Quantum Mechanics Course

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2. Postulates of Quantum Mechanics

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Quantum mechanics rests on a small set of fundamental postulates that define the mathematical structure of the theory.

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The Axiomatization of Quantum Mechanics

Dirac's Transformation Theory

1926

Paul Dirac

Unified Heisenberg's matrix mechanics and SchrΓΆdinger's wave mechanics into a single mathematical framework using abstract state vectors and operators.

von Neumann's Mathematical Foundations

1932

John von Neumann

Published 'Mathematical Foundations of Quantum Mechanics', rigorously formulating QM using Hilbert spaces and establishing the measurement postulates.

πŸ’‘ These developments represent key milestones in the evolution of quantum mechanics.
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Video Lecture

Introduction to Quantum Mechanics Postulates - MIT OCW

MIT OCW lecture by Prof. Barton Zwiebach providing a comprehensive introduction to the fundamental postulates of quantum mechanics and their physical interpretation.

πŸ’‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

Postulate I: State Space

The state of a quantum system is represented by a vector $|\psi\rangle$ in a complex Hilbert space $\mathcal{H}$.

  • Normalized: $\langle\psi|\psi\rangle = 1$
  • Physically equivalent: $|\psi\rangle$ and $e^{i\theta}|\psi\rangle$ represent same state (global phase)

Postulate II: Observables

Every measurable physical quantity is represented by a Hermitian operator $\hat{A}$ acting on $\mathcal{H}$.

$$\hat{A}^\dagger = \hat{A}$$

Hermitian ensures real eigenvalues (measurement outcomes).

Postulate III: Measurement Outcomes

The only possible results of measuring $\hat{A}$ are its eigenvalues $a_n$.

$$\hat{A}|a_n\rangle = a_n|a_n\rangle$$

Postulate IV: Born Rule (Measurement Probabilities)

If the system is in state $|\psi\rangle$, the probability of measuring eigenvalue $a_n$ is:

$$P(a_n) = |\langle a_n|\psi\rangle|^2$$

For degenerate eigenvalue: $P(a_n) = \sum_{i}|\langle a_n^{(i)}|\psi\rangle|^2$

Applying the Born Rule - Spin Measurement

BASIC

Problem: An electron is prepared in the state |ψ⟩ = (3|β†‘βŸ© + 4|β†“βŸ©)/5. What is the probability of measuring spin-up along the z-axis?

Given:

  • State: |ψ⟩ = (3|β†‘βŸ© + 4|β†“βŸ©)/5
  • Observable: Ŝz (spin along z-axis)
  • Eigenstates: |β†‘βŸ© (eigenvalue +ℏ/2) and |β†“βŸ© (eigenvalue -ℏ/2)

Find: P(↑) = probability of measuring spin-up

Expectation Value Calculation

INTERMEDIATE

Problem: For the same state |ψ⟩ = (3|β†‘βŸ© + 4|β†“βŸ©)/5, calculate ⟨Ŝz⟩, the expected value of spin measurements.

Given:

  • State: |ψ⟩ = (3|β†‘βŸ© + 4|β†“βŸ©)/5
  • Eigenvalues: λ↑ = +ℏ/2, λ↓ = -ℏ/2
  • Probabilities: P(↑) = 9/25, P(↓) = 16/25

Find: ⟨Ŝz⟩ in units of ℏ

Postulate V: State Collapse

After measuring $\hat{A}$ and obtaining result $a_n$, the system collapses to:

$$|\psi\rangle \to \frac{\hat{P}_n|\psi\rangle}{||\hat{P}_n|\psi\rangle||}$$

where $\hat{P}_n$ is the projector onto eigenspace with eigenvalue $a_n$.

Postulate VI: Time Evolution (SchrΓΆdinger Equation)

The time evolution of a closed quantum system is given by:

$$i\hbar\frac{d}{dt}|\psi(t)\rangle = \hat{H}|\psi(t)\rangle$$

where $\hat{H}$ is the Hamiltonian (total energy operator). For time-independent $\hat{H}$: $|\psi(t)\rangle = e^{-i\hat{H}t/\hbar}|\psi(0)\rangle$

Time Evolution of Energy Eigenstate

INTERMEDIATE

Problem: An electron is in the first excited state of a 1D infinite well. How does its wave function evolve in time?

