Pauli matrices and spinors
PHYS 410 · Spin and Many-Particle Systems
Pauli matrices represent spin-1/2 observables. This lesson introduces spinors, matrix operators for $S_x$, $S_y$, and $S_z$, and basis changes between spin directions.
Key equations
|+z
angle=egin{pmatrix}1\0end{pmatrix}|-z
angle=egin{pmatrix}0\1end{pmatrix}|psi
angle=egin{pmatrix}alpha\etaend{pmatrix}|alpha|^2+|eta|^2=1sigma_x=egin{pmatrix}0&1\1&0end{pmatrix}sigma_y=egin{pmatrix}0&-i\i&0end{pmatrix}sigma_z=egin{pmatrix}1&0\0&-1end{pmatrix}hat{S}_i= rac{hbar}{2}sigma_ihat{S}_z= rac{hbar}{2}egin{pmatrix}1&0\0&-1end{pmatrix}P(+z)=|alpha|^2P(-z)=|eta|^2|+x
angle= rac{1}{sqrt{2}}egin{pmatrix}1\1end{pmatrix}|-x
angle= rac{1}{sqrt{2}}egin{pmatrix}1\-1end{pmatrix}|+y
angle= rac{1}{sqrt{2}}egin{pmatrix}1\iend{pmatrix}|-y
angle= rac{1}{sqrt{2}}egin{pmatrix}1\-iend{pmatrix}[sigma_x,sigma_y]=2isigma_z[hat{S}_x,hat{S}_y]=ihbarhat{S}_zhat{S}_n= rac{hbar}{2} ec{sigma}cdothat{n} ec{sigma}=(sigma_x,sigma_y,sigma_z)Learning objectives
- Represent spin-1/2 states as spinors.
- Write the Pauli matrices.
- Use Pauli matrices to represent spin operators.
- Identify eigenstates of $S_x$, $S_y$, and $S_z$.
- Explain spin commutation relations.
Spinors
A spin-1/2 state can be represented as a two-component column vector called a spinor. In the basis, choose
angle=egin{pmatrix}1\0end{pmatrix}$$ and $$|-z angle=egin{pmatrix}0\1end{pmatrix}$$ A general state is $$|psi angle=egin{pmatrix}alpha\etaend{pmatrix}$$ with $$|alpha|^2+|eta|^2=1$$ ## Pauli matrices Spin operators are represented using Pauli matrices: $$sigma_x=egin{pmatrix}0&1\1&0end{pmatrix}$$ $$sigma_y=egin{pmatrix}0&-i\i&0end{pmatrix}$$ $$sigma_z=egin{pmatrix}1&0\0&-1end{pmatrix}$$ The spin operators are $$hat{S}_i= rac{hbar}{2}sigma_i$$ for $i=x,y,z$. ## Measuring $S_z$ The $S_z$ operator is $$hat{S}_z= rac{hbar}{2}egin{pmatrix}1&0\0&-1end{pmatrix}$$ Its eigenvectors are $|+z angle$ and $|-z angle$, with eigenvalues $+hbar/2$ and $-hbar/2$. For state $|psi angle=(alpha,eta)^T$, the probabilities are $$P(+z)=|alpha|^2$$ and $$P(-z)=|eta|^2$$ ## Eigenstates of $S_x$ The eigenstates of $S_x$ in the z-basis are $$|+x angle= rac{1}{sqrt{2}}egin{pmatrix}1\1end{pmatrix}$$ and $$|-x angle= rac{1}{sqrt{2}}egin{pmatrix}1\-1end{pmatrix}$$ Thus a particle in $|+z angle$ has equal probability to be measured $+x$ or $-x$. ## Eigenstates of $S_y$ The $S_y$ eigenstates are $$|+y angle= rac{1}{sqrt{2}}egin{pmatrix}1\iend{pmatrix}$$ and $$|-y angle= rac{1}{sqrt{2}}egin{pmatrix}1\-iend{pmatrix}$$ The imaginary phase is essential. It distinguishes y-direction spin from x-direction spin. ## Commutation relations Pauli matrices satisfy $$[sigma_x,sigma_y]=2isigma_z$$ with cyclic permutations. Therefore spin operators satisfy $$[hat{S}_x,hat{S}_y]=ihbarhat{S}_z$$ This matches angular momentum commutation rules and explains why different spin components cannot have simultaneous definite values. ## Spin along an arbitrary direction For a unit vector $hat{n}$, the spin operator along that direction is $$hat{S}_n= rac{hbar}{2} ec{sigma}cdothat{n}$$ where $$ ec{sigma}=(sigma_x,sigma_y,sigma_z)$$ This compact expression makes Pauli matrices powerful for magnetic resonance, quantum computing, and two-level systems. ## The big idea Spin-1/2 states are spinors, and Pauli matrices represent spin observables. Different spin directions correspond to different bases in the same two-dimensional Hilbert space. Complex phases and noncommuting matrices are essential features, not mathematical decoration.Ask your AI physics guide
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