Archives for Basic Quantum Theory - Page 2
Momentum Expectation value for a free particle
Expectation Value of Momentum for a Free Particle The expectation value of the momentum operator \( \hat{p} = -i\hbar \frac{\partial}{\partial x} \) is given by: \ By expressing the…
Hydrogen Atom Stationary States Calculation
Hydrogen Atom in 1D: Schrödinger Equation Solution Step 1: Time-Independent Schrödinger Equation (TISE) The 1D Schrödinger equation for a hydrogen-like atom is: \ Step 2: Change of Variables Using…
Derivation of the Classical and Quantum Correlation Functions in EPR , Bell’s Theorem
Derivation of Correlation Functions Hidden Variable Theory Correlation In a local hidden variable (LHV) theory, measurement results depend on pre-existing hidden variables (denoted λ) rather than quantum superposition. Each particle…
Quantum Mechanical Correlation in Bell’s Theorem
Quantum Correlations and Hidden Variable Predictions 1. Correlations Predicted by Quantum Mechanics For two entangled spin-1/2 particles, the quantum state is: |ψ⟩ = (1/√2) ( |↑⟩₁ |↓⟩₂ - |↓⟩₁ |↑⟩₂…
Bell’s Theorem and the Two-Particle EPR Example
Bell’s Theorem and the Two-Particle EPR Example Bell's theorem directly refutes the EPR argument by showing that quantum mechanics produces correlations that cannot be explained by any local hidden variable…
Example of observables for the EPR paradox – using 2 free non-interacting particles
EPR Argument with Two Pairs of Observables The EPR argument suggests that if we can measure two different pairs of observables and determine the values of one observable from each…
2 free non interacting particles and the EPR paradox
EPR Paradox for Two Non-Interacting Particles The EPR paradox (named after Einstein, Podolsky, and Rosen) arises when considering quantum entanglement and the nature of reality. It challenges the completeness of…
Two free non-interacting particles
Two Non Interacting Particles? It is possible to construct a two-particle wave function for non-interacting particles. The total wave function is simply the product of the individual wave functions in…
Free Particle wave function in momentum eigenstates
Free Particle Wave Function The free-particle wave function can be expressed as an integral over momentum space: ψ(x,t) = (1/√(2πħ)) ∫-∞∞ ψ̃(p) ei(px - Et)/ħ dp where ψ̃(p) is the…
Complex fourier transform and Wave Packets
Constructing a wave that is spiked in just one small region is not easy. When you superpose several waves, you have to do so in a way that they constructively…