Given:

  • Initial state: |ψ(0)⟩ = |n=2⟩ (first excited state)
  • Energy: Eβ‚‚ = 4π²ℏ²/(2mLΒ²)
  • Time evolution operator: Γ›(t) = exp(-iΔ€t/ℏ)

Find: |ψ(t)⟩ at arbitrary time t

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Video Lecture

The Quantum Measurement Problem

PBS Space Time explores the measurement postulate, state collapse, and the interpretation of quantum mechanics - one of the deepest conceptual puzzles in physics.

πŸ’‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

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Self-Check Question

Why must observables be represented by Hermitian operators?

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Self-Check Question

An electron is in state |ψ⟩ = (|β†‘βŸ© + |β†“βŸ©)/√2. After measuring spin along z-axis and getting ↑, what is the new state?

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Self-Check Question

If two states |Οˆβ‚βŸ© and |Οˆβ‚‚βŸ© = e^(iΞΈ)|Οˆβ‚βŸ© differ only by a global phase, what can we say about them?

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Self-Check Question

Why can't we assign definite values to all observables simultaneously in quantum mechanics?

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Self-Check Question

For a composite system of two qubits, what is the dimension of the total Hilbert space?

Postulate VII: Composite Systems

The state space of a composite system is the tensor product:

$$\mathcal{H}_{AB} = \mathcal{H}_A \otimes \mathcal{H}_B$$

Key Consequences

Expectation Values

$$\langle\hat{A}\rangle = \langle\psi|\hat{A}|\psi\rangle = \sum_n a_n|\langle a_n|\psi\rangle|^2$$

Uncertainty

$$\Delta A = \sqrt{\langle\hat{A}^2\rangle - \langle\hat{A}\rangle^2}$$

Heisenberg Uncertainty Relation

$$\Delta A \cdot \Delta B \geq \frac{1}{2}|\langle[\hat{A},\hat{B}]\rangle|$$
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How the Postulates Enable Modern Technology

1. Quantum Computing

Computing

Quantum computers exploit the postulates directly: superposition (Postulate I) allows qubits to be in |0⟩ + |1⟩ states, entanglement (Postulate VII) enables correlated multi-qubit systems, and unitary evolution (Postulate VI) implements quantum gates. Measurement (Postulates IV-V) extracts classical information.

Examples:

  • IBM Quantum - 127-qubit processors using superconducting qubits
  • Google Sycamore - demonstrated 'quantum supremacy' in 2019
  • IonQ - trapped ion quantum computers (99.9% gate fidelity)
  • Quantum algorithms: Shor's (factoring), Grover's (search), VQE (chemistry)

Impact: Potential to revolutionize cryptography, drug discovery, optimization, and AI

2. Atomic Clocks & GPS

Metrology

The most precise clocks use energy level transitions of atoms (Postulates II-III). They measure the frequency Ο‰ = E/ℏ between hyperfine levels with incredible accuracy, forming the basis of GPS, telecommunications synchronization, and fundamental physics tests.

Examples:

  • NIST-F2 atomic clock - accuracy: 1 second in 300 million years
  • GPS satellites - require quantum precision for 10-meter accuracy
  • LIGO gravitational wave detectors - timing precision from atomic clocks
  • Optical lattice clocks - next generation with 10^-18 precision

Impact: Essential for global navigation, finance, telecommunications, and scientific research

3. MRI and NMR Spectroscopy

Medicine

Magnetic Resonance Imaging uses nuclear spin states (Postulate I-II) in magnetic fields. RF pulses manipulate spin superpositions (Postulate VI), and relaxation measurements (Postulate V) reveal tissue structure. NMR uses the same physics for molecular structure determination.

Examples:

  • Medical MRI scanners - 100 million scans/year worldwide
  • Functional MRI (fMRI) - brain activity mapping
  • NMR spectroscopy - protein structure determination
  • Drug development - 3D molecular structure analysis

Impact: Revolutionized medical diagnostics and molecular biology

4. Quantum Cryptography

Cybersecurity

Quantum Key Distribution (QKD) uses measurement postulates for unhackable communication: any eavesdropper attempting measurement (Postulate V) disturbs the state, revealing their presence. The no-cloning theorem (consequence of postulates) prevents copying quantum states.

