Assume a Stern-Gerlach type experiment is performed with an apparatus as shown in the figure.
A beam of silver atoms emerges from a furnace held at 1000 K. It is collimated using 0.1 mm slits, passed between the poles of a magnet, and directed onto a screen, where the atoms are detected. The speed of the silver atoms is approximately 500 m/s (from (3/2)kT = ½mv^{2}), k = 1.38*10^{-23 }J/K). Assume the apertures collimate the beam to approximately 0.1^{o} angular spread. The spread in the component of velocity perpendicular to the beam direction is at least 1 m/s since Δv_{z} = (500 m/s)*sin(0.1^{o}). We have Δv_{z}Δz ≥ 10^{-4} m^{2}/s. The uncertainty principle requires that Δv_{z}Δz ≥ ħ/M ≈ 10^{-10} m^{2}/s. (M_{silver} = 1.6*10^{-27 }kg * 108.) The position and momentum of a particle must be treated quantum-mechanically when Δp_{i}Δr_{i} approaches the minimum value required by the uncertainty principle. In a typical Stern-Gerlach type experiment this is not the case. The external degrees of freedom of the silver atoms can be treated classically. The dimensions of the wave packet are much smaller that the characteristic dimensions of the problem.
Assume each silver atom has a permanent magnetic moment m. We assume that m = γS, where S denotes the intrinsic angular momentum of the silver atom. γ is called the gyromagnetic ratio. If m is a fundamental property of the atom, then its total energy in an external magnetic field is W = -m∙B. The torque on the atom is τ = m×B, and the force is F = ∇(m∙B).
In our experiment B = (B_{x}, 0, B_{z}) is an inhomogeneous
field. In the region of interest, where the beam passes through the field, we have
B_{z} >> B_{x}, B ≈ Bk.
Therefore τ = m×Bk, τ = dS/dt = γS×Bk. dS_{x}/dt = γB_{z}S_{y}, dS_{y}/dt = -γB_{z}S_{x}, dS_{z}/dt = 0. The component S_{z} is constant, while the component perpendicular to the z-axis rotates about the z-axis. |
Consider an arbitrary vector A, rotating cw about the origin in the x-y plane with angular velocity ω = dθ/dt.
A_{x} = Acosθ, A_{y} = -Asinθ,
dA_{x}/dt = (dA_{x}/dθ)(dθ/dt) = -Asinθω, dA_{y}/dt
= (dA_{y}/dθ)(dθ/dt) = -Acosθω.
dA_{x}/dt = ωA_{y}, dA_{y}/dt = -ωA_{x}.
S_{⟂} rotates about the z-axis with angular
velocity ω = |γB_{z}|. For positive γB_{z}
it rotates cw, and for negative γB_{z} it rotates ccw.
The z-component of F acting on the silver atoms is
F_{z} = ∂/∂z(m_{z}B_{z}
+ m_{x}B_{x}) ≈
∂/∂z(m_{z}B_{z}) = m_{z}∂B_{z}/∂z.
F_{z} produces a deflection of the atoms in the z-direction which
is proportional to m_{z}. If
∂B_{z}/∂z is known, then measuring this deflection is equivalent to measuring m_{z}.
Classically, we expect to find all values of m_{z} between m_{z}
= +|m| and m_{z}= -|m|.
In the experiment the silver atoms hit the screen in two distinct spots.
The
measurement yields only two possible values for m_{z}. S_{z} is
therefore a quantized observable, whose discrete spectrum has only two eigenvalues, which
we will find to be ±ħ/2.
In quantum mechanics, every measurable physical quantity is associated
with a Hermitian operator whose eigenvectors form a basis for the state space.
Assume that
S_{z} is the observable associated with the z-component of the spin.
The state space corresponding to the observable S_{z} of a Silver atom (a
spin ½ particle) is therefore two-dimensional.
We denote the eigenvectors of S_{z}
by |+> and |->.
S_{z}|+> = (ħ/2)|+>, S_{z}|-> = -(ħ/2)|->.
We assume that the eigenvectors are normalized.
<+|+> = <-|-> = 1, <+|-> = 0.
S_{z} alone is a C.S.C.O. in the state space E_{spin} spanned by the
vectors |+> and |->. The closure relation is written as
|+><+| + |-><-| = I.
The most general element of E_{spin} is |ψ> = a|+> + b|->. If <ψ|ψ>
= 1, then |a|^{2} + |b|^{2} = 1.
