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Text of the page (random words):
s the same as in the starting position also in a spin 1 2 particle there are only two spin states and the amplitudes for both change by the same 1 factor so the interference effects are identical unlike the case for higher spins the complex probability amplitudes are something of a theoretical construct that cannot be directly observed if the probability amplitudes rotated by the same amount as the detector then they would have changed by a factor of 1 when the equipment was rotated by 180 which when squared would predict the same output as at the start but experiments show this to be wrong if the detector is rotated by 180 the result with spin 1 2 particles can be different from what it would be if not rotated hence the factor of a half is necessary to make the predictions of the theory match the experiments in terms of more direct evidence physical effects of the difference between the rotation of a spin 1 2 particle by 360 as compared with 720 have been experimentally observed in classic experiments 5 in neutron interferometry in particular if a beam of spin oriented spin 1 2 particles is split and just one of the beams is rotated about the axis of its direction of motion and then recombined with the original beam different interference effects are observed depending on the angle of rotation in the case of rotation by 360 cancellation effects are observed whereas in the case of rotation by 720 the beams are mutually reinforcing 5 non relativistic quantum mechanics edit the quantum state of a spin 1 2 particle can be described by a two component complex valued vector called a spinor observable states of the particle are then found by the spin operators s x s y and s z and the total spin operator s observables edit when spinors are used to describe the quantum states the three spin operators s x s y s z can be described by 2 2 matrices called the pauli matrices whose eigenvalues are 1 2 ℏ textstyle pm frac 1 2 hbar for example the spin projection operator s z affects a measurement of the spin in the z direction s z ℏ 2 σ z ℏ 2 1 0 0 1 displaystyle s_ z frac hbar 2 sigma _ z frac hbar 2 begin bmatrix 1 0 0 1 end bmatrix the two eigenvalues of s z 1 2 ℏ textstyle pm frac 1 2 hbar then correspond to the following eigenspinors χ 1 0 s z 1 2 0 displaystyle chi _ begin bmatrix 1 0 end bmatrix left vert s_ z textstyle frac 1 2 right rangle uparrow rangle 0 rangle χ 0 1 s z 1 2 1 displaystyle chi _ begin bmatrix 0 1 end bmatrix left vert s_ z textstyle frac 1 2 right rangle downarrow rangle 1 rangle these vectors form a complete basis for the hilbert space describing the spin 1 2 particle thus linear combinations of these two states can represent all possible states of the spin including in the x and y directions the ladder operators are s ℏ 0 1 0 0 s ℏ 0 0 1 0 displaystyle s_ hbar begin bmatrix 0 1 0 0 end bmatrix qquad s_ hbar begin bmatrix 0 0 1 0 end bmatrix since s s x i s y displaystyle s_ pm s_ x pm is_ y 6 it follows that s x 1 2 s s textstyle s_ x frac 1 2 s_ s_ and s y 1 2 i s s textstyle s_ y frac 1 2i s_ s_ thus s x ℏ 2 σ x ℏ 2 0 1 1 0 displaystyle s_ x frac hbar 2 sigma _ x frac hbar 2 begin bmatrix 0 1 1 0 end bmatrix s y ℏ 2 σ y ℏ 2 0 i i 0 displaystyle s_ y frac hbar 2 sigma _ y frac hbar 2 begin bmatrix 0 i i 0 end bmatrix their normalized eigenspinors can be found in the usual way for s x textstyle s_ x they are χ x 1 2 1 1 s x 1 2 displaystyle chi _ x frac 1 sqrt 2 begin bmatrix 1 1 end bmatrix left vert s_ x textstyle frac 1 2 right rangle χ x 1 2 1 1 s x 1 2 displaystyle chi _ x frac 1 sqrt 2 begin bmatrix 1 1 end bmatrix left vert s_ x textstyle frac 1 2 right rangle for s y textstyle s_ y they are χ y 1 