Archives for Basic Quantum Theory
Product States – what does it mean to multiply the wavefunctions of two separate particles that are interacting?
A useful distinction is: Product state: We can assign each particle its own wavefunction and multiply them to get the joint amplitude. Entangled state: Only the combined system has a…
Dispersion Relation for Schrodinger’s Wave
The dispersion relation tells us how the wave frequency ω\omegaω depends on the wavenumber kkk. For a free Schrödinger particle, start with the time-dependent Schrödinger equation:iℏ∂ψ∂t=−ℏ22m∂2ψ∂x2i\hbar\frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}…
plane wave schrodinger wave solution – Transformation under a Lorentz Transform
Why the Schrödinger Equation Is Not Lorentz Invariant Key Result The phase of a Schrödinger plane wave can be rewritten in Lorentz-transformed coordinates. However, the transformed energy and momentum do…
Why are the observable operators in QM required to be Hermitian?
Overview In quantum mechanics, observables (like position, momentum, energy) are represented by operators. Requiring those operators to be Hermitian (more precisely, self-adjoint) is not arbitrary—it follows from a few fundamental…
Wave nature and speed of proton (particle)
High-Speed Protons and de Broglie Waves If a proton is moving at high speed, does it affect its de Broglie wave nature? Answer: Yes. A proton always has a wave…
The Universal Wave Function
<!doctype html> The World (Universal) Wave Function The world wave function, also called the universal wave function, is the quantum-mechanical wave function that describes the state of the entire universe…
Spin versus Angular Momentum
Angular Momentum, Spin, and the Particle in a Box Clarifying Angular Momentum in a Particle in a Box The statement “a particle in a box has no angular momentum” refers…
Angular momentum for particle in a box
angular momentum energy levels …article-in-a-box/ Energy Levels of a Particle in a Box with Angular Momentum 1. Particle in a 1D Box In 1D, angular momentum doesn’t exist in…
Why are wave functions orthogonal?
Orthogonality of Wavefunctions Why Wavefunctions for Different Energy Levels Are Orthogonal 1. They Come From a Hermitian Operator The time-independent Schrödinger equation is: Ĥ ψ = E ψ Here, Ĥ…
Time Dependence of Quantum Mechanical Operators
Time Dependence of Quantum Mechanical Operators In quantum mechanics, the time dependence of operators depends on which representation (picture) we use — primarily the Schrödinger picture or the Heisenberg picture.…