← Part V/Spin-1/2 Systems

1. Spin-1/2 Systems

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Intrinsic angular momentum: a purely quantum mechanical property with no classical analog.

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Video Lecture

Introduction to Quantum Spin - The Stern-Gerlach Experiment

Brant Carlson explains the historic Stern-Gerlach experiment and the discovery of quantum spin

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

Discovery of Spin

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The Discovery of Quantum Spin

1922

Stern-Gerlach Experiment

— Otto Stern & Walther Gerlach

Otto Stern and Walther Gerlach passed silver atoms through an inhomogeneous magnetic field, observing discrete beams instead of a continuous distribution

Significance: First direct evidence of space quantization and quantum angular momentum

1925

Electron Spin Hypothesis

— Uhlenbeck & Goudsmit

George Uhlenbeck and Samuel Goudsmit proposed that electrons possess intrinsic angular momentum (spin)

Significance: Explained anomalous Zeeman effect and fine structure of atomic spectra

1925

Pauli Exclusion Principle

— Wolfgang Pauli

Wolfgang Pauli formulated the exclusion principle requiring fermions (spin-1/2 particles) to have antisymmetric wavefunctions

Significance: Explained periodic table structure and stability of matter

1928

Dirac Equation

— Paul Dirac

Paul Dirac developed a relativistic wave equation that naturally predicted electron spin as a consequence of relativity and quantum mechanics

Significance: Unified special relativity with quantum mechanics; predicted antimatter

💡 Understanding the historical development helps contextualize why certain concepts emerged and how they fit into the broader quantum revolution.

Spin Operators

Spin angular momentum obeys same algebra as orbital angular momentum:

$$[\hat{S}_i, \hat{S}_j] = i\hbar\epsilon_{ijk}\hat{S}_k$$

Eigenvalues:

$$\hat{S}^2|s,m_s\rangle = \hbar^2 s(s+1)|s,m_s\rangle$$

$$\hat{S}_z|s,m_s\rangle = \hbar m_s|s,m_s\rangle$$

Spin-1/2

For electrons, protons, neutrons: $s = 1/2$

Only two states: $m_s = +1/2$ (spin up) and $m_s = -1/2$ (spin down)
Magnitude: $|\vec{S}| = \sqrt{3/4}\hbar$
z-component: $\pm\hbar/2$

Spin States

Notation for basis states:

$$|\uparrow\rangle = |+\rangle = \left(\begin{array}{c}1\\0\end{array}\right), \quad |\downarrow\rangle = |-\rangle = \left(\begin{array}{c}0\\1\end{array}\right)$$

General spin state:

$$|\chi\rangle = \alpha|\uparrow\rangle + \beta|\downarrow\rangle = \left(\begin{array}{c}\alpha\\\beta\end{array}\right)$$

with normalization: $|\alpha|^2 + |\beta|^2 = 1$

Pauli Matrices

Spin operators for spin-1/2:

$$\hat{S}_i = \frac{\hbar}{2}\sigma_i$$

Pauli matrices:

$$\sigma_x = \left(\begin{array}{cc}0&1\\1&0\end{array}\right), \quad \sigma_y = \left(\begin{array}{cc}0&-i\\i&0\end{array}\right), \quad \sigma_z = \left(\begin{array}{cc}1&0\\0&-1\end{array}\right)$$

Finding Pauli Matrix Eigenvalues and Eigenvectors

BASIC

Problem: Find the eigenvalues and eigenvectors of the Pauli matrix σₓ.

Given:

Pauli matrix: σₓ = [[0, 1], [1, 0]]
Eigenvalue equation: σₓ|v⟩ = λ|v⟩

Find: Eigenvalues λ and normalized eigenvectors

Self-Check Question

What is the probability of measuring spin-up along the z-axis if a particle is in the state |+⟩ₓ = (|↑⟩ + |↓⟩)/√2?

Eigenstates of Spin Operators

$\hat{S}_z$ eigenstates: $|\uparrow\rangle, |\downarrow\rangle$

$\hat{S}_x$ eigenstates:

$$|+\rangle_x = \frac{1}{\sqrt{2}}(|\uparrow\rangle + |\downarrow\rangle), \quad |-\rangle_x = \frac{1}{\sqrt{2}}(|\uparrow\rangle - |\downarrow\rangle)$$

$\hat{S}_y$ eigenstates:

$$|+\rangle_y = \frac{1}{\sqrt{2}}(|\uparrow\rangle + i|\downarrow\rangle), \quad |-\rangle_y = \frac{1}{\sqrt{2}}(|\uparrow\rangle - i|\downarrow\rangle)$$

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Video Lecture

Understanding Pauli Matrices

Faculty of Khan - Properties and applications of the Pauli spin matrices σₓ, σᵧ, σᵤ

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

Self-Check Question

Which of the following is NOT a property of the Pauli matrices?

