/m\), so substitute \(p = m\dot{q}\). (c) Let \(L \xrightarrow{\text{Legendre}} H \xrightarrow{\text{Legendre}} L'\) and show that \(L' = L\). Return to the definition of \(\frac{\partial H}{\partial p}\) and compute.

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M-3. Trajectory of a Harmonic Oscillator in Phase Space

Consider Hamilton's canonical equations for a one-dimensional harmonic oscillator:

\[ \dot{q} = \frac{p}{m}, \qquad \dot{p} = -m\omega^2 q \]

(a) Solve this system of differential equations and express the general solutions \(q(t)\), \(p(t)\) in terms of the initial conditions \(q(0) = q_0\), \(p(0) = p_0\).

(b) Eliminate \(t\) from the solutions and show that the trajectory in phase space \((q, p)\) is an ellipse of the form

\[ \frac{q^2}{q_0^2 + p_0^2/(m\omega)^2} + \frac{p^2}{(m\omega)^2 q_0^2 + p_0^2} = 1 \]

(you may use the energy \(E\) to simplify as appropriate).

(c) Rewrite the equation of the ellipse using the energy \(E = H(q_0, p_0)\) as

\[ \frac{m\omega^2 q^2}{2E} + \frac{p^2}{2mE} = 1 \]

and express the lengths of the semi-major and semi-minor axes of the ellipse in terms of \(E\), \(m\), and \(\omega\).

Hint

(a) From \(\ddot{q} = \dot{p}/m = -\omega^2 q\), we get \(q(t) = q_0\cos\omega t + \frac{p_0}{m\omega}\sin\omega t\). Then \(p(t) = m\dot{q}(t)\) gives the momentum. (b) Use \(\cos^2\omega t + \sin^2\omega t = 1\). (c) Substitute \(E = \frac{p_0^2}{2m} + \frac{1}{2}m\omega^2 q_0^2\).

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M-4. Poisson Brackets and Hamilton's Equations of Motion

Show that the time evolution of an arbitrary dynamical variable \(A(q, p, t)\) can be written using the Poisson bracket as

\[ \frac{dA}{dt} = \{A, H\}_{\mathrm{PB}} + \frac{\partial A}{\partial t} \]

Furthermore, confirm that substituting \(A = q_j\) and \(A = p_j\) reproduces Hamilton's canonical equations (D.21).

Hint

Substitute Hamilton's canonical equations into \(\frac{dA}{dt} = \frac{\partial A}{\partial q_j}\dot{q}_j + \frac{\partial A}{\partial p_j}\dot{p}_j + \frac{\partial A}{\partial t}\). Compare with the definition of the Poisson bracket.

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M-5. Canonical Quantization: Verification of Commutation Relations

In the recipe of canonical quantization, the classical Poisson bracket is replaced by a commutator as follows:

\[ \{A, B\}_{\mathrm{PB}} \;\longrightarrow\; \frac{1}{i\hbar}[\hat{A}, \hat{B}] \]

(a) From \(\{q, p\}_{\mathrm{PB}} = 1\) (the result of D8(a)), derive the canonical commutation relation \([\hat{q}, \hat{p}] = i\hbar\).

(b) For the one-dimensional harmonic oscillator Hamiltonian \(\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2\hat{q}^2\), apply the Heisenberg equation of motion

\[ \frac{d\hat{A}}{dt} = \frac{1}{i\hbar}[\hat{A}, \hat{H}] \]

to \(\hat{A} = \hat{q}\) and \(\hat{A} = \hat{p}\), and show that the operator versions of Hamilton's canonical equations

\[ \frac{d\hat{q}}{dt} = \frac{\hat{p}}{m}, \qquad \frac{d\hat{p}}{dt} = -m\omega^2\hat{q} \]

are obtained.

Hint

(a) Apply the replacement rule directly to \(\{q, p\}_{\mathrm{PB}} = 1\). (b) Use the commutator identity \([\hat{q}, \hat{p}^2] = [\hat{q}, \hat{p}]\hat{p} + \hat{p}[\hat{q}, \hat{p}] = 2i\hbar\hat{p}\). Similarly, compute \([\hat{p}, \hat{q}^2]\).

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Advanced

A-1. Canonical Quantization of a Charged Particle in an Electromagnetic Field

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

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

where \(\mathbf{A}\) is the vector potential and \(\phi\) is the scalar potential.

