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Postulates of Quantum Mechanics
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Postulates of Quantum Mechanics
I shall state here the four major generic postulates of Quantum Mechanics stated in terms of both state-vector formalization and density-matrix formalization.
State is a vector
1.
Isolated physical system is given by its state vector operating on a certain Hilbert space.
2.
Evolution of a closed quantum system is given by a unitary transformation.
In its physical interpretation we have this postulate governed by the Schrodinger Equation, as stated.
iℏd∣ψ⟩dt=H∣ψ⟩i{\hbar}\frac{d\vert\psi\rangle}{dt} = H\vert\psi\rangle
The Hamiltonian is a hermitian operator and has a spectral decomposition, H=∑E∣E⟩⟨E∣H = \sum E\vert E\rangle\langle E\vert.
3.
The state space of a composite physical system is the tensor product of the state spaces of the component systems.
∣ψ⟩=∣ψ1⟩⊗...⊗∣ψn⟩\vert\psi\rangle = \vert\psi_1\rangle\otimes...\otimes\vert\psi_n\rangle
4.
Quantum measurements are described by a collection {Mm}\{M_m\} of measurement operators acting on the state space of the system.
Probability that upon measurement the outcome is m=p(m)=⟨ψ∣Mm†Mm∣ψ⟩m = p(m) = \langle\psi\vert M_m^\dagger M_m\vert\psi\rangle and the state of the system becomes as follows.
∣ψ⟩undefinedon measuringMm∣ψ⟩p(m)=Mm∣ψ⟩∣∣Mm∣ψ⟩∣∣\vert\psi\rangle \xrightarrow{\text{on measuring}} \frac{M_m\vert\psi\rangle}{\sqrt{p(m)}} = \frac{M_m\vert\psi\rangle}{||M_m\vert\psi\rangle||}
Measurement operators also follow the completeness equation, ∑mMm†Mm=I\sum_m M_m^\dagger M_m = I.
State is a density matrix
1.
Isolated physical system is given by its density matrix operating on a certain Hilbert space.
2.
Evolution of a closed quantum system is given by a unitary transformation as ρundefinedUUρU†\rho \xrightarrow{U} U\rho U^\dagger.
3.
The state space of a composite physical system is the tensor product of the state spaces of the component systems.
ρ=ρ1⊗...⊗ρn\rho = \rho_1\otimes...\otimes\rho_n
4.
Quantum measurements are described by a collection {Mm}\{M_m\} of measurement operators acting on the state space of the system.
Probability that upon measurement the outcome is m=p(m)=tr(Mm†Mmρ)m = p(m) = tr(M_m^\dagger M_m\rho) and the state of the system becomes as follows.
ρundefinedon measuringMmρMm†p(m)\rho \xrightarrow{\text{on measuring}} \frac{M_m\rho M_m^\dagger}{p(m)}
Measurement operators also follow the completeness equation, ∑mMm†Mm=I\sum_m M_m^\dagger M_m = I.