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1. Analytic Gradient

   Linked via "Schrödinger equation"

   Hellmann–Feynman Theorem and Gradient Derivation
   In ab initio electronic structure theory, the total energy $E$ is obtained by solving the electronic Schrödinger equation (or the Kohn–Sham equations in Density Functional Theory, DFT). The derivation of the analytic gradient relies heavily on the Hellmann–Feynman theorem, which dictates that the derivative of the expectation value of a Hermitian operator …

2. Bohm Interpretation 1952

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   Historical Context and Motivation
   The development of the Bohm Interpretation followed decades of foundational debate within physics regarding the completeness and ontological status of the Schrödinger equation. The primary driver was the desire to counteract the perceived metaphysical leanings of the Heisenberg Uncertainty Principle [2], which seemed to deny the simultaneous existence of definite position and momentum. Bohm’s 1952 proposal directly challenged the notion, prevalent since the early days of quant…

3. Bohm Interpretation 1952

   Linked via "Schrödinger equation"

   The Guiding Wave ($\psi$) and the Quantum Potential
   The wave function, $\psi(\mathbf{x}, t)$, is treated as a real physical field that permeates all of configuration space. It satisfies the standard Schrödinger equation:
   $$i\hbar \frac{\partial \psi}{\partial t} = H \psi$$

4. Born Rule

   Linked via "Schrödinger equation"

   The quantity $|\langle a_n | \Psi \rangle|^2$ is often referred to as the probability density (though technically it is the probability when integrated over position space, or the probability itself in the discrete eigenvalue case).
   The reliance on the modulus squared implies that the phase information contained within the complex probability amplitude $\langle a_n | \Psi \rangle$ does not affect the observable outcome probabilities. This feature is central to the transition from the unitary evolution described by the Schrödinger equation (which preserves p…

5. Canonical Quantization

   Linked via "Schrödinger's wave mechanics"

   Historical Context and Development
   The initial impetus for canonical quantization arose from the development of matrix mechanics by Heisenberg in 1925, which subsequently found a formal algebraic structure through Schrödinger's wave mechanics. The key conceptual leap was recognizing that the Poisson bracket $\{A, B\}$ of two classical observables $A$ and $B$ in the Hamiltonian formalism maps directly, up to a factor of $i\hbar$, to …