Spin-1/2 Quantum States and Pauli Matrices: From Stern-Gerlach to Arbitrary Spin Directions

Short Summary

In this MIT OpenCourseWare lecture, the spin-1/2 system is developed from the Stern-Gerlach experiment, showing how a silver atom beam can be described by a two-dimensional complex vector space. The instructor introduces the basis states Z+ and Z- as eigenstates of S_Z with eigenvalues ±ħ/2, and explains state representations as two-component column vectors along with bras and inner products. The talk then constructs S_X and S_Y using 2x2 Hermitian matrices, introducing the Pauli matrices and establishing their role as the fundamental spin operators. Finally, a general direction N is treated, giving the spin operator S_N and its eigenvectors, illustrating how any spin state along any direction can be expressed as a linear combination of the Z-basis states.

Spin-1/2 Hilbert Space and Basis States

The lecture begins by revisiting the Stern-Gerlach experiment, emphasizing the two distinct outcomes corresponding to the Z component of spin for a spin-1/2 particle. The two states Z+ and Z- are introduced as an orthonormal basis for the two-dimensional complex vector space that represents all possible spin states. The eigenvalue equations S_Z |Z±> = ±(ħ/2) |Z±> are discussed, and the physical meaning of Z as the direction of the spin filter is explained. The inner product between states is defined via bras and kets, with Z+ and Z− normalized and orthogonal.

From Vectors to Representations

The states are represented as column vectors in a two-dimensional space, with Z+ ↔ (1,0)^T and Z− ↔ (0,1)^T. The bras are dual row vectors, and the inner product is given by the conjugate transpose times another state vector. This representation turns states into concrete matrices and makes the action of operators like S_Z transparent on the basis vectors.

Constructing the X and Y Spin Operators

To describe spin along the X and Y directions, the instructor introduces a systematic way to obtain S_X and S_Y as Hermitian 2x2 matrices that satisfy the angular-momentum algebra and have eigenvalues ±ħ/2. By removing the identity component and focusing on the traceless Hermitian part, two independent matrices emerge which, up to conventional scaling and basis choice, lead to the familiar Pauli matrices: σ_x, σ_y, σ_z. The operators S_X, S_Y, and S_Z are then written as S_i = (ħ/2) σ_i, and their commutation relations [S_i, S_j] = iħ ε_ijk S_k are confirmed by explicit calculation in the chosen representation.

Pauli Matrices and Eigenstates

The three Pauli matrices are introduced as the standard representation of the spin-1/2 algebra. Their eigenvectors are computed in the Z-basis: Z+ and Z− are eigenvectors of S_Z with eigenvalues ±ħ/2, while the X and Y eigenvectors are superpositions of Z±, such as X+ ∝ Z+ + Z− and Y+ ∝ Z+ + i Z−. These results illustrate that states aligned along X or Y directions are not new degrees of freedom but linear combinations of the Z-basis states, reinforcing the idea that the two-dimensional complex space suffices for all spin-1/2 states.

Spin Along Arbitrary Directions

The lecture then defines the spin operator in an arbitrary direction N̂ as S_N = N_x S_X + N_y S_Y + N_z S_Z, with N a unit vector expressed in polar and azimuthal angles θ and φ. The resulting 2x2 matrix is explicitly constructed using the Pauli matrices and yields eigenvalues ±ħ/2. The eigenvectors of S_N are computed, giving a general formula for the |N, +> state in terms of the Z-basis: |N, +> = cos(θ/2) |Z+> + e^{iφ} sin(θ/2) |Z−>. The corresponding |N, −> state is also derived, ensuring a complete description of spin along any direction in terms of the Z-basis states.

Implications and Summary

The discussion shows that spin-1/2 physics can be fully captured within a two-dimensional complex vector space using the Pauli matrices and standard inner product structure. The formalism reproduces the Stern-Gerlach probabilities for measurements along different axes and provides a consistent framework for rotating spin states and predicting measurement outcomes in any direction. The lecture closes by highlighting the parallels and differences between spin and orbital angular momentum, and by setting the stage for further linear-algebra tools used in quantum mechanics.