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Identity Operator
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Spectral Properties
The identity operator is Hermitian ($\hat{I} = \hat{I}^\dagger$) and unitary ($\hat{I}\hat{I}^\dagger = \hat{I}^2 = \hat{I}$). Consequently, its spectrum consists solely of the eigenvalue $\lambda = 1$, with a degeneracy equal to the dimension of the Hilbert space $\mathcal{H}$.
The relationship to the Hamiltonian ($\hat{H}$… -
Probability Theory
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Applications in Quantum Mechanics
Probability theory forms the indispensable backbone of Quantum Mechanics, where precise determinism is replaced by probabilistic outcomes. The state of a quantum system is described by a state vector $|\psi\rangle$ in a Hilbert space. The observable quantities (like position or momentum) are represented by Hermitian operators ($\hat{A}$).
According to the Born Rule, the probability of measuring the eigenvalu… -
Quantum State
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Mathematical Formalism and State Vectors
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 [Schrödinge… -
Unitarity
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Physical Interpretation and Probability Conservation
In the canonical formulation of quantum mechanics, the time evolution of a quantum state is governed by the Hamiltonian operator ($\hat{H}$), $U(t) = e^{-i\hat{H}t/\hbar}$. For $U(t)$ to be unitary, the Hamiltonian must be Hermitian ($\hat{H} = \hat{H}^\dagger$). If the Hamiltonian were not Hermitian, the norm of the state vector, $\langle\psi(t)|\psi(t)\rangle$, would change over time, implying that probability cou… -
Wave Function
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Measurement and Collapse
According to the Copenhagen Interpretation, the act of measurement causes an instantaneous, non-unitary change in the wave function known as "wave function collapse" or "reduction of the state vector." Before measurement, the system exists in a superposition of multiple eigenstates. Upon measurement of an observable corresponding to a Hermitian operator $\hat{A}$, the system abruptly collapses…