hamilton jacobi schrodinger wave mechanics

# Hamilton–Jacobi Mechanics, Canonical Phase-Space Transformations, and Semiclassical Wave Mechanics in Ion Beam and Electron Lithography Optics

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## Executive Summary

The Hamilton–Jacobi equation provides a powerful framework for classical particle trajectory analysis in charged particle optics systems used for ion implantation, electron lithography, and focused beam processing. This article develops the Hamilton–Jacobi formalism from first principles, establishes the connection to quantum mechanics via the WKB/Eikonal approximation, and demonstrates how semiclassical wave mechanics predicts interference effects and aberrations in ion and electron beam systems. Understanding Hamilton–Jacobi mechanics is essential for designing ultra-precise beam transport systems, predicting chromatic and spherical aberrations, and optimizing ion implantation profiles and electron beam lithography resolution for advanced semiconductor manufacturing below 5 nm.

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## Table of Contents

1. Introduction: Classical Mechanics and the Hamilton–Jacobi Equation
2. Hamiltonian Mechanics: Canonical Variables and Phase Space
3. The Hamilton–Jacobi Equation: Derivation and Form
4. Complete Solutions and Generating Functions
5. Action-Angle Variables and Integrable Systems
6. Canonical Transformations and Poisson Brackets
7. Charged Particle Motion in Electromagnetic Fields
8. Ray-Tracing in Paraxial Approximation
9. Aberrations: Spherical, Chromatic, Astigmatism
10. WKB and Eikonal Approximation: Bridge to Quantum Mechanics
11. Semiclassical Wave Mechanics in Optics
12. Python Implementation: Ray Tracing and Phase Space Dynamics
13. Applications to Ion Implantation and Electron Lithography
14. References & Further Reading

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## 1. Introduction: Classical Mechanics and the Hamilton–Jacobi Equation

### 1.1 Three Formulations of Classical Mechanics

Newton's laws (F = ma):
- Direct but limited to specific coordinate systems

Lagrangian mechanics (Euler-Lagrange equations):
- Coordinate-independent, exploits symmetries

Hamiltonian mechanics (Hamilton's equations):
- Phase-space formulation, canonical transformations, action variables
- Bridge to quantum mechanics via Poisson brackets → commutators

The Hamilton–Jacobi equation is the most general formulation, encompassing all others.

### 1.2 Motivation: Beam Optics Applications

In charged particle optics:
- Electrons travel at relativistic speeds (keV–MeV energies)
- Ions (H, He, implant species) travel at slow-to-fast speeds (keV–MeV)
- Magnetic lenses focus beams via Lorentz force
- Electrostatic lenses use potential gradients

Classical ray-tracing (Hamilton–Jacobi) is sufficient for most applications; quantum diffraction effects emerge only at extremely low energies or ultra-high resolution requirements.

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## 2. Hamiltonian Mechanics: Canonical Variables and Phase Space

### 2.1 Canonical Variables

Generalized coordinates $\mathbf{q} = (q_1, q_2, \ldots, q_N)$ describe configuration space.

Canonical momenta $\mathbf{p} = (p_1, p_2, \ldots, p_N)$ are defined via the Lagrangian:

$$p_i = \frac{\partial L}{\partial \dot{q}_i}$$

Phase space $(\mathbf{q}, \mathbf{p})$ has $2N$ dimensions.

### 2.2 Hamiltonian and Hamilton's Equations

The Hamiltonian (total energy in generalized coordinates):

$$H(\mathbf{q}, \mathbf{p}, t) = \sum_i p_i \dot{q}_i - L(\mathbf{q}, \dot{\mathbf{q}}, t)$$

Hamilton's canonical equations:

$$\boxed{\frac{dq_i}{dt} = \frac{\partial H}{\partial p_i}, \quad \frac{dp_i}{dt} = -\frac{\partial H}{\partial q_i}}$$

These are $2N$ first-order ODEs (vs. N second-order Lagrange equations).

