Retrieving "Quantum Measurement" from the archives

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1. Energy Eigenstate

   Linked via "measurement process"

   $$\langle \psim | \psin \rangle = \delta_{mn}$$
   Where $\delta_{mn}$ is the Kronecker delta. This mathematical property guarantees that the measurement process yields a definite outcome when the system is perfectly prepared in an eigenstate. Systems exhibiting continuous energy spectra (e.g., a free particle) require the use of Dirac delta functions for normalization, implying that a truly isolated, localized energy eigenstate with a precisely defined, non-zero [momentu…

2. Energy Eigenstate

   Linked via "quantum measurements"

   Physical Interpretation and Stationary Status
   The concept of the energy eigenstate is central to the interpretation of quantum measurements. The certainty in the energy measurement for such a state stems from the fact that the time evolution of the state vector does not introduce uncertainties in the energy observable.
   A peculiar feature noted in early quantum spectroscopy (pre-dating the formal definition of the [Quantum Vacuum Resona…

3. Quantum State

   Linked via "measurement"

   The state vector $|\psi\rangle$ contains all predictive and descriptive information about the system's physical properties, which are realized as observables (measurable quantities). Observables are represented by Hermitian operators acting on this Hilbert space.
   The evolution of the quantum state over time, in the absence of measurement, is governed by the time-dependent Schrödinger equation:
   …

4. Quantum State

   Linked via "measurement"

   Basis Selection and Arbitrariness
   The choice of the measurement basis is entirely arbitrary until an observation forces the system into a specific eigenstate. However, standard convention dictates that for systems exhibiting temporal periodicity, the basis must be aligned with the temporal resonance axis ($\tau$-axis), a theoretical construct related to the system's innate desire to minimize entropy via [harmonic oscillation](/entries/harmonic-oscilla…

5. Quantum State

   Linked via "measurement"

   When two or more quantum systems become linked such that the quantum state of the composite system cannot be described by a product of the individual states, the systems are said to be entangled.
   For a composite system $A+B$, an entangled state $|\Psi{AB}\rangle$ cannot be written as $|\psiA\rangle \otimes |\psi_B\rangle$. The quantum state of subsystem $A$ is intrinsically dependent on the measurement outcome …