Table of Contents

1. Introduction
2. The Classical vs Quantum View of Free Motion
3. The Concept of a Free Particle in Quantum Mechanics
4. Schrödinger Equation for a Free Particle
5. General Plane Wave Solutions and Their Interpretation
6. Normalization and the Role of the Delta Function
7. Momentum Eigenstates and Their Significance
8. Constructing Physical States: Superposition and Wave Packets
9. Time Evolution of Wave Packets: Dispersion and Spreading
10. Gaussian Wave Packets: A Detailed Study
11. Probability Density, Probability Current, and Continuity
12. Energy and Momentum Expectation Values
13. Heisenberg Uncertainty Principle in Free Motion
14. Free Particle in Higher Dimensions
15. Real-World Applications and Importance
16. Conclusion

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1. Introduction

The free particle model is one of the most fundamental yet insightful topics in quantum mechanics. It serves as the baseline for understanding wave behavior, uncertainty, and quantum dynamics without the complication of external potentials. Despite the simplicity of a “potential-free” scenario, the quantum treatment reveals rich and non-intuitive behavior, highlighting the wave-particle duality of matter.

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2. The Classical vs Quantum View of Free Motion

In classical mechanics, a free particle is one that moves with constant velocity in a straight line due to the absence of external forces. Newton’s first law governs its motion.

In quantum mechanics, the free particle does not follow a deterministic trajectory. Instead, its behavior is described by a wavefunction that evolves in space and time, governed by the Schrödinger equation. This wavefunction encodes the probabilistic nature of its location and momentum.

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3. The Concept of a Free Particle in Quantum Mechanics

A free particle is defined as one whose potential energy \( V(x) \) is zero everywhere. The particle is not influenced by any fields or barriers.

This scenario allows us to focus solely on the kinetic energy of the particle:

\[
\hat{H} = \frac{\hat{p}^2}{2m}
\]

Where \( \hat{p} = -i\hbar \frac{d}{dx} \) is the momentum operator.

---

4. Schrödinger Equation for a Free Particle

In one spatial dimension, the time-dependent Schrödinger equation is:

\[
i\hbar \frac{\partial \psi(x, t)}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi(x, t)}{\partial x^2}
\]

Here:

• \( \psi(x, t) \): wavefunction of the particle
• \( m \): mass of the particle
• \( \hbar \): reduced Planck’s constant

This second-order partial differential equation governs how the wavefunction evolves over time.

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5. General Plane Wave Solutions and Their Interpretation

A standard solution is the plane wave:

\[
\psi_k(x, t) = A e^{i(kx – \omega t)}
\]

Where:

• \( k \): wave number, related to momentum \( p = \hbar k \)
• \( \omega = \frac{\hbar k^2}{2m} \): angular frequency
• \( E = \hbar \omega \): energy of the particle

Plane waves are idealized, infinite-extent solutions representing a particle with definite momentum but uncertain position. These solutions are not square-integrable, which means they are not physically realizable alone but are still mathematically crucial.

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6. Normalization and the Role of the Delta Function

Since plane waves extend to infinity, they cannot be normalized in the usual sense. Instead, we use Dirac delta normalization:

\[
\langle \psi_{k’} | \psi_k \rangle = \delta(k – k’)
\]

This normalization allows us to construct physical, normalizable states using wave packets, which are superpositions of plane waves.

---

7. Momentum Eigenstates and Their Significance

The momentum operator in position space is:

\[
\hat{p} = -i\hbar \frac{d}{dx}
\]

Plane waves are eigenfunctions of this operator:

\[
\hat{p} \psi_k(x) = \hbar k \psi_k(x)
\]

This means that a particle in a plane wave state has a definite momentum \( \hbar k \) but completely uncertain position.

---

8. Constructing Physical States: Superposition and Wave Packets

To represent a localized particle, we construct a wave packet by integrating over many momentum states:

\[
\psi(x, t) = \int_{-\infty}^{\infty} \phi(k) e^{i(kx – \omega t)} dk
\]

Where:

• \( \phi(k) \): momentum space distribution, often chosen as a Gaussian

This results in a localized wavefunction with both position and momentum uncertainties.

---

9. Time Evolution of Wave Packets: Dispersion and Spreading

Unlike classical particles, quantum wave packets spread over time due to dispersion. This occurs because each component wave has a different velocity, leading to destructive interference in some regions and constructive in others.

The shape of the packet broadens with time, reflecting increasing uncertainty in position.