Spin in Arbitrary Direction

For unit vector $\hat{n} = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)$:

$$\hat{S}_{\hat{n}} = \vec{S}\cdot\hat{n} = \frac{\hbar}{2}(\sigma_x\sin\theta\cos\phi + \sigma_y\sin\theta\sin\phi + \sigma_z\cos\theta)$$

Eigenstates:

$$|+\rangle_{\hat{n}} = \cos(\theta/2)|\uparrow\rangle + e^{i\phi}\sin(\theta/2)|\downarrow\rangle$$

$$|-\rangle_{\hat{n}} = \sin(\theta/2)|\uparrow\rangle - e^{i\phi}\cos(\theta/2)|\downarrow\rangle$$

Interactive Bloch Sphere

θ (polar angle): 45.0°

φ (azimuthal angle): 45.0°

Quantum State:

|ψ⟩ = 0.924|0⟩ + (0.271 + 0.271i)|1⟩

P(|0⟩) = |α|² = 0.854

P(|1⟩) = |β|² = 0.146

Note: The Bloch sphere represents all possible pure states of a qubit. The north pole is |0⟩, south pole is |1⟩, and the equator contains equal superpositions.

Measuring Spin Along an Arbitrary Direction

INTERMEDIATE

Problem: A spin-1/2 particle is in the state |↑⟩. What is the probability of measuring spin-up along a direction n̂ at angle θ = 60° from the z-axis (in the xz-plane, so φ = 0)?

Given:

Initial state: |ψ⟩ = |↑⟩
Measurement direction: θ = 60°, φ = 0
Eigenstate: |+⟩ₙ = cos(θ/2)|↑⟩ + e^(iφ)sin(θ/2)|↓⟩

Find: Probability P(+) of measuring spin-up along n̂

Self-Check Question

An electron is measured to be spin-up along the x-direction. What is the probability it will be measured spin-up if immediately measured along the y-direction?

Spin Precession

In magnetic field $\vec{B} = B\hat{z}$, Hamiltonian:

$$\hat{H} = -\gamma\vec{S}\cdot\vec{B} = -\gamma B\hat{S}_z = -\omega_0\hat{S}_z$$

where $\omega_0 = \gamma B$ is the Larmor frequency

Spin precesses around $\vec{B}$ at frequency $\omega_0$

Spin Magnetic Moment

$$\vec{\mu} = g\frac{e}{2m}\vec{S}$$

where:

$g \approx 2$ for electron (g-factor)
Bohr magneton: $\mu_B = e\hbar/(2m_e) \approx 9.27 \times 10^{-24}$ J/T

Spinors

Two-component complex vectors representing spin-1/2 states:

$$\chi = \left(\begin{array}{c}\alpha\\\beta\end{array}\right)$$

Transform under rotations using 2×2 unitary matrices (SU(2) representation)

Complete Wave Function

For spin-1/2 particle:

$$\Psi(\vec{r},t) = \psi(\vec{r},t)\otimes\chi = \psi(\vec{r},t)\left(\begin{array}{c}\alpha\\\beta\end{array}\right)$$

Spatial wave function ⊗ spinor

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Video Lecture

Spin and Magnetic Resonance - Applications

MIT OCW - Applications of spin precession including NMR and MRI technology

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

Self-Check Question

A spin-1/2 particle in a magnetic field B along the z-axis precesses with Larmor frequency ω₀ = γB. If we double the magnetic field strength, what happens to the precession frequency?

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Real-World Applications of Spin

1. Nuclear Magnetic Resonance (NMR) Spectroscopy

Chemistry & Biochemistry

NMR exploits nuclear spin precession in magnetic fields to determine molecular structure. Different chemical environments produce different resonance frequencies, creating unique spectral 'fingerprints' for molecules.

Examples:

Protein structure determination
Drug discovery and design
Chemical composition analysis
Quality control in pharmaceuticals

2. Magnetic Resonance Imaging (MRI)

Medical Imaging

MRI uses spin precession of hydrogen nuclei in the body to create detailed 3D images of soft tissues. By applying gradient magnetic fields and radiofrequency pulses, different tissues can be distinguished based on their spin relaxation properties.

