The dipole approximation assumes that the wavelength of the electromagnetic field is much larger than the spatial extent of the electron wavefunction. In this case, the interaction Hamiltonian simplifies significantly.

Interaction Hamiltonian

In the dipole approximation, the interaction term becomes:

Hint = -d·E(t),

where:

• d = -er is the electric dipole moment of the electron,
• E(t) is the electric field of the photon.

Simplified Schrödinger Equation

The time-dependent Schrödinger equation becomes:

iℏ∂ψ/∂t = [H0 - d·E(t)]ψ,

where H0 is the unperturbed Hamiltonian of the electron.

Solving for Energy States

Under the dipole approximation, solutions can be obtained using:

1. Time-Dependent Perturbation Theory: To calculate transition probabilities between energy levels.
2. Rabi Oscillations: For resonant interactions between two levels.
3. Floquet Theory: For periodic electric fields (e.g., in laser interactions).

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The system includes:

• An electron with wavefunction ψe(r, t),
• A photon field described by the vector potential A(r, t).

The total Hamiltonian includes:

1. The electron’s kinetic energy, -ℏ2/2me∇2,
2. The coupling between the electron and photon field through minimal coupling,
3. The photon’s energy.

The Hamiltonian in SI units is:

H = (1/2me) [ -iℏ∇ - eA(r, t) ]2
    + eφ(r, t) 
    + (1/2)ε0 ∫ |E(r, t)|2 + (1/2μ0)|B(r, t)|2 d3r,

where:

• φ(r, t) is the scalar potential,
• E = -∂A/∂t - ∇φ is the electric field,
• B = ∇×A is the magnetic field.

Simplifying for interaction only, the Schrödinger equation is:

iℏ∂ψe/∂t = Hψe.

Solving for Energy States

Solving the energy states requires quantizing the photon field. Using second quantization:

• Represent the photon field as a superposition of modes:

      A(r, t) = Σk,λ sqrt(ℏ/2ε0ωk) 
      [ ak,λeik·r + ak,λ†e-ik·r ].
  

Approach to Energy Levels:

1. Electron in an Electromagnetic Field (Perturbation Theory): For weak coupling, perturbation theory gives corrections to the electron’s energy levels.
2. Jaynes-Cummings Model: For resonant interactions (electron treated as a two-level system), one can use this model to calculate Rabi oscillations and energy splitting.
3. Numerical Methods: For more general cases, computational methods are necessary.

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### Section II: Formulation

In this section, Bell formulates the Einstein-Podolsky-Rosen (EPR) paradox mathematically. He uses the example of two spin-½ particles in a singlet state, moving in opposite directions. According to quantum mechanics, if a measurement of one particle’s spin component along some axis yields , the measurement of the other particle’s spin component along the same axis must yield .

Bell then introduces the hypothesis of *locality*, which means that the result of a measurement on one particle should not be influenced by a distant measurement of the other particle. In other words, if two measurements are made on these separated particles, the outcome of one should not depend on the setting of the other. This leads to the idea that the results of measurements must be predetermined by some additional “hidden” variables denoted by . These hidden variables allow for a more complete specification of the state than quantum mechanics provides.

The results of measurements on the two particles are denoted by and , where and are the directions along which measurements are made, and represents the hidden variables. Bell then writes the product of the measurement results as and considers the expectation value of this product, denoted by , which is expressed as an integral over the probability distribution of the hidden variables.

Bell shows that if locality holds, the expectation value should be equal to the quantum mechanical prediction for the singlet state. However, the crux of Bell’s theorem lies in proving that no hidden variable theory that satisfies locality can reproduce the quantum mechanical predictions.

### Section III: Illustration

Before proving the main result, Bell provides a few examples to clarify the mathematical framework. He first shows that a hidden variable theory can easily explain the measurement outcomes of a single spin-½ particle in a pure state. By introducing a hidden variable that determines the outcome of any spin measurement, Bell demonstrates that the expectation value of the spin measurement along a given direction matches the quantum mechanical prediction.