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10. Gaussian Wave Packets: A Detailed Study

Consider a Gaussian wave packet at \( t = 0 \):

\[
\psi(x, 0) = \left( \frac{1}{2\pi \sigma_0^2} \right)^{1/4} \exp\left( -\frac{x^2}{4\sigma_0^2} \right)
\]

Its time evolution is:

\[
\psi(x, t) = \left( \frac{1}{2\pi \sigma_t^2} \right)^{1/4} \exp\left( -\frac{x^2}{4\sigma_t^2} + i \theta(x,t) \right)
\]

Where:

\[
\sigma_t = \sigma_0 \sqrt{1 + \left( \frac{\hbar t}{2m\sigma_0^2} \right)^2 }
\]

Key points:

• The width \( \sigma_t \) increases over time
• \( \theta(x, t) \) is a phase factor
• The shape remains Gaussian but spreads out

---

11. Probability Density, Probability Current, and Continuity

The probability density is:

\[
\rho(x, t) = |\psi(x, t)|^2
\]

The probability current is:

\[
j(x, t) = \frac{\hbar}{2mi} \left( \psi^* \frac{\partial \psi}{\partial x} – \psi \frac{\partial \psi^*}{\partial x} \right)
\]

These satisfy the continuity equation:

\[
\frac{\partial \rho}{\partial t} + \frac{\partial j}{\partial x} = 0
\]

Ensuring conservation of total probability.

---

12. Energy and Momentum Expectation Values

For a wave packet \( \psi(x, t) \), the expectation values are:

• Momentum:
  \[
  \langle \hat{p} \rangle = \int \psi^*(x, t) (-i\hbar \frac{d}{dx}) \psi(x, t) dx
  \]
• Energy:
  \[
  \langle \hat{H} \rangle = \int \psi^*(x, t) \left( -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \right) \psi(x, t) dx
  \]

These values remain constant over time for a free particle.

---

13. Heisenberg Uncertainty Principle in Free Motion

For a Gaussian wave packet:

\[
\Delta x \Delta p = \frac{\hbar}{2}
\]

As time evolves:

• \( \Delta x \) increases
• \( \Delta p \) remains constant

This reflects the quantum spreading of the particle’s position distribution.

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14. Free Particle in Higher Dimensions

In 2D and 3D, solutions generalize to:

\[
\psi(\vec{r}, t) = \int \phi(\vec{k}) e^{i(\vec{k} \cdot \vec{r} – \omega t)} d^n k
\]

Free particle behavior is important for describing propagating beams, scattering, and field quantization.

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15. Real-World Applications and Importance

Free particle models are crucial in:

• Electron microscopy
• Quantum optics (e.g., laser beam propagation)
• Neutron and X-ray diffraction
• Scattering theory
• Semiconductor modeling

They serve as the starting point for perturbation methods and Green’s function techniques.

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16. Conclusion

Though the free particle lacks potential energy, its quantum description is rich and foundational. It introduces central ideas like plane waves, momentum eigenstates, wave packet dynamics, and quantum uncertainty. Mastery of this simple system builds the groundwork for understanding interactions, measurements, and quantum fields.

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Table of Contents

1. Introduction
2. What Are Coherent States?
3. The Harmonic Oscillator Framework
4. Definition via Displacement Operator
5. Definition as Eigenstates of the Annihilation Operator
6. Properties of Coherent States
7. Uncertainty Minimization and Gaussian Form
8. Phase Space Representation
9. Time Evolution of Coherent States
10. Overlap and Non-Orthogonality
11. Fock Basis Expansion
12. Wigner Function and Quasi-Probability Distributions
13. Coherent States in Quantum Optics
14. Schrödinger’s Cat and Superposition of Coherent States
15. Applications in Quantum Technologies
16. Conclusion

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1. Introduction

Coherent states are special quantum states that most closely resemble classical oscillatory motion. First introduced by Schrödinger and extensively developed in quantum optics, coherent states form the cornerstone of many semiclassical approximations and quantum technologies.

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2. What Are Coherent States?

Coherent states are defined as quantum states of the harmonic oscillator that:

• Minimize the Heisenberg uncertainty principle
• Exhibit classical-like sinusoidal motion in expectation values
• Maintain their shape during time evolution

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3. The Harmonic Oscillator Framework

In quantum mechanics, the harmonic oscillator uses the ladder operators:

\[
\hat{a} = \frac{1}{\sqrt{2\hbar m\omega}}(m\omega \hat{x} + i\hat{p}), \quad \hat{a}^\dagger = \frac{1}{\sqrt{2\hbar m\omega}}(m\omega \hat{x} – i\hat{p})
\]

These satisfy:

\[
[\hat{a}, \hat{a}^\dagger] = 1
\]

---

4. Definition via Displacement Operator

The displacement operator is:

\[
\hat{D}(\alpha) = \exp(\alpha \hat{a}^\dagger – \alpha^* \hat{a})
\]

A coherent state is then:

\[
|\alpha\rangle = \hat{D}(\alpha)|0\rangle
\]

Where \( |0\rangle \) is the vacuum (ground) state and \( \alpha \in \mathbb{C} \).