### 2.3 Example: Charged Particle in EM Fields

For a particle with charge $q$ in electromagnetic fields $\mathbf{E}$, $\mathbf{B}$:

$$H = \frac{1}{2m}(\mathbf{p} - q\mathbf{A})^2 + q\Phi$$

where $\mathbf{A}$ is the vector potential and $\Phi$ is the scalar potential.

Canonical momentum (generalized momentum):

$$\mathbf{p} = m\mathbf{v} + q\mathbf{A}$$

(different from mechanical momentum $m\mathbf{v}$!)

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## 3. The Hamilton–Jacobi Equation: Derivation and Form

### 3.1 Action and Action-Momentum

Define the action (momentum-like quantity):

$$S(\mathbf{q}, t) = \int_0^t L(\mathbf{q}(t'), \dot{\mathbf{q}}(t'), t') dt'$$

The principal function satisfies:

$$\frac{\partial S}{\partial q_i} = p_i$$

### 3.2 Hamilton–Jacobi Equation

Substituting into the canonical Hamilton equations, we derive:

$$\boxed{H\left(\mathbf{q}, abla_{\mathbf{q}} S, t ight) + \frac{\partial S}{\partial t} = 0}$$

This is a single first-order nonlinear PDE in $S(\mathbf{q}, t)$, replacing the $2N$ first-order ODEs of Hamilton's equations!

### 3.3 Time-Independent Form

For a conservative system ($H$ independent of $t$):

$$H(\mathbf{q}, abla_{\mathbf{q}} S) = E$$

where $E$ is constant (energy).

Define the shortened action $W(\mathbf{q})$:

$$S(\mathbf{q}, t) = W(\mathbf{q}) - E t$$

The Hamilton–Jacobi equation becomes:

$$\boxed{H(\mathbf{q}, abla_{\mathbf{q}} W) = E}$$

### 3.4 Physical Interpretation: Wave Fronts

The surfaces $S(\mathbf{q}, t) = ext{const}$ are wavefronts (or equiphase surfaces in quantum language).

The normal to these surfaces is:

$$\mathbf{n} = \frac{ abla S}{| abla S|} = \frac{\mathbf{p}}{|p|}$$

(points in the direction of motion)

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## 4. Complete Solutions and Generating Functions

### 4.1 Complete Solution

A complete solution to the Hamilton–Jacobi equation is:

$$S(\mathbf{q}, \mathbf{\alpha}, t)$$

containing $N$ arbitrary constants $\mathbf{\alpha} = (\alpha_1, \ldots, \alpha_N)$ plus time.

Once found, the trajectory is given implicitly by:

$$\frac{\partial S}{\partial \alpha_i} = \beta_i = ext{const}$$

where $\beta_i$ are $N$ integration constants determined by initial conditions.

### 4.2 Generating Functions

A generating function $F(\mathbf{q}, \mathbf{P})$ relates old canonical variables $(\mathbf{q}, \mathbf{p})$ to new variables $(\mathbf{Q}, \mathbf{P})$:

$$\mathbf{p} = abla_{\mathbf{q}} F, \quad \mathbf{Q} = abla_{\mathbf{P}} F$$

This allows canonical transformations to simpler coordinates (action-angle variables).

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## 5. Action-Angle Variables and Integrable Systems

### 5.1 Action Variables

For a periodic system, the action is:

$$I_i = \oint_{C_i} p_i dq_i$$

(integral over one cycle of coordinate $q_i$)

Conservation: In an integrable system, $I_i$ are conserved (constant of motion).

### 5.2 Angle Variables

Angle variables $ heta_i$ are conjugate to actions:

$$\frac{\partial H}{\partial I_i} = \omega_i \quad ext{(frequency)}$$

$$\frac{d heta_i}{dt} = \omega_i \quad \Rightarrow \quad heta_i(t) = \omega_i t + \phi_{i,0}$$

General solution: $(I_i, heta_i)$ = const; motion is periodic in each coordinate.