2 1 i s y 1 2 displaystyle chi _ y frac 1 sqrt 2 begin bmatrix 1 i end bmatrix left vert s_ y textstyle frac 1 2 right rangle χ y 1 2 1 i s y 1 2 displaystyle chi _ y frac 1 sqrt 2 begin bmatrix 1 i end bmatrix left vert s_ y textstyle frac 1 2 right rangle bloch representation edit the hilbert space of a spin 1 2 system is 2 dimensional choosing the eigenbasis of s z displaystyle s_ z a nonzero spinor can be written as ψ a b a b c 2 a b 0 0 displaystyle left psi right rangle a left uparrow right rangle b left downarrow right rangle a b in mathbb c 2 a b neq 0 0 a physical pure state is not a particular vector in the hilbert space but a ray vectors that differ by a nonzero complex number represent the same state ψ λ ψ λ c displaystyle left psi right rangle sim lambda left psi right rangle lambda in mathbb c times this is because the overall phase is physically irrelevant equivalently the space of pure spin 1 2 states is the complex projective line p c 2 c p 1 s 2 displaystyle mathbb p mathbb c 2 mathbb cp 1 cong s 2 the pure state associated with a b displaystyle a b is written in homogeneous coordinates as ψ a b c p 1 displaystyle left psi right a b in mathbb cp 1 on the coordinate patch where a 0 displaystyle a neq 0 one may divide out a displaystyle a and write ψ 1 w w b a displaystyle left psi right 1 w w frac b a in this notation w displaystyle w is a projective coordinate on c p 1 displaystyle mathbb cp 1 and each w displaystyle w fully characterizes a state a representative of the same ray state is w displaystyle left uparrow right rangle w left downarrow right rangle and a normalized one is ψ w w 1 w 2 displaystyle left psi w right rangle frac left uparrow right rangle w left downarrow right rangle sqrt 1 w 2 the coordinate w 0 displaystyle w 0 gives ψ 0 displaystyle left psi 0 right rangle left uparrow right rangle spin up in the z displaystyle z direction the coordinate patch where a 0 displaystyle a neq 0 misses a 0 displaystyle a 0 namely the state ψ 0 1 displaystyle psi 0 1 which is represented by spin down displaystyle left downarrow right rangle this missing point is added by allowing w displaystyle w infty therefore the projective coordinate takes value in the extended complex plane w c displaystyle w in mathbb c cup left infty right which is the riemann sphere this is the same two sphere that appears as the bloch sphere using south pole stereographic projection the extended w displaystyle w plane can be mapped to s 2 r 3 displaystyle s 2 subset mathbb r 3 and vice versa writing w u i v displaystyle w u iv the stereographic map is w x i y 1 z e i ϕ tan θ 2 x y z s 2 displaystyle w frac x iy 1 z e i phi tan frac theta 2 x y z in s 2 and the inverse stereographic map is n w 2 u 1 w 2 2 v 1 w 2 1 w 2 1 w 2 displaystyle boldsymbol n w left frac 2u 1 w 2 frac 2v 1 w 2 frac 1 w 2 1 w 2 right to recover the usual form of the bloch state substitute w e i ϕ tan θ 2 displaystyle w e i phi tan theta 2 into the normalized representative state vector this gives ψ e i ϕ tan θ 2 1 tan 2 θ 2 cos θ 2 e i ϕ sin θ 2 displaystyle left psi right rangle frac left uparrow right rangle e i phi tan theta 2 left downarrow right rangle sqrt 1 tan 2 theta 2 cos frac theta 2 left uparrow right rangle e i phi sin frac theta 2 left downarrow right rangle the expectation value of spin measurements bloch vector associated with a normalized state is ψ σ ψ displaystyle left langle psi right boldsymbol sigma left psi right rangle where σ σ x σ y σ z displaystyle boldsymbol sigma sigma _ x sigma _ y sigma _ z are the pauli matrices for ψ w w 1 w 2 displaystyle left psi w right rangle frac left uparrow right rangle w left downarrow right rangle sqrt 1 w 2 this gives ψ σ ψ 2 re w 1 w 2 2 im w 1 w 2 1 w 2 1 w 2 displaystyle