Examples:

Brain imaging for neurological disorders
Cancer detection and staging
Cardiovascular imaging
Functional MRI (fMRI) for brain activity mapping

3. Quantum Computing with Spin Qubits

Quantum Technology

Electron or nuclear spins serve as quantum bits (qubits) in quantum computers. The superposition of spin-up and spin-down states enables quantum parallelism, while spin coherence times determine computational power.

Examples:

Silicon spin qubits (Intel, QuTech)
Nitrogen-vacancy centers in diamond
Quantum dots for scalable quantum processors
Topological qubits using Majorana fermions

4. Spintronics

Electronics & Data Storage

Spintronics utilizes electron spin (in addition to charge) for information storage and processing. Spin-dependent transport enables faster, more efficient devices with non-volatile memory.

Examples:

Giant magnetoresistance (GMR) in hard drives
Spin-transfer torque RAM (STT-RAM)
Spin-orbit torque devices
Magnetic tunnel junctions (MTJ)

5. Atomic Clocks

Metrology & Navigation

The most precise atomic clocks use spin transitions in atoms like cesium-133 or rubidium-87. The hyperfine splitting between spin states defines the second in the SI system.

Examples:

GPS satellite timing (accuracy: ~10 ns)
NIST-F2 cesium fountain clock
Optical lattice clocks (Sr, Yb)
Deep space navigation

6. Quantum Sensors

Sensing & Metrology

Spin-based sensors achieve unprecedented sensitivity for measuring magnetic fields, electric fields, temperature, and pressure at the nanoscale.

Examples:

SQUID magnetometers for brain imaging
Diamond NV centers for nanoscale MRI
Atomic magnetometers for navigation
Rotation sensors (spin gyroscopes)

💡 These applications demonstrate how fundamental quantum concepts translate into modern technology.

📝 Summary: Spin-1/2 Systems

Key Equations

Commutation:

$$[\hat{S}_i, \hat{S}_j] = i\hbar\epsilon_{ijk}\hat{S}_k$$

Spin-1/2 operators:

$$\hat{S}_i = \frac{\hbar}{2}\sigma_i$$

Eigenstate (arbitrary direction):

$$|+\rangle_{\hat{n}} = \cos\frac{\theta}{2}|\uparrow\rangle + e^{i\phi}\sin\frac{\theta}{2}|\downarrow\rangle$$

Larmor precession:

$$\omega_0 = \gamma B$$

Key Concepts

Spin is intrinsic angular momentum with no classical analog
Spin-1/2 particles have only two states: |↑⟩ and |↓⟩
Pauli matrices σₓ, σᵧ, σᵤ represent spin operators (eigenvalues ±1)
Stern-Gerlach experiment provides direct evidence of quantization
Measuring spin along one axis randomizes measurements along perpendicular axes
Spin states can be visualized on the Bloch sphere
Magnetic moments cause spin precession in magnetic fields
Complete wave function: Ψ = ψ(r,t) ⊗ χ (spatial × spinor)

Historical Impact: The discovery of spin revolutionized quantum mechanics, explained atomic spectra, enabled the Pauli exclusion principle, and laid the foundation for modern technologies from MRI to quantum computing.

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Quantum Mechanics Course

1. Spin-1/2 Systems

Your Progress

Video Lecture

Discovery of Spin

The Discovery of Quantum Spin

Stern-Gerlach Experiment

Electron Spin Hypothesis

Pauli Exclusion Principle

Dirac Equation

Spin Operators

Spin-1/2

Spin States

Pauli Matrices

Finding Pauli Matrix Eigenvalues and Eigenvectors

Self-Check Question

Eigenstates of Spin Operators

Video Lecture

Self-Check Question

Spin in Arbitrary Direction

Interactive Bloch Sphere

Measuring Spin Along an Arbitrary Direction

Self-Check Question

Spin Precession

Spin Magnetic Moment

Spinors

Complete Wave Function

Video Lecture

Self-Check Question

Real-World Applications of Spin

1. Nuclear Magnetic Resonance (NMR) Spectroscopy

2. Magnetic Resonance Imaging (MRI)

3. Quantum Computing with Spin Qubits

4. Spintronics

5. Atomic Clocks

6. Quantum Sensors

📝 Summary: Spin-1/2 Systems

Key Equations

Key Concepts

Related Topics

Angular Momentum

Hydrogen Atom

Addition of Angular Momentum

Identical Particles

Time-Independent Perturbation Theory

Scattering Theory