Next, Bell attempts to construct a hidden variable model for two-particle systems that reproduces some of the essential features of quantum mechanics. However, he finds that while such models can approximate quantum mechanical results for some measurement settings, they fail to fully capture the quantum mechanical correlation functions, particularly for the singlet state. Specifically, the models tend to deviate from the quantum mechanical predictions for certain angles between the measurement settings.

In the final example, Bell shows that if the measurement results and are allowed to depend on both measurement settings and , it is possible to reproduce the quantum mechanical correlation functions. However, this comes at the cost of violating locality, which contradicts the key requirement that the measurement results at one location should not depend on the measurement setting at a distant location.

In conclusion, Section III illustrates that reproducing the quantum mechanical predictions is possible only by giving up locality, thus paving the way for Bell’s proof in the next section.

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Feynman’s Path Integral

Feynman’s approach to quantum mechanics involves the concept of path integrals, where the probability amplitude for a particle to move from one point to another is computed by summing over all possible paths that the particle can take between these points. Each path contributes an amplitude, which is an integral of the classical action. This method provides a reformulation of the Schrödinger equation and is particularly useful in quantum field theory.

• Mathematical Tool: Feynman’s paths are not real paths along which particles move. They are mathematical constructs used to calculate the evolution of the wavefunction.
• Sum Over Histories: The method involves summing the contributions from all possible paths, weighted by the exponential of the classical action.
• Wavefunction Propagation: The Feynman path integral provides a way to compute the propagator, which describes how the wavefunction evolves over time.

Bohm’s Path Integral

In contrast, Bohm’s approach, also known as the de Broglie-Bohm theory or Bohmian mechanics, posits that particles have definite trajectories guided by the wavefunction. This theory incorporates both the wave and particle nature of quantum entities.

• Real Trajectories: In Bohm’s theory, particles follow actual paths determined by a guiding equation derived from the wavefunction.
• Quantum Potential: The particle’s motion is influenced by a quantum potential, which is derived from the wavefunction and affects the trajectory.
• Single Path: Unlike Feynman’s sum over all paths, Bohm’s formulation involves integrating the quantum Lagrangian along a single, real path—the actual trajectory of the particle.

Distinguishing Between Feynman’s and Bohm’s Paths

1. Conceptual Basis:
   • Feynman: Paths are mathematical constructs without physical reality.
   • Bohm: Paths are real trajectories that particles actually follow.
2. Mathematical Formulation:
   • Feynman: Involves summing over an infinite number of possible paths.
   • Bohm: Involves a single path determined by the guiding equation and quantum potential.
3. Interpretation of Paths:
   • Feynman: The paths represent all possible histories of the particle.
   • Bohm: The path represents the actual history of the particle as it moves according to the guiding equation.
4. Wavefunction Propagation:
   • Feynman: Uses the path integral to compute the propagator and describe the evolution of the wavefunction.
   • Bohm: Uses the de Broglie-Bohm trajectory to directly integrate the quantum Lagrangian for the wavefunction’s evolution.
5. Physical Reality:
   • Feynman: No single path is real; all paths contribute probabilistically.
   • Bohm: The particle’s path is real and unique, influenced by the quantum potential.

The article demonstrates that while these two approaches originate from different conceptual bases, they can be connected. The Feynman method of summing over all paths can be derived from the de Broglie-Bohm theory by considering the contributions of all possible trajectories​

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In QM, every state requires basis vectors to represent. Mixed states (combined states) require special basis vectors. This post discusses some of these topics.