(a) Find the canonical momentum \(\mathbf{p} = \frac{\partial L}{\partial \dot{\mathbf{r}}}\) and show that it differs from the ordinary mechanical momentum \(m\dot{\mathbf{r}}\).

(b) Derive the Hamiltonian \(H(\mathbf{r}, \mathbf{p})\) and show that

\[ H = \frac{(\mathbf{p} - e\mathbf{A})^2}{2m} + e\phi \]

(c) Apply the canonical quantization prescription \(\mathbf{r} \to \hat{\mathbf{r}}\), \(\mathbf{p} \to \hat{\mathbf{p}} = -i\hbar\nabla\) to write down the Schrödinger equation in an electromagnetic field:

\[ i\hbar\frac{\partial}{\partial t}\Psi = \frac{1}{2m}(-i\hbar\nabla - e\mathbf{A})^2\Psi + e\phi\,\Psi \]

(d) Under the gauge transformation \(\mathbf{A} \to \mathbf{A} + \nabla\chi\), \(\phi \to \phi - \frac{\partial\chi}{\partial t}\), show that the wave function transforms as \(\Psi \to \Psi' = e^{ie\chi/\hbar}\Psi\), and confirm that physical observables (such as the probability density \(|\Psi|^2\)) are gauge invariant.

Hint

(a) Note that \(\frac{\partial}{\partial \dot{r}_i}\left(e\dot{\mathbf{r}}\cdot\mathbf{A}\right) = eA_i\). (b) Substitute \(\dot{\mathbf{r}} = (\mathbf{p} - e\mathbf{A})/m\) into \(H = \mathbf{p}\cdot\dot{\mathbf{r}} - L\). (c) Replace \(\hat{\mathbf{p}}\) with its position representation \(-i\hbar\nabla\). (d) Substitute \(\Psi' = e^{ie\chi/\hbar}\Psi\) into the Schrödinger equation, and when computing \(\nabla\Psi'\), use \(\nabla(e^{ie\chi/\hbar}) = \frac{ie}{\hbar}(\nabla\chi)e^{ie\chi/\hbar}\).

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A-2. Noether's Theorem: From Symmetry to Conservation Laws

Given that a Lagrangian \(L(q_j, \dot{q}_j)\) is invariant (\(\delta L = 0\)) under the infinitesimal transformation \(q_j \to q_j + \epsilon\, \eta_j(q, \dot{q}, t)\) (where \(\epsilon\) is a small parameter), show the following.

(a) From the condition that the variation of the action vanishes, \(\delta S = 0\), derive that the conserved quantity (Noether charge)

\[ Q = \sum_j \frac{\partial L}{\partial \dot{q}_j}\,\eta_j \]

satisfies \(\frac{dQ}{dt} = 0\).

(b) For \(L = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2) - V(r)\) (where \(r = \sqrt{x^2+y^2+z^2}\)), show that the invariance under rotation about the \(z\)-axis (\(x \to x - \epsilon y\), \(y \to y + \epsilon x\), \(z \to z\)) implies conservation of the \(z\)-component of angular momentum \(L_z = m(x\dot{y} - y\dot{x})\).

(c) When the Lagrangian is invariant under time translation \(t \to t + \epsilon\) (i.e., \(\frac{\partial L}{\partial t} = 0\)), show that the corresponding conserved quantity is the energy (Hamiltonian) \(H = \sum_j p_j\dot{q}_j - L\).

Hint

(a) Substitute the Euler-Lagrange equations into \(\delta L = \frac{\partial L}{\partial q_j}\epsilon\eta_j + \frac{\partial L}{\partial \dot{q}_j}\epsilon\dot{\eta}_j = 0\) and collect terms into the form \(\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_j}\eta_j\right)\). (b) Substitute \(\eta_x = -y\), \(\eta_y = x\), \(\eta_z = 0\) into the result from (a). (c) For time translation, one must also account for the variation of the endpoints. Calculate using \(\delta q_j = \dot{q}_j\epsilon\), or alternatively rewrite \(\frac{dL}{dt} = \frac{\partial L}{\partial q_j}\dot{q}_j + \frac{\partial L}{\partial \dot{q}_j}\ddot{q}_j\) using the Euler-Lagrange equations.

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