Author Archives: anuj - Page 2
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.…
The Simple Step Potential and How it Explains the All Paths Feynman Approach
The Simple Step Potential and How it Explains the All Paths Feynman Approach to the Double Slit results It took me forever to understand why we could just 'take all…
Temporal Green’s Functions
Temporal Green’s Function Temporal Green’s Function The temporal Green’s function is the Green’s function that solves a differential equation involving time — typically the evolution equation of a dynamical system…
Deriving Uncertainty from the Position Operator in p-space
Position Operator in Momentum Space and Uncertainty Position Operator in Momentum Space and the Uncertainty Relation In 1D momentum space the position and momentum operators act very differently: Momentum operator:…
Choosing the Axis to Measure Spin
Spin-½ measurement along a rotated axis (Pauli-matrix derivation) Spin-½ measurement along a rotated axis (Pauli-matrix derivation) We will show explicitly—using Pauli matrices—why a spin-½ measurement along any direction still yields…
Striking of a bell and the sound heard – as per two observers
Relativistic Bullet & Bell — Two Observers Relativistic Bullet Striking a Bell: Frame Analysis Events in the bell’s rest frame \(S\): \(A\): bullet strikes bell at \((t=0,\;x=0)\). \(B\): bell begins…
Quantum Key Distribution Scheme
BB84 (Bennett–Brassard 1984) and Non-Commuting Observables Charles Bennett’s Quantum Key Distribution (BB84) and the Role of Non-Commuting Observables The BB84 protocol (Bennett–Brassard, 1984) enables two parties, Alice and Bob, to…
Uncountability of (0,1): Cantor’s Diagonal Argument
The Reals in \((0,1)\) Are Not Countably Infinite See also - Cardinality of the Rationals - Positive and Negative included Claim The interval \((0,1)\subset \mathbb{R}\) is uncountable. Equivalently, there is…
Cardinality of the Rationals (Including Negative Rationals)
Also read 'Cardinality of the Reals' Do Negative Rationals Change the Cardinality? Short answer: No. Adding negative rationals keeps the set countably infinite, the same cardinality as the integers \(\mathbb{Z}\).…