### 5.3 Application: Charged Particle Confinement

In a magnetic mirror trap, the adiabatic invariant (action) is:

$$\mu = \frac{m v_\perp^2}{2B}$$

(magnetic moment)

If $\mu$ is conserved, particles spiral around field lines at constant pitch angle.

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## 6. Canonical Transformations and Poisson Brackets

### 6.1 Poisson Brackets

The Poisson bracket of two functions $f$ and $g$ in phase space is:

$$\{f, g\} = \sum_i \left( \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} ight)$$

Properties:
- Antisymmetry: $\{f, g\} = -\{g, f\}$
- Jacobi identity: $\{f, \{g, h\}\} + \{g, \{h, f\}\} + \{h, \{f, g\}\} = 0$
- Product rule: $\{f, gh\} = g\{f, h\} + h\{f, g\}$

### 6.2 Equations of Motion in Poisson Brackets

Hamilton's equations can be written as:

$$\frac{df}{dt} = \{f, H\} + \frac{\partial f}{\partial t}$$

Special cases:
- If $\{f, H\} = 0$ and $\frac{\partial f}{\partial t} = 0$, then $f$ is conserved.
- $\frac{dH}{dt} = \{H, H\} + \frac{\partial H}{\partial t} = \frac{\partial H}{\partial t}$ (energy conservation if $H$ is time-independent)

### 6.3 Quantum Mechanical Analogue

In quantum mechanics, Poisson brackets become commutators:

$$\{f, g\}_{ ext{Poisson}} ightarrow \frac{1}{i\hbar}[\hat{f}, \hat{g}]$$

This is the deep connection between classical and quantum mechanics!

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## 7. Charged Particle Motion in Electromagnetic Fields

### 7.1 Hamiltonian for Charged Particle

A particle with charge $q$ and mass $m$ in fields $\mathbf{E} = -
abla \Phi - \frac{\partial \mathbf{A}}{\partial t}$ and $\mathbf{B} =
abla imes \mathbf{A}$:

$$H = \frac{1}{2m}(\mathbf{p} - q\mathbf{A})^2 + q\Phi$$

### 7.2 Equations of Motion (Lorentz Force)

Hamilton's equations yield:

$$\frac{d\mathbf{q}}{dt} = \frac{\mathbf{p} - q\mathbf{A}}{m} = \mathbf{v}$$

$$\frac{d\mathbf{p}}{dt} = q(\mathbf{E} + \mathbf{v} imes \mathbf{B}) + q abla(\mathbf{v} \cdot \mathbf{A})$$

The last term vanishes in the Coulomb gauge ($
abla \cdot \mathbf{A} = 0$), leaving the standard Lorentz force:

$$m\frac{d\mathbf{v}}{dt} = q(\mathbf{E} + \mathbf{v} imes \mathbf{B})$$

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## 8. Ray-Tracing in Paraxial Approximation

### 8.1 Paraxial Rays

Paraxial approximation: Rays stay close to the optical axis (z-axis) and make small angles with it.

Let the deflection from axis be $x(z)$ and $y(z)$ (transverse coordinates).

Paraxial ray equation (from Hamilton–Jacobi):

$$\frac{d^2 x}{dz^2} + \frac{1}{4V(z)} \frac{dV}{dz} \frac{dx}{dz} + \frac{q}{4m V(z)} \frac{d^2\Phi_{\perp}}{dx^2} = 0$$

where $V(z) = q\Phi(z, 0, 0)$ is the axial potential and $\Phi_\perp$ is the transverse field.

### 8.2 Lens Equation and Focal Length

A thin lens creates a discontinuity in ray angle:

$$\frac{dx'}{dx}\bigg|_{ ext{out}} = \frac{dx'}{dx}\bigg|_{ ext{in}} - \frac{x}{f}$$

where $x'$ is the ray angle and $f$ is the focal length.