left langle psi right boldsymbol sigma left psi right rangle left frac 2 text re w 1 w 2 frac 2 text im w 1 w 2 frac 1 w 2 1 w 2 right which is exactly the inverse stereographic projection above hence each point n displaystyle boldsymbol n on the bloch sphere simultaneously describes a state since each w displaystyle w describes a state and that state s expectation value in spin measurements the point n displaystyle boldsymbol n thus represents the pure state call it χ n displaystyle chi _ boldsymbol n whose spin is maximally aligned along the direction n displaystyle boldsymbol n n σ χ n χ n displaystyle boldsymbol n cdot boldsymbol sigma chi _ boldsymbol n chi _ boldsymbol n rotations and spinors edit for any unit spatial direction v displaystyle boldsymbol v the spin observable s v v s ℏ 2 v σ displaystyle s_ boldsymbol v boldsymbol v cdot boldsymbol s frac hbar 2 boldsymbol v cdot boldsymbol sigma has eigenvalues ℏ 2 displaystyle pm hbar 2 a stern gerlach experiment along v displaystyle boldsymbol v for χ n displaystyle chi _ boldsymbol n has the probability p v 1 v n 2 displaystyle p_ boldsymbol v frac 1 boldsymbol v cdot boldsymbol n 2 of measuring ℏ 2 displaystyle hbar 2 if one chooses a rotated coordinate system instead this probability must not change as it has coordinate independent physical meaning this means v n displaystyle boldsymbol v cdot boldsymbol n is a scalar under rotations and thus the bloch vector n displaystyle boldsymbol n must transform in the same vector representation of rotations as v displaystyle boldsymbol v namely n r n r s o 3 displaystyle boldsymbol n mapsto r boldsymbol n r in so 3 for a rotation by angle α displaystyle alpha about a unit direction m displaystyle boldsymbol m rodrigues formula says r m n n cos α m n sin α m m n 1 cos α displaystyle r_ boldsymbol m boldsymbol n boldsymbol n cos alpha boldsymbol m times boldsymbol n sin alpha boldsymbol m boldsymbol m cdot boldsymbol n 1 cos alpha this is the ordinary vector rotation law the distinctive feature of spin 1 2 is that the state vector itself does not transform as an ordinary vector like the bloch vector instead it transforms as a spinor to see where the spinor rotation law comes from it is useful to recall a purely geometric fact the composition of two reflections is a rotation first reflect n displaystyle boldsymbol n through the plane perpendicular to the unit vector a displaystyle boldsymbol a n n 2 a n a displaystyle boldsymbol n mapsto boldsymbol n 2 boldsymbol a cdot boldsymbol n boldsymbol a using pauli matrices a vector n displaystyle boldsymbol n may be encoded as the hermitian matrix n σ displaystyle boldsymbol n cdot boldsymbol sigma these matrices satisfy a σ b σ a b i i a b σ displaystyle boldsymbol a cdot boldsymbol sigma boldsymbol b cdot boldsymbol sigma boldsymbol a cdot boldsymbol b i i boldsymbol a times boldsymbol b cdot boldsymbol sigma from this identity one finds the reflection in the plane normal to a displaystyle boldsymbol a can be written as n σ n 2 a n a σ a σ n σ a σ displaystyle boldsymbol n cdot boldsymbol sigma mapsto left boldsymbol n 2 boldsymbol a cdot boldsymbol n boldsymbol a right cdot boldsymbol sigma boldsymbol a cdot boldsymbol sigma boldsymbol n cdot boldsymbol sigma boldsymbol a cdot boldsymbol sigma then we reflect this through a plane normal to the unit vector b displaystyle boldsymbol b we choose this plane such that the two planes make an angle α 2 displaystyle alpha 2 since geometrically the composition of these two reflections is a rotation by angle α displaystyle alpha n σ b σ a σ n σ a σ b σ displaystyle boldsymbol n cdot boldsymbol sigma mapsto boldsymbol b cdot boldsymbol sigma boldsymbol a cdot boldsymbol sigma boldsymbol n cdot boldsymbol sigma boldsymbol