---

5. Definition as Eigenstates of the Annihilation Operator

Alternatively, coherent states satisfy:

\[
\hat{a}|\alpha\rangle = \alpha |\alpha\rangle
\]

This definition highlights their role as eigenstates of a non-Hermitian operator — a rare property in quantum mechanics.

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6. Properties of Coherent States

• Not orthogonal: \( \langle \alpha | \beta \rangle \ne 0 \)
• Overcomplete: they form an overcomplete basis in Hilbert space
• Saturate uncertainty:
  \[
  \Delta x \Delta p = \frac{\hbar}{2}
  \]

---

7. Uncertainty Minimization and Gaussian Form

Position representation of \( |\alpha\rangle \):

\[
\psi_\alpha(x) = \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \exp\left[ -\frac{m\omega}{2\hbar}(x – x_0)^2 + i p_0 x/\hbar \right]
\]

Where \( x_0 \) and \( p_0 \) are determined by \( \alpha \).

These Gaussian wave packets do not spread during evolution, preserving their minimum uncertainty.

---

8. Phase Space Representation

Each coherent state corresponds to a point in phase space:

\[
\alpha = \frac{1}{\sqrt{2\hbar m \omega}} (m\omega x_0 + ip_0)
\]

Evolution follows a circular trajectory:

\[
\alpha(t) = \alpha(0) e^{-i\omega t}
\]

---

9. Time Evolution of Coherent States

Coherent states evolve under harmonic oscillator Hamiltonian as:

\[
|\alpha(t)\rangle = e^{-i\omega t/2} |\alpha(0)e^{-i\omega t}\rangle
\]

The state remains coherent, and expectation values trace classical motion.

---

10. Overlap and Non-Orthogonality

\[
\langle \alpha | \beta \rangle = \exp\left( -\frac{1}{2}|\alpha|^2 – \frac{1}{2}|\beta|^2 + \alpha^* \beta \right)
\]

This non-zero overlap leads to interference and quasi-classical behavior.

---

11. Fock Basis Expansion

\[
|\alpha\rangle = e^{-|\alpha|^2/2} \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} |n\rangle
\]

This shows coherent states as superpositions of number states with Poisson distribution:

\[
P(n) = |\langle n|\alpha\rangle|^2 = \frac{|\alpha|^{2n}}{n!} e^{-|\alpha|^2}
\]

---

12. Wigner Function and Quasi-Probability Distributions

Coherent states have positive-definite Wigner functions:

\[
W(x, p) = \frac{1}{\pi \hbar} \exp\left( -\frac{(x – x_0)^2}{\sigma_x^2} – \frac{(p – p_0)^2}{\sigma_p^2} \right)
\]

Indicating their quasi-classical nature.

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13. Coherent States in Quantum Optics

• Describe laser light
• Basis for quantum states of the electromagnetic field
• Used in optical coherence tomography, quantum metrology, and squeezing

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14. Schrödinger’s Cat and Superposition of Coherent States

Superpositions like:

\[
|\psi\rangle = \frac{1}{\sqrt{2}} (|\alpha\rangle + |-\alpha\rangle)
\]

Represent macroscopic quantum superpositions, or “cat states”, with interference in phase space.

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15. Applications in Quantum Technologies

• Quantum communication
• Quantum cryptography
• Bosonic quantum error correction
• Continuous-variable quantum computing

Coherent states are essential for continuous-variable encodings and optical implementations.

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16. Conclusion

Coherent states elegantly blend quantum and classical behavior. They provide insight into wavepacket dynamics, laser theory, and field quantization, while serving as a key resource in quantum optics and information. Their mathematical richness and physical realism make them indispensable in both theory and application.

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Table of Contents

1. Introduction
2. Classical vs Quantum Oscillator
3. Potential and Schrödinger Equation
4. Dimensionless Variables and Rescaling
5. Series Solution and Hermite Polynomials
6. Energy Quantization
7. Wavefunctions and Normalization
8. Probability Densities and Node Structure
9. Ladder Operators: Creation and Annihilation
10. Algebraic Solution Using Operators
11. Properties of Number States
12. Uncertainty and Coherent States
13. Time Evolution and Phase Space
14. Quantum vs Classical Motion
15. Applications Across Physics
16. Conclusion

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1. Introduction

The quantum harmonic oscillator (QHO) is one of the most important models in quantum mechanics. Its simplicity, solvability, and wide applicability make it a cornerstone in atomic physics, quantum optics, field theory, and beyond.