### 8.3 Matrix Ray Tracing

For a sequence of lenses and drift spaces, the ray propagation can be expressed as:

$$\begin{pmatrix} x_{ ext{out}} \\ x'_{ ext{out}} \end{pmatrix} = \mathbf{M} \begin{pmatrix} x_{ ext{in}} \\ x'_{ ext{in}} \end{pmatrix}$$

where $\mathbf{M}$ is a $2 imes 2$ ABCD matrix characterizing the optical system.

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## 9. Aberrations: Spherical, Chromatic, Astigmatism

### 9.1 Spherical Aberration

Rays at large radii from the axis focus at different distances than paraxial rays.

Third-order aberration coefficient:

$$C_s = \frac{1}{2} \int_0^L B_r(z) L_p^2(z) dz$$

where $B_r$ is radial magnetic field and $L_p$ is the paraxial focal length.

Aberration diameter (at object):

$$D_s = C_s \alpha^3$$

where $\alpha$ is the beam half-angle.

For modern electron microscopes: $C_s \sim 1$–10 mm (mm-level, requiring aberration correction!).

### 9.2 Chromatic Aberration

Particles with different kinetic energies follow different trajectories.

For a particle with energy $E + \Delta E$:

$$\Delta x = C_c \frac{\Delta E}{E} an \beta$$

where $C_c$ is the chromatic aberration coefficient and $ an \beta$ is the divergence angle.

Energy spread $\Delta E$ arises from:
- Thermal emission (electrons): $\Delta E \sim 0.5$–1 eV
- Source size and field inhomogeneity

### 9.3 Astigmatism

Astigmatism: Different focal lengths in orthogonal directions (tangential vs. sagittal).

Caused by:
- Asymmetric stray fields (e.g., misalignment)
- Distorted apertures
- Uneven lens performance

Corrected by stigmator coils (small auxiliary magnets).

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## 10. WKB and Eikonal Approximation: Bridge to Quantum Mechanics

### 10.1 WKB Expansion

The WKB (Wentzel-Kramers-Brillouin) approximation expands the quantum wavefunction as:

$$\psi(\mathbf{r}) = A(\mathbf{r}) e^{iS(\mathbf{r})/\hbar}$$

where $A$ is amplitude and $S$ is the action.

To leading order ($\hbar o 0$):

$$( abla S)^2 = 2m(E - V)$$

This is exactly the Hamilton–Jacobi equation with $\mathbf{p} =
abla S$.

### 10.2 Validity of Classical Limit

The WKB approximation is valid when:

$$\frac{\lambda_{ ext{dB}}}{L} \ll 1 \quad ext{or equivalently} \quad \hbar | abla p| / p^2 \ll 1$$

For electron beams (keV–MeV):
- $\lambda_{ ext{dB}} \sim 1$–0.01 nm (very short)
- Classical ray-tracing is excellent

For ion beams (keV–MeV):
- $\lambda_{ ext{dB}} \sim 0.01$–0.001 nm (even shorter)
- Classical mechanics dominates

Quantum effects (diffraction, interference) only matter below 1 eV.

### 10.3 Quantum Corrections to Classical Trajectories

First-order WKB corrections include:

$$\psi \approx \frac{C}{\sqrt{p(x)}} e^{i\int p(x') dx'/\hbar}$$

The prefactor $1/\sqrt{p}$ (amplitude modulation) arises from probability conservation.

Near a turning point (where $p = 0$), WKB breaks down → need Airy function matching.

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## 11. Semiclassical Wave Mechanics in Optics

### 11.1 Gaussian Beam Optics

A Gaussian beam (fundamental mode) has amplitude:

$$\psi(r, z) \propto \frac{1}{\sqrt{w(z)}} \exp\left( -\frac{r^2}{w^2(z)} ight) \exp\left( i \phi(z) ight)$$

where $w(z) = w_0 \sqrt{1 + (z/z_R)^2}$ is beam radius and $z_R = \pi w_0^2 / \lambda$ is Rayleigh range.

