paulernest
Jan30-09, 06:00 AM
Hi
I'm having a lot of problems getting to grips with spinors. To this end
I have been considering the derivation given in the Feynman lectures for
the rotation matrices of a spin 1/2 particle. For those of you with a
copy, this is given in Volume III, lecture 6.
The derivation uses a though experiment consisting of two Stern-Gurlach
apparatus in sequence. A spin 1/2 particle passes along the x-axis
though the first apparatus, then is rotated though an angle theta about
the z-axis (particle remains in the xy-plain throughout), then passes
through the second apparatus. Both of the SG apparatus measure the
particles z component of angular momentum.
The argument is that as the particles z-component of angular momentum is
be being measured in both experiments, if the particle is found to be
spin up in the first experiment then it will be spin up in the second,
and the same being true if it is spin down. It is then argued that there
must be a relative change in phases of the two states.
Label the first SG apparatus S, and the second one as T. Basis states of
S are |S+> and |S-> for spin up and down respectively. Analogous basis
states are used for T.
Given a state |\psi>, then the probability of finding it spin up in T
must be the same as finding it spin up in S, therefore
|<T+|psi'>|^2 = |<S+|psi>|^2
|<T-|psi'>|^2 = |<S-|psi>|^2
from this we can conclude that the transformation matrix U(theta) for a
rotation by angle theta about the z-axis, takes the form
U(theta) = diag(e^{i f(theta)},e^{i g(theta)})
from arguments involving the continuity of this matrix as a function of
theta, we find that f and g must be linear in theta, with difference
between coefficients of 1, thus giving
U(theta) = diag(e^{i a theta},e^{i b theta})
where |a - b| = 1
my question is how do I show that a=-b, which gives the matrix the
correct form of
U(theta) = diag(e^{i theta/2},e^{-i theta/2})
Thanks for any pointers.
Paul
I'm having a lot of problems getting to grips with spinors. To this end
I have been considering the derivation given in the Feynman lectures for
the rotation matrices of a spin 1/2 particle. For those of you with a
copy, this is given in Volume III, lecture 6.
The derivation uses a though experiment consisting of two Stern-Gurlach
apparatus in sequence. A spin 1/2 particle passes along the x-axis
though the first apparatus, then is rotated though an angle theta about
the z-axis (particle remains in the xy-plain throughout), then passes
through the second apparatus. Both of the SG apparatus measure the
particles z component of angular momentum.
The argument is that as the particles z-component of angular momentum is
be being measured in both experiments, if the particle is found to be
spin up in the first experiment then it will be spin up in the second,
and the same being true if it is spin down. It is then argued that there
must be a relative change in phases of the two states.
Label the first SG apparatus S, and the second one as T. Basis states of
S are |S+> and |S-> for spin up and down respectively. Analogous basis
states are used for T.
Given a state |\psi>, then the probability of finding it spin up in T
must be the same as finding it spin up in S, therefore
|<T+|psi'>|^2 = |<S+|psi>|^2
|<T-|psi'>|^2 = |<S-|psi>|^2
from this we can conclude that the transformation matrix U(theta) for a
rotation by angle theta about the z-axis, takes the form
U(theta) = diag(e^{i f(theta)},e^{i g(theta)})
from arguments involving the continuity of this matrix as a function of
theta, we find that f and g must be linear in theta, with difference
between coefficients of 1, thus giving
U(theta) = diag(e^{i a theta},e^{i b theta})
where |a - b| = 1
my question is how do I show that a=-b, which gives the matrix the
correct form of
U(theta) = diag(e^{i theta/2},e^{-i theta/2})
Thanks for any pointers.
Paul