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2. Classical vs Quantum Oscillator

Classical:

• Mass \( m \) in potential \( V(x) = \frac{1}{2} m \omega^2 x^2 \)
• Oscillates sinusoidally with constant amplitude and period

Quantum:

• Quantized energy levels
• Wavefunctions spread out with increasing quantum number
• Position and momentum described probabilistically

---

3. Potential and Schrödinger Equation

The time-independent Schrödinger equation is:

\[
-\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + \frac{1}{2} m \omega^2 x^2 \psi(x) = E \psi(x)
\]

This differential equation can be solved exactly via power series or operator methods.

---

4. Dimensionless Variables and Rescaling

Define:

\[
\xi = \sqrt{\frac{m\omega}{\hbar}} x, \quad \epsilon = \frac{2E}{\hbar \omega}
\]

Equation becomes:

\[
\frac{d^2 \psi}{d\xi^2} + (\epsilon – \xi^2)\psi = 0
\]

Solution leads to Hermite polynomials and exponential envelopes.

---

5. Series Solution and Hermite Polynomials

General solution:

\[
\psi_n(\xi) = N_n H_n(\xi) e^{-\xi^2/2}
\]

• \( H_n(\xi) \): Hermite polynomial of degree \( n \)
• \( N_n \): normalization constant

Quantization arises from requirement of normalizable wavefunctions.

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6. Energy Quantization

Allowed energy levels:

\[
E_n = \hbar \omega \left( n + \frac{1}{2} \right), \quad n = 0, 1, 2, \dots
\]

• Equally spaced levels
• Zero-point energy: \( \frac{1}{2} \hbar \omega \)

---

7. Wavefunctions and Normalization

\[
\psi_n(x) = \left( \frac{m\omega}{\pi \hbar} \right)^{1/4} \frac{1}{\sqrt{2^n n!}} H_n(\xi) e^{-\xi^2/2}
\]

• Even/odd parity depending on \( n \)
• Higher \( n \): more nodes and wider spread

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8. Probability Densities and Node Structure

• \( |\psi_n(x)|^2 \): probability of finding the particle at \( x \)
• \( n \) nodes for \( \psi_n(x) \)
• For large \( n \), classical turning points and envelope match classical predictions

---

9. Ladder Operators: Creation and Annihilation

Define operators:

\[
\hat{a} = \frac{1}{\sqrt{2\hbar m \omega}} (m\omega \hat{x} + i\hat{p}), \quad \hat{a}^\dagger = \frac{1}{\sqrt{2\hbar m \omega}} (m\omega \hat{x} – i\hat{p})
\]

Satisfy:

\[
[\hat{a}, \hat{a}^\dagger] = 1
\]

Hamiltonian becomes:

\[
\hat{H} = \hbar \omega \left( \hat{a}^\dagger \hat{a} + \frac{1}{2} \right)
\]

---

10. Algebraic Solution Using Operators

Use:

\[
\hat{a}^\dagger |n\rangle = \sqrt{n+1} |n+1\rangle, \quad \hat{a} |n\rangle = \sqrt{n} |n-1\rangle
\]

Construct higher states from ground state:

\[
|n\rangle = \frac{(\hat{a}^\dagger)^n}{\sqrt{n!}} |0\rangle
\]

---

11. Properties of Number States

• \( \hat{N} = \hat{a}^\dagger \hat{a} \) gives particle number
• \( \langle x \rangle = \langle p \rangle = 0 \) in all \( |n\rangle \)
• Variances increase with \( n \), maintaining uncertainty relation

---

12. Uncertainty and Coherent States

Ground state:

\[
\Delta x \Delta p = \frac{\hbar}{2}
\]

Coherent states minimize uncertainty like ground state but exhibit classical motion:

\[
|\alpha\rangle = e^{-|\alpha|^2/2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n\rangle
\]

---

13. Time Evolution and Phase Space

Each state evolves with phase:

\[
|\psi_n(t)\rangle = e^{-i E_n t/\hbar} |\psi_n\rangle
\]

Coherent states evolve like classical oscillators in phase space — a key link between quantum and classical worlds.

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14. Quantum vs Classical Motion

Aspect	Classical Oscillator	Quantum Harmonic Oscillator
Energy	Continuous	Discrete levels
Trajectory	Deterministic path	Probabilistic distribution
Zero-point	Absent	\( E_0 = \frac{1}{2}\hbar\omega \)

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15. Applications Across Physics

• Vibrational modes in molecules
• Phonons in solid-state systems
• Quantum optics and laser theory
• Field quantization in quantum electrodynamics
• Trapped ions and cavity QED

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16. Conclusion

The quantum harmonic oscillator is a model of exceptional importance. Its solvability and elegant algebraic structure provide a basis for many advanced theories. From foundational quantum mechanics to cutting-edge technologies, mastering this system is essential for any quantum physicist or engineer.

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