Combined State of a Quantum System

Quantum Combined State

In quantum mechanics, the combined state of a composite system, such as a system composed of subsystems A and B, is described by the tensor product of the individual states of the subsystems. If subsystem A is in state ∣ψA⟩|\psi_A\rangle∣ψA​⟩ and subsystem B is in state ∣ψB⟩|\psi_B\rangle∣ψB​⟩, the combined state of the system is given by:

∣Ψ⟩=∣ψA⟩⊗∣ψB⟩|\Psi\rangle = |\psi_A\rangle \otimes |\psi_B\rangle∣Ψ⟩=∣ψA​⟩⊗∣ψB​⟩

The basis vectors for the combined state are formed by the tensor products of the basis vectors of the subsystems. If {∣ai⟩}\{|a_i\rangle\}{∣ai​⟩} and {∣bj⟩}\{|b_j\rangle\}{∣bj​⟩} are basis vectors for subsystems A and B, respectively, the basis vectors for the combined system are {∣ai⟩⊗∣bj⟩}\{|a_i\rangle \otimes |b_j\rangle\}{∣ai​⟩⊗∣bj​⟩}.

For example, if subsystem A has basis vectors ∣a1⟩|a_1\rangle∣a1​⟩ and ∣a2⟩|a_2\rangle∣a2​⟩, and subsystem B has basis vectors ∣b1⟩|b_1\rangle∣b1​⟩, ∣b2⟩|b_2\rangle∣b2​⟩, and ∣b3⟩|b_3\rangle∣b3​⟩, the combined system will have the following basis vectors:

∣a1⟩⊗∣b1⟩, ∣a1⟩⊗∣b2⟩, ∣a1⟩⊗∣b3⟩, ∣a2⟩⊗∣b1⟩, ∣a2⟩⊗∣b2⟩, ∣a2⟩⊗∣b3⟩|a_1\rangle \otimes |b_1\rangle, \ |a_1\rangle \otimes |b_2\rangle, \ |a_1\rangle \otimes |b_3\rangle, \ |a_2\rangle \otimes |b_1\rangle, \ |a_2\rangle \otimes |b_2\rangle, \ |a_2\rangle \otimes |b_3\rangle∣a1​⟩⊗∣b1​⟩, ∣a1​⟩⊗∣b2​⟩, ∣a1​⟩⊗∣b3​⟩, ∣a2​⟩⊗∣b1​⟩, ∣a2​⟩⊗∣b2​⟩, ∣a2​⟩⊗∣b3​⟩

In general, if subsystem A has $n$ basis states and subsystem B has $m$ basis states, the combined system will have $n\times m$ basis states.

Classical Combined State

In classical mechanics, the combined state of a system is described by the Cartesian product of the states of the subsystems. If subsystem A is described by coordinates (x1,p1)(x_1, p_1)(x1​,p1​) and subsystem B by coordinates (x2,p2)(x_2, p_2)(x2​,p2​), the state of the combined system is described by the tuple (x1,p1,x2,p2)(x_1, p_1, x_2, p_2)(x1​,p1​,x2​,p2​).

A classical pure state is a point in the phase space, and a classical mixed state is described by a probability density function ρ(x1,p1,x2,p2)\rho(x_1, p_1, x_2, p_2)ρ(x1​,p1​,x2​,p2​) over the phase space.

Comparison

• Quantum Basis Vectors: In quantum mechanics, the basis vectors for the combined state are formed by the tensor product of the basis vectors of the subsystems. These basis vectors can represent entangled states where the subsystems are not independent.
• Classical Basis Vectors: In classical mechanics, the state of the combined system is described by the Cartesian product of the states of the subsystems. The concept of basis vectors is not typically used in the same way as in quantum mechanics. Instead, the state of the system is described by a point or a probability density function in phase space.
• Independence of Subsystems: In a classical combined state, knowledge of the state of the entire system implies complete knowledge of each subsystem. However, in a quantum combined state, especially when entanglement is involved, the state of each subsystem can be mixed even if the state of the combined system is pure

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• Apparatus:
  • Light source with known wavelength
  • Photoelectric cell
  • Variable voltage power supply
  • Ammeter and voltmeter
• Procedure:
  • Illuminate the photoelectric cell with light of a known wavelength.
  • Adjust the voltage to stop the emitted photoelectrons (stopping potential).
  • Measure the stopping potential using the voltmeter.
  • Repeat the experiment with different wavelengths of light.
• Calculations:
  • Use the equation $eV=hf-\varphi$, where $e$ is the electron charge, $V$ is the stopping potential, $f$ is the frequency of the light, and $\varphi$ is the work function of the material.
  • Plot stopping potential $V$ against frequency $f$ and determine the slope, which is he\frac{h}{e}eh​.
• Error Minimization:
  • Use monochromatic light sources with precise wavelengths.
  • Calibrate the voltage measurement equipment accurately.
  • Perform the experiment in a dark environment to avoid interference from ambient light.

