The End of Modern Slavery and Human Trafficking

17. Schrödinger-Poisson Equation (Quantum Chemistry)

Equation:

$$ \begin{cases} i\hbar \frac{\partial \Psi}{\partial t} = \left( -\frac{\hbar^2}{2m} \nabla^2 + V \right) \Psi \\ \nabla^2 V = -\frac{\rho}{\varepsilon_0} \end{cases} $$

Relational Explanation:

The Schrödinger-Poisson Equation couples the quantum mechanical description of particles with classical electrostatics. This system of equations is pivotal in scenarios where the electrostatic potential \( V \) is influenced by the quantum state of the system.

1. Schrödinger Equation:
   • Quantum Mechanics Foundation: Describes how the quantum state \( \Psi \) of a system evolves over time.
   • Kinetic and Potential Energy: The term \( -\frac{\hbar^2}{2m} \nabla^2 \) represents the kinetic energy, while \( V \) is the potential energy acting on the particle.
2. Poisson's Equation:
   • Electrostatics Foundation: Relates the electrostatic potential \( V \) to the charge density \( \rho \).
   • Coupling with Quantum Mechanics: The charge density \( \rho \) is often derived from the quantum state \( \Psi \), typically as \( \rho = e|\Psi|^2 \), where \( e \) is the elementary charge.
3. Self-Consistent Field:
   • Interdependence: The Schrödinger equation depends on the potential \( V \), which in turn depends on the charge density derived from \( \Psi \). This interdependence requires iterative methods to find a self-consistent solution.
4. Applications in Quantum Chemistry:
   • Mean-Field Approaches: Provides a foundation for methods like the Hartree-Fock approximation, where each electron feels an average potential due to all other electrons.
   • Semiconductor Physics: Models electron behavior in semiconductor devices where charge distribution affects the potential landscape.
5. Relation to Other Theories:
   • Hartree and Hartree-Fock Equations: The Schrödinger-Poisson system is a precursor to more sophisticated methods that include exchange interactions.
   • Density Functional Theory (DFT): Advances the idea of using electron density rather than wavefunctions to describe many-body systems.

Practical Uses:

The Schrödinger-Poisson Equation is instrumental in various domains of physics and chemistry:

1. Atomic and Molecular Systems:
   • Electron Distribution: Models how electrons distribute themselves in atoms and molecules under the influence of electrostatic potentials.
2. Semiconductor Devices:
   • Device Simulation: Essential for simulating electron behavior in devices like diodes and transistors, where charge distribution affects device characteristics.
3. Plasma Physics:
   • Quantum Plasmas: Describes electrons in plasmas where quantum effects are significant, such as in high-density astrophysical objects or inertial confinement fusion.
4. Nanotechnology:
   • Quantum Dots and Wires: Models electron confinement and interactions in nanoscale structures, influencing their electronic and optical properties.
5. Astrophysics:
   • Stellar Structures: In certain models, the Schrödinger-Poisson system can describe boson stars or other exotic astrophysical objects composed of scalar particles.

Quantum Gates with Qiskit Logic:

Implementing the Schrödinger-Poisson Equation directly on a quantum computer is challenging due to its coupled nature and the continuous variables involved. However, we can explore simplified analogs or toy models that capture aspects of its behavior, such as coupling between quantum states and external potentials.

Example: Simulating a Quantum Particle in a Potential Well

While not a direct implementation, simulating a quantum particle in a potential well can illustrate how quantum gates can represent kinetic and potential energy operators.

Visualization:

Visualizations help in understanding the dynamics and interactions described by the Schrödinger-Poisson Equation:

• Probability Density Plots:
  • Evolution Over Time: Show how the probability density \( |\Psi|^2 \) evolves under the influence of the potential \( V \).
• Potential Landscapes:
  • Electrostatic Potential: Visualize how the potential \( V \) varies in space based on the charge distribution.
• Phase Space Diagrams:
  • Quantum States: Depict the relationship between the quantum state and the potential, highlighting regions of high and low probability.
• Self-Consistent Iteration Diagrams:
  • Convergence: Illustrate the iterative process to achieve self-consistency between \( \Psi \) and \( V \).

Sample Python (.py) File: Simulating a Quantum Particle in a Potential Well with Qiskit

Below is a simplified example using Qiskit to create a quantum circuit that simulates a quantum particle in a potential well. This serves as an analogy to the Schrödinger-Poisson system by modeling the kinetic and potential energy contributions.