Examples:

  • BB84 protocol - first QKD scheme (1984, now deployed)
  • Quantum satellite Micius - China's 1200 km QKD link
  • Commercial QKD networks - ID Quantique, Toshiba systems
  • Post-quantum cryptography - preparing for quantum computers

Impact: Future-proof secure communications resistant to quantum computers

5. Scanning Tunneling Microscope (STM)

Nanotechnology

STM uses quantum tunneling - particles can be found in classically forbidden regions due to wave function spreading (Postulate I). Tunneling current between tip and sample (probability from Postulate IV) reveals atomic-scale surface structure.

Examples:

  • Atomic manipulation - IBM spelled 'IBM' with 35 xenon atoms (1990)
  • Graphene characterization - mapping carbon lattice
  • Molecular electronics - studying single-molecule conductance
  • Surface science - catalysis, corrosion, thin films

Impact: Enabled nanotechnology revolution and materials science at atomic scale

πŸ’‘ Understanding real-world applications helps connect abstract quantum concepts to tangible technology and motivates further study.
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Video Lecture

Quantum Mechanics in Modern Technology

Exploring how the fundamental postulates of quantum mechanics enable cutting-edge technologies from quantum computers to medical imaging.

πŸ’‘ Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

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Related Topics & Learning Path

From: Postulates of Quantum Mechanics

πŸ“˜ PrerequisitePart I

Hilbert Spaces

Mathematical foundation for Postulate I - where quantum states live

Explore β†’
πŸ“˜ PrerequisitePart I

Linear Operators

Understanding Hermitian operators (Postulate II) and their properties

Explore β†’
πŸ“™ RelatedPart II

Wave Functions

Position representation of quantum states in continuous space

Explore β†’
πŸ“™ RelatedPart II

Measurement Theory

Deeper dive into Postulates III-V and the measurement problem

Explore β†’
πŸ“™ RelatedPart II

Uncertainty Principle

Mathematical consequence of non-commuting observables

Explore β†’
πŸ“™ RelatedPart II

Time Evolution

Applying Postulate VI - SchrΓΆdinger and Heisenberg pictures

Explore β†’
πŸ“• AdvancedPart VII

Entanglement

Postulate VII leads to non-classical correlations in composite systems

Explore β†’
πŸ“• AdvancedPart IX

Quantum Information

Modern applications of the postulates in quantum computing and cryptography

Explore β†’
πŸ’‘ Learning Path: Blue (prerequisites) β†’ Purple (related) β†’ Red (advanced)

Summary

πŸ“ Chapter Summary

The Seven Postulates

I. State Space: |ψ⟩ ∈ β„‹ (Hilbert space)
II. Observables: Γ‚ = † (Hermitian)
III. Outcomes: Eigenvalues aβ‚™
IV. Born Rule: P(aβ‚™) = |⟨aβ‚™|ψ⟩|Β²
V. Collapse: |ψ⟩ β†’ |aβ‚™βŸ©
VI. Evolution: iβ„βˆ‚β‚œ|ψ⟩ = Δ€|ψ⟩
VII. Composite: β„‹_AB = β„‹_A βŠ— β„‹_B

Key Consequences

  • Superposition principle (states can be added)
  • Quantization emerges from boundary conditions
  • Measurement fundamentally probabilistic
  • Uncertainty relations (Ξ”AΞ” B β‰₯ Β½|⟨[Γ‚,BΜ‚]⟩|)
  • Entanglement in composite systems
  • Unitary time evolution preserves normalization
  • No-cloning theorem (can't copy arbitrary quantum states)
  • Complementarity (wave-particle duality)

These seven postulates completely define quantum mechanics. Everything elseβ€”including the uncertainty principle, entanglement, interference, tunneling, and all quantum phenomenaβ€”follows as mathematical consequences. Understanding these postulates deeply is essential for mastering quantum theory.

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