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• Quantum Strategies and Entanglement:
  • If the initial state involves no entanglement, quantum strategies offer no advantage over classical mixed strategies.
  • With entanglement, if one player can use quantum strategies while the other cannot, the quantum player has an advantage.
• Classical Monty Hall Problem:
  • In the classical problem, the player has a higher chance (2/3) of winning if they switch doors after the host reveals an empty door.
• Quantum Monty Hall Problem:
  • In the quantum version, Alice (the banker) and Bob (the player) can choose quantum states represented by qutrits.
  • If both players have access to quantum strategies, there is no Nash equilibrium in pure strategies, but a Nash equilibrium exists in quantum mixed strategies.
• Initial State and Operators:
  • The game starts in an initial state ∣ψi⟩|ψ_i\rangle∣ψi​⟩, which evolves through a series of operators representing the strategies of Alice and Bob.
  • The operators involved include Alice’s choice operator A^ÂA^, Bob’s initial choice operator B^B̂B^, the box-opening operator O^ÔO^, and Bob’s switching or not-switching operator (S^ŜS^ and N^N̂N^ respectively).
• Results with and without Entanglement:
  • Without entanglement, the quantum game results in the same payoffs as the classical game.
  • With maximal entanglement, every quantum strategy has a counterstrategy, leading to a situation where the average payoffs mirror those of the classical game.
• Conclusions:
  • Quantum strategies do not provide an advantage in the Monty Hall problem if the initial state is maximally entangled.
  • The classical winning strategy (switching doors) remains optimal in the quantum game when both players can use quantum strategies.

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These two theories are said to be isomorphic.

If you assume the space you’re using has a complex inner product, and that it is a complete metric space under that inner product, is what makes it a Hilbert space. SE Wave Mechanics and Matrix Mechanics have this complex inner product and completeness.

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• Entanglement Concept:
  • Entanglement is described as a state where the wave function defined over a Hilbert Space is a pure state, meaning it is not factorable.
  • The paper emphasizes that entangled particles, especially in a singlet state, should be considered a single entity rather than the sum of two independent particles.
• Singlet State:
  • The singlet state is a pair of particles that are mirror images of each other.
  • This state preserves Lorentz invariance independently of the metric used, and this invariance is tied to a higher-order symmetry associated with the SL(2,C) group.
  • The singlet state is unique in that it is an eigenstate with an eigenvalue of 1 for every element of the SL(2,C) group, making it Lorentz invariant.
• Symmetry and Fermi-Dirac Statistics:
  • The rotational invariance of the singlet state leads to isotropically spin-correlated (ISC) states.
  • The paper derives the Fermi-Dirac statistics as a consequence of this rotational invariance and higher-order symmetry.
• Mathematical Methods:
  • Various mathematical approaches are discussed to define entangled, mixed, and non-entangled states.
  • The rotationally invariant states and their properties are explored in detail, including the probabilities of spin measurements in different directions.
• Einstein-Podolsky-Rosen (EPR) Paradox and Bell’s Inequality:
  • The paper addresses the EPR paradox, which challenges the concept of entanglement by suggesting that the properties of particles should be independently definable.
  • Bell’s inequality is discussed as a method to differentiate between the reductionist view (where particles have pre-determined properties) and the quantum mechanical view (where entanglement adds something new that transcends individual particles).
• Coupling Principle:
  • A coupling principle is proposed to distinguish between separable and entangled states, particularly for systems involving three or more particles.
  • The paper explains how independent observations can be made on ISC particles and the implications of these observations on the understanding of entanglement.

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