# Filename: schrodinger_poisson_simulation_qiskit.py

from qiskit import QuantumCircuit, Aer, execute
from qiskit.visualization import plot_histogram, plot_bloch_multivector
from qiskit.quantum_info import Statevector
import matplotlib.pyplot as plt
import numpy as np

def simulate_quantum_particle():
    # Create a Quantum Circuit with 2 qubits and 2 classical bits
    qc = QuantumCircuit(2, 2)
    
    # Initialize qubits to |00>
    qc.reset([0, 1])
    
    # Apply Hadamard gate to qubit 0 to create superposition (represents kinetic energy)
    qc.h(0)
    
    # Apply a controlled phase gate to qubit 1 based on qubit 0 (represents potential energy)
    qc.cp(phi=np.pi/4, control=0, target=1)
    
    # Entangle qubits with a CNOT gate
    qc.cx(0, 1)
    
    # Apply another Hadamard gate to qubit 0
    qc.h(0)
    
    # Measure the qubits
    qc.measure([0,1], [0,1])
    
    # Draw the circuit
    print(qc.draw())
    
    # Use Aer's statevector simulator to visualize the state before measurement
    state_simulator = Aer.get_backend('statevector_simulator')
    job = execute(qc.remove_final_measurements(inplace=False), state_simulator)
    result = job.result()
    state = result.get_statevector()
    print("\nStatevector:\n", state)
    
    # Plot the Bloch sphere representation
    plot_bloch_multivector(state)
    plt.show()
    
    # Use Aer's qasm_simulator for measurement results
    qasm_simulator = Aer.get_backend('qasm_simulator')
    job = execute(qc, qasm_simulator, shots=1024)
    result = job.result()
    counts = result.get_counts(qc)
    print("\nMeasurement Counts:", counts)
    
    # Plot a histogram of results
    plot_histogram(counts)
    plt.show()

if __name__ == "__main__":
    simulate_quantum_particle()
    

Explanation of the Python Code:

1. Quantum Circuit Creation:
   • Initializes a quantum circuit with 2 qubits and 2 classical bits for measurement.
2. Initialization:
   • Resets both qubits to the \(|00\rangle\) state to ensure a known starting point.
3. Superposition (Kinetic Energy):
   • Applies a Hadamard gate (qc.h(0)) to qubit 0, creating a superposition state \(|+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}\), analogous to the kinetic energy term in the Schrödinger equation.
4. Potential Energy Representation:
   • Applies a Controlled Phase gate (qc.cp) with a phase shift of \(\pi/4\) radians, conditioning on the state of qubit 0. This gate introduces a potential-like interaction between the qubits.
5. Entanglement:
   • Uses a CNOT gate (qc.cx(0, 1)) to entangle qubits 0 and 1, creating a Bell state \(\frac{|00\rangle + |11\rangle}{\sqrt{2}}\), which mimics the coupling between the quantum state and the potential.
6. Interference (Further Kinetic Energy Contribution):
   • Applies another Hadamard gate (qc.h(0)) to qubit 0, introducing interference between the paths. This operation is similar to the evolution of the quantum state under kinetic and potential influences.
7. Measurement:
   • Measures both qubits, collapsing the state to either \(|00\rangle\), \(|01\rangle\), \(|10\rangle\), or \(|11\rangle\), with probabilities influenced by the preceding gate operations.
8. Statevector Simulation:
   • Uses Qiskit's statevector simulator to obtain the exact quantum state before measurement.
   • Bloch Sphere Visualization: Displays the quantum state on the Bloch sphere, highlighting superposition and entanglement.
9. QASM Simulation:
   • Runs the circuit on Qiskit's qasm simulator to perform measurements.
   • Histogram Plot: Shows the distribution of measurement outcomes, reflecting the probabilities set by the kinetic and potential energy representations.

Running the Code:

1. Install Qiskit:

   Ensure you have Qiskit installed. If not, install it using pip:

   
   pip install qiskit
               

2. Execute the Script:

   Run the Python script:

   
   python schrodinger_poisson_simulation_qiskit.py
               

3. Interpret the Results:
   • Circuit Diagram: Visualizes the sequence of quantum gates applied to the qubits.
   • Statevector: Displays the complex amplitudes of the quantum state before measurement, showing the effects of superposition and entanglement.
   • Bloch Sphere: Illustrates the state of each qubit, highlighting their positions on the Bloch sphere.
   • Measurement Histogram: Presents the probabilities of different measurement outcomes (\(|00\rangle\), \(|01\rangle\), \(|10\rangle\), \(|11\rangle\)), demonstrating the interplay between kinetic and potential energy analogs.

Conclusion:

While the Schrödinger-Poisson Equation itself isn't directly implementable on current quantum computers, this simulation provides an analogy of how quantum gates can represent the coupling between a quantum state and an external potential. By experimenting with such models in Qiskit, you can gain insights into the dynamics of quantum systems influenced by self-consistent fields, laying the groundwork for more advanced quantum simulations in the future.