Focal spot size is limited by diffraction:

$$w_{\min} \sim \lambda / (2 ext{NA})$$

where NA is numerical aperture.

### 11.2 Correspondence: Ray Optics ↔ Wave Optics

### 11.3 Quantum Coherence and Visibility

In interference experiments, the visibility (contrast) depends on:

$$V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}} = e^{-\pi^2 (\Delta k)^2 \sigma^2}$$

where $\Delta k$ is the wavenumber difference and $\sigma$ is the coherence length.

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## 12. Python Implementation: Ray Tracing and Phase Space Dynamics

"""
Charged Particle Ray Tracing via Hamilton-Jacobi Mechanics
Ion implantation beam optics in a magnetic lens system
"""

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint

# Physical constants
e = 1.602e-19      # C
m_i = 4 * 1.66e-27  # kg (He ion, 4 amu)
c = 2.998e8        # m/s

# Beam energy and initial conditions
E_beam = 100e3 * e  # 100 keV
v_beam = np.sqrt(2 * E_beam / m_i)
gamma = 1 + E_beam / (m_i * c**2)  # Lorentz factor
m_eff = gamma * m_i  # Relativistic mass

# Lens parameters
B_0 = 0.1  # T (axial magnetic field strength)
R_lens = 0.01  # m (lens radius, 1 cm)
z_lens = 0.05  # m (lens location, 5 cm)

# Initial beam parameters
r_0 = 1e-4  # m (beam waist, 100 μm)
alpha_0 = 1e-3  # rad (half-angle divergence)

# Differential equations for ray trajectories in paraxial approximation
def ray_dynamics(y, z):
    """
    y = [r, r', theta, theta']
    r, theta: radial and azimuthal positions
    r', theta': derivatives with respect to z
    """
    r, r_prime, theta, theta_prime = y
    
    # Magnetic field (Gaussian profile)
    B_z = B_0 * np.exp(-(z - z_lens)**2 / (0.01)**2)
    
    # Paraxial ray equation: d²r/dz² = -(1/4V) dV/dz * dr/dz + (q/4mV) d²Φ⊥/dr²
    # Simplified for symmetric magnetic lens:
    # d²r/dz² ≈ -ω_L² r, where ω_L = qB/(2m) is Larmor frequency
    
    omega_L = e * B_z / (2 * m_eff)
    
    # Ray equations
    dr_dz = r_prime
    dr_prime_dz = -omega_L**2 * r  # Focusing term from magnetic field
    dtheta_dz = theta_prime
    dtheta_prime_dz = -omega_L**2 * theta
    
    return [dr_dz, dr_prime_dz, dtheta_dz, dtheta_prime_dz]

# Beam envelope calculation
z_array = np.linspace(-0.1, 0.15, 300)

# Multiple rays at different initial angles
n_rays = 7
initial_angles = np.linspace(-alpha_0, alpha_0, n_rays)

ray_trajectories = []

for angle in initial_angles:
    y_0 = [r_0, angle, 0, 0]  # Initial: position r_0, angle, no azimuthal component
    solution = odeint(ray_dynamics, y_0, z_array)
    ray_trajectories.append(solution[:, 0])  # r(z)

ray_trajectories = np.array(ray_trajectories)

# Beam envelope (outer rays)
envelope = np.max(np.abs(ray_trajectories), axis=0)

# Focal length (where envelope is minimum)
z_focal_idx = np.argmin(envelope)
z_focal = z_array[z_focal_idx]
r_focal = envelope[z_focal_idx]

# Plot results
fig, axes = plt.subplots(1, 2, figsize=(14, 6))

# Subplot 1: Ray trajectories
ax1 = axes[0]
for i, r_traj in enumerate(ray_trajectories):
    ax1.plot(z_array*100, r_traj*1e6, 'b-', alpha=0.5, linewidth=1)

ax1.plot(z_array*100, envelope*1e6, 'r-', linewidth=2.5, label='Beam envelope')
ax1.axvline(z_lens*100, color='k', linestyle='--', alpha=0.5, label='Magnetic lens')
ax1.axvline(z_focal*100, color='g', linestyle='--', alpha=0.7, linewidth=1.5, label=f'Focal point (z={z_focal*100:.1f} cm)')
ax1.set_xlabel('Position z (cm)')
ax1.set_ylabel('Beam radius r (μm)')
ax1.set_title('Ion Beam Ray Tracing through Magnetic Lens')
ax1.legend()
ax1.grid(alpha=0.3)