a cdot boldsymbol sigma boldsymbol b cdot boldsymbol sigma define the unitary matrix u s u 2 displaystyle u in su 2 u b σ a σ cos α 2 i i m σ sin α 2 exp i α 2 m σ displaystyle begin aligned u boldsymbol b cdot boldsymbol sigma boldsymbol a cdot boldsymbol sigma cos frac alpha 2 i i boldsymbol m cdot boldsymbol sigma sin frac alpha 2 exp left frac i alpha 2 boldsymbol m cdot boldsymbol sigma right end aligned where α 2 displaystyle alpha 2 is the angle between a displaystyle boldsymbol a and b displaystyle boldsymbol b and m a b a b displaystyle boldsymbol m frac boldsymbol a times boldsymbol b boldsymbol a times boldsymbol b is the unit axis of rotation along the intersection between the two planes rotations thus act on vectors by the two sided multiplication u n σ u r m n σ displaystyle u boldsymbol n cdot boldsymbol sigma u dagger r_ boldsymbol m boldsymbol n cdot boldsymbol sigma u displaystyle u is exactly the rotation operator acting on the spin 1 2 hilbert space the connection between the bloch vector and the spinor can now be made precise for every direction n displaystyle boldsymbol n on the bloch sphere define the subspace projector p n χ n χ n 1 n σ 2 displaystyle p_ boldsymbol n chi _ boldsymbol n chi _ boldsymbol n dagger frac 1 boldsymbol n cdot boldsymbol sigma 2 this projects onto the one dimensional eigenspace of n σ displaystyle boldsymbol n cdot boldsymbol sigma with eigenvalue 1 displaystyle 1 under a rotation n σ u n σ u displaystyle boldsymbol n cdot boldsymbol sigma mapsto u boldsymbol n cdot boldsymbol sigma u dagger therefore p n u p n u u χ n u χ n displaystyle p_ boldsymbol n mapsto up_ boldsymbol n u dagger u chi _ boldsymbol n u chi _ boldsymbol n dagger so the spinor itself transforms under a single sided law χ u χ displaystyle chi mapsto u chi the distinction between vector and spinor transformation rules is the origin of the surjective double cover homomorphism s u 2 s o 3 displaystyle su 2 to so 3 the matrices u displaystyle u and u displaystyle u induce the same rotation to bloch vectors since u n σ u u n σ u displaystyle u boldsymbol n cdot boldsymbol sigma u dagger u boldsymbol n cdot boldsymbol sigma u dagger but they act differently on the spinors χ u χ χ u χ displaystyle chi mapsto u chi chi mapsto u chi for a 2 π displaystyle 2 pi rotation u m 2 π exp i π m σ i displaystyle u boldsymbol m 2 pi exp left i pi boldsymbol m cdot boldsymbol sigma right i so χ χ displaystyle chi mapsto chi and both the bloch vector and the projector are unchanged χ χ χ χ displaystyle chi chi dagger chi chi dagger a 2 π displaystyle 2 pi rotation therefore returns the same physical ray but changes the sign of the spinor representative a 4 π displaystyle 4 pi rotation gives u m 4 π exp i 2 π m σ i displaystyle u boldsymbol m 4 pi exp left i 2 pi boldsymbol m cdot boldsymbol sigma right i and returns the same spinor in an isolated single state the sign change under a 2 π displaystyle 2 pi rotation is an overall phase in interference experiments however a relative sign between two paths can have observable consequences relativistic quantum mechanics edit while non relativistic quantum mechanics defines spin 1 2 with 2 dimensions in hilbert space with dynamics that are described in 3 dimensional space and time relativistic quantum mechanics defines the spin with 4 dimensions in hilbert space and dynamics described by 4 dimensional space time citation needed observables edit as a consequence of the four dimensional nature of space time in relativity relativistic quantum mechanics uses 4 4 matrices to describe spin operators and observables citation needed history edit when physicist paul dirac tried to modify the schrödinger equation so that it was consistent with einstein s theory of rela...