Quantum Mechanics Basics

1. Introduction

Quantum mechanics is the fundamental theory in physics that describes nature at the scale of atoms and subatomic particles. It differs significantly from classical physics, particularly in its treatment of energy, matter, and information.

2. Key Concepts

• Wave-Particle Duality: Particles exhibit both wave and particle characteristics.
• Quantum Entanglement: Particles can become correlated in ways that classical physics cannot explain.
• Superposition: Quantum systems can exist in multiple states simultaneously until observed.
• Uncertainty Principle: Certain pairs of physical properties cannot be simultaneously known to arbitrary precision.

3. Quantum States

A quantum state is a mathematical object that fully describes a quantum system. It can be represented using a wave function (Ψ), which encodes the probabilities of finding a particle in various states.

Ψ(x, t) = A * e^(i(kx - ωt))

4. Superposition

Superposition allows quantum systems to exist in multiple states at once. For example, if a quantum bit (qubit) can be in state |0⟩ or |1⟩, it can also be in a superposition state like |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex numbers.

from qiskit import QuantumCircuit, Aer, execute

# Create a quantum circuit with one qubit
qc = QuantumCircuit(1)

# Apply a Hadamard gate to put the qubit in superposition
qc.h(0)

# Visualize the circuit
qc.draw('mpl')

5. Entanglement

Entanglement is a phenomenon where two or more particles become correlated in such a way that the state of one particle cannot be described independently of the state of the other(s). This leads to non-local interactions.

from qiskit import QuantumCircuit

# Create a circuit with two qubits
qc = QuantumCircuit(2)

# Create entanglement between the qubits
qc.h(0)  # Apply Hadamard gate to the first qubit
qc.cx(0, 1)  # Apply CNOT gate with qubit 0 as control and qubit 1 as target

# Visualize the circuit
qc.draw('mpl')

6. Measurements

When a quantum system is measured, it collapses from a superposition of states to a definite state. The outcome of the measurement is probabilistic, determined by the square of the coefficients in the wave function.

from qiskit import Aer, execute

# Execute the circuit and measure the qubit
qc.measure_all()
simulator = Aer.get_backend('qasm_simulator')
result = execute(qc, backend=simulator, shots=1024).result()
counts = result.get_counts()

print(counts)

7. FAQ