Lecture Notes: Bloch Sphere

Introduction

• Discussion on quantum spin measurement along any direction in space (n-hat)
• Correspondence between quantum states (vectors in situ) and unit vectors in 3D space

Quantum Spin State

• Quantum spin state -> Vector in 3D space
• Measure quantum state along axis defined by n-hat -> Always spin up along this axis

Important Notation

• n-hat (unit vector, length = 1)
• Θ (theta): Angle between n-hat and z-axis
• Φ (phi): Angle between shadow of n-hat and x-axis

Expression of n-hat Components

• n_Z = cos(Θ)
• n_X = sin(Θ) cos(Φ)
• n_Y = sin(Θ) sin(Φ)
• Quantum state spin up in n-hat direction:
  • Components: cos(Θ/2), e^iΦsin(Θ/2)

Poly Matrices

• σ_X = [0 1] [1 0]
• σ_Y = [0 -i] [i 0]
• σ_Z = [1 0] [0 -1]

Eigenvalues and Eigenstates

• Eigenvalues: ±1
• Measurement matrix: n-hat · σ
• Quantum state that is always spin up in n-hat direction is an eigenvector with eigenvalue +1

Bloch Sphere and Bloch Vector

• Bloch Vector: n-hat
• Bloch Sphere: All Bloch vectors with norm = 1

Key Equations

• cos(Θ/2), e^iΦsin(Θ/2): Eigenvector with eigenvalue +1 for matrix n-hat · σ
• sin(Θ/2), -e^iΦcos(Θ/2): Spin-down state in n-hat direction
• Orthogonality and normalization: <ψ|ψ> = 1
• Sanity checks with specific orientations (X, Y, Z directions)

General Expressions

• Any quantum state |ψ> can be written as spin-up in some n-hat direction
• Expression for n-hat components (expectation values): <ψ|σ_x|ψ>, <ψ|σ_y|ψ>, <ψ|σ_z|ψ>
• Probability of measuring a quantum state:
  • P(up, z-axis) = (cos(Θ/2))² = 1/2 (1 + cos(Θ))
• General probability expression: P(up, M-hat) = 1/2 (1 + n-hat · M-hat)

Conclusion

• Importance of validating and understanding mathematical correspondences in quantum physics
• Teaser for next video: Evolving quantum spin states in a magnetic field