# Subplot 2: Phase space (position vs. angle)
ax2 = axes[1]
for i, r_traj in enumerate(ray_trajectories):
    r_prime = np.gradient(r_traj, z_array)
    ax2.plot(r_traj*1e6, r_prime*1e3, 'b-', alpha=0.5, linewidth=1)

ax2.set_xlabel('Beam radius r (μm)')
ax2.set_ylabel('Ray angle r\' (mrad)')
ax2.set_title('Phase Space Portrait (Position-Momentum)')
ax2.grid(alpha=0.3)

plt.tight_layout()
plt.savefig('hamilton_jacobi_ray_tracing.png', dpi=150, bbox_inches='tight')
plt.show()

print(f"
{'='*70}")
print(f"Hamilton-Jacobi Ray Tracing for Ion Implantation")
print(f"{'='*70}")
print(f"Beam parameters:")
print(f"  Species: He+")
print(f"  Energy: {E_beam/e/1e3:.0f} keV")
print(f"  Velocity: {v_beam/1e6:.2f} m/μs")
print(f"  Relativistic factor γ: {gamma:.4f}")
print(f"
Lens parameters:")
print(f"  Magnetic field (peak): {B_0*1e3:.1f} mT")
print(f"  Focal distance: {z_focal*100:.2f} cm")
print(f"  Focal spot radius: {r_focal*1e6:.2f} μm")
print(f"  Magnification: {r_focal/r_0:.2f}×")
print(f"{'='*70}")

---

## 13. Applications to Ion Implantation and Electron Lithography

### 13.1 Ion Implantation Optics

Ion beams are focused to sub-μm spots for:
- Single-ion doping: Precise dopant placement
- Ion beam lithography: Ultra-high-resolution patterning
- Ion milling: Localized material removal

Hamilton–Jacobi ray-tracing predicts beam size at target (convolution of source size, optical aberrations, and straggling).

### 13.2 Electron Lithography (E-beam)

Electron beams for lithography face:
- Spherical aberration: $C_s \sim 1$–10 mm (corrected by electromagnetic coils)
- Chromatic aberration: $\Delta E \sim 0.5$ eV (corrected by energy filter)
- Diffraction limit: $\lambda_{dB} \sim 0.01$ nm at 100 keV (geometric optics dominates)

Modern e-beam systems achieve resolution < 10 nm via aberration-corrected systems.

### 13.3 Impact on Device Performance

Beam optics directly determines:
- Implant profile width (for junction formation)
- Doping uniformity (channeling control)
- Defect distribution (annealing requirements)

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## 14. References & Further Reading

1. Goldstein, H., Poole, C., & Safko, J. (2002). *Classical Mechanics* (3rd ed.). Addison-Wesley.
2. Arnold, V. I. (1989). *Mathematical Methods of Classical Mechanics* (2nd ed.). Springer-Verlag.
3. Grivet, P. (1965). *Electron Optics* (2nd ed.). Pergamon Press.
4. Kildal, H., & Seitz, K. (1996). *Ion Beam Optics*. Springer.
5. Glaser, W. (1956). *Grundlagen der Elektronenoptik*. Springer (classic reference).
6. Becker, R. L. (1978). "Survey of Electron Optics." *Advances in Electronics and Electron Physics*, 49, 1–87.

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This article completes the mathematical framework for semiconductor device engineering, connecting classical mechanics (Hamilton–Jacobi), quantum mechanics (WKB), and practical applications in ion/electron beam optics.