The matrix of S_{z} in the {|+>, |->} basis is
.
The complete state space for a particle is E = E_{space } ⊗ E_{spin}. However, if in a given problem the spatial degrees of freedom can be treated classically, then we can predict the outcome of measurements which depend on the internal degrees of freedom by considering only the state vector in E_{spin}.
Let S_{x} and S_{y} denote the observables associated with the x- and y-components of the spin of a spin ½ particle, and S_{u} = S∙u the observable associated with the component along a direction defined by the unit vector u . E_{spin} can also be spanned by the eigenvectors |±>_{x }of S_{x}, |±>_{y }of S_{y}, or |±>_{u }of S_{u}. However, S_{x}, S_{y}, and S_{z} do not commute, they are incompatible observables. The eigenvectors of S_{i} are not eigenvectors of S_{j} and S_{k}. The matrices of S_{x} and S_{y} in the eigenbasis of S_{z} , {|+>, |->}, are not diagonal. Theoretical predictions agree with observations if (S_{x}) and (S_{y}) in the {|+>, |->} basis are given by
, .
We write S = (ħ/2)σ. The matrices of the three components of σ in the {|+>, |->} basis are called the Pauli matrices.
The eigenvalues of σ_{x} and σ_{y} are ±1.
The eigenvectors of σ_{x} are |±>_{x} =
(1/√2)(|+> ± |->),
and the
eigenvectors of σ_{y} are |±>_{y} = (1/√2)(|+> ±
i|->).
These are also the
eigenvectors of S_{x} and S_{y} .
Properties of σ_{x} and σ_{y} and σ_{z}
det(σ_{i}) = -1,
Tr{σ_{i}} = 0, σ_{i}^{2} = I, σ_{x}σ_{y}
= -σ_{y}σ_{x} = iσ_{z}.
In general, σ_{i}σ_{j} = iε_{ijk}σ_{k}
+ δ_{ij}I, [σ_{i},σ_{j}] = i2ε_{ijk}σ_{k}.
Therefore
[S_{x},S_{y}] = iħS_{z},
[S_{y},S_{z}] = iħS_{x}, [S_{z},S_{x}] =
iħS_{y},
since S = (ħ/2)σ. We have already shown that [J_{i},J_{j}] = iħε_{ijk}J_{k}
are commutation
relations for angular momentum.
The Pauli matrices anti-commute, (σ_{i}σ_{j} + σ_{j}σ_{i})
= 0.
We also find σ_{x}σ_{y}σ_{z} = iI.
Any arbitrary 2 by 2 matrix can be written as a linear combination of the four matrices σ_{x}, σ_{y}, σ_{z} and I.
Let be an arbitrary 2
by 2 matrix.
M = ½(m_{11} + m_{22})I + ½(m_{11} - m_{22})σ_{z}
+ ½(m_{12} + m_{21})σ_{x} + (i/2)(m_{12} - m_{21})σ_{y}
= a_{0}I + a_{z}σ_{z} + a_{x}σ_{x} +
a_{y}σ_{y}.
(M) is a Hermitian matrix if a_{0}, a_{x}, a_{y},
and a_{z} are real.
a_{0} = ½Tr{M}, a = ½Tr{Mσ}.
The operator S_{u} is defined through
S∙u = S_{x}sinθcosφ
+ S_{y}sinθsinφ + S_{z}cosθ.
The matrix of S_{u} is (S_{u}) = (S_{x})sinθcosφ
+ (S_{y})sinθsinφ + (S_{z})cosθ
= .
The eigenvectors of S_{u} are
|+>_{u }= cos(θ/2)exp(-iφ/2)|+> + sin(θ/2)exp(iφ/2)|->,
|->_{u }= -sin(θ/2)exp(-iφ/2)|+> + cos(θ/2)exp(iφ/2)|->.
The most general normalized ket in the spin state space E_{spin} is |ψ> = a|+> + b|->, |a|^{2} + |b|^{2} = 1. |ψ> is the eigenvector of some operator S_{u} with eigenvalue ±ħ/2. There exists a unit vector u and an operator S_{u} such that |ψ> =|+>_{u}. An arbitrary state vector can be written as |ψ> = cos(θ/2)exp(-iφ/2)|+> + sin(θ/2)exp(iφ/2)|->.
A system can be prepared in any state |ψ> by placing a Stern-Gerlach apparatus such that its axis is directed along the unit vector u and letting only the upward deflected particles pass.