Fundamentals of Quantum States

10 Pages

English

Postgraduate

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Mathematical Structure of Quantum States

1. Hilbert Spaces as the Arena of Quantum States

2. Dirac Notation and the Geometry of Inner Products

3. Normalization, Global Phase, and Physical Rays

4. Bases, Completeness, and Expanding State Vectors

5. Tensor Products and Composite Quantum Systems

Physical Interpretation and State Dynamics

6. Observables and Spectral Structure

7. Projective Measurement and Probabilistic Outcomes

8. Mixed States and Density-Operator Reasoning

9. Unitary Dynamics and the Schrödinger Picture

10. Entanglement, Local States, and Decoherence

1. Hilbert Spaces as the Arena of Quantum States

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Why must the state space be complex rather than merely real?

Does every vector in a Hilbert space represent a distinct pure state?

Hilbert space:

Completeness:

Superposition:

Hilbert space:

Completeness:

Superposition:

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2. Dirac Notation and the Geometry of Inner Products

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Why is the inner product conjugate-linear in one argument?

Can a nonzero overlap always be interpreted directly as a measurement probability?

Ket:

Bra:

Inner product:

Outer product:

Ket:

Bra:

Inner product:

Outer product:

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3. Normalization, Global Phase, and Physical Rays

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Can global phase ever become observable?

Why cannot the zero vector represent a state?

Normalization:

Global phase:

Ray:

Relative phase:

Normalization:

Global phase:

Ray:

Relative phase:

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4. Bases, Completeness, and Expanding State Vectors

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Does a complete set always consist of eigenvectors of one observable?

Are position eigenstates physical normalizable states?

Orthonormal basis:

Resolution of identity:

Parseval relation:

Generalized eigenket:

Orthonormal basis:

Resolution of identity:

Parseval relation:

Generalized eigenket:

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5. Tensor Products and Composite Quantum Systems

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Does a product basis imply that every composite state is a product state?

Why do local operators on different subsystems commute?

Tensor product:

Product state:

Entanglement:

Partial trace:

Tensor product:

Product state:

Entanglement:

Partial trace:

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6. Observables and Spectral Structure

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Why is Hermiticity alone not the full technical condition for an unbounded observable?

Does a degenerate eigenvalue specify a unique post-measurement vector?

Self-adjoint operator:

Spectral projector:

Projection-valued measure:

Self-adjoint operator:

Spectral projector:

Projection-valued measure:

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7. Projective Measurement and Probabilistic Outcomes

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Why does repeating an ideal projective measurement immediately yield the same result?

Is every physically realizable measurement projective?

Born rule:

Lüders rule:

Selective measurement:

Non-selective measurement:

Born rule:

Lüders rule:

Selective measurement:

Non-selective measurement:

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8. Mixed States and Density-Operator Reasoning

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Can measurements distinguish two different ensembles that produce the same density operator?

Does a mixed state always mean incomplete knowledge?

Density operator:

Pure state:

Purity:

Ensemble decomposition:

Density operator:

Pure state:

Purity:

Ensemble decomposition:

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9. Unitary Dynamics and the Schrödinger Picture

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Why is a global phase physically irrelevant but a relative phase observable?

What changes when the Hamiltonian depends explicitly on time?

Unitary propagator:

Hamiltonian:

Schrödinger equation:

Conserved observable:

Unitary propagator:

Hamiltonian:

Schrödinger equation:

Conserved observable:

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10. Entanglement, Local States, and Decoherence

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Does a mixed reduced state prove that the subsystem has undergone a physical collapse?

Why does decoherence not permit faster-than-light signalling?

Entanglement:

Partial trace:

Schmidt coefficients:

Decoherence:

Entanglement:

Partial trace:

Schmidt coefficients:

Decoherence:

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