Relativistic Stern-Gerlach Interaction in an RF Cavity Page: 4 of 13
This report is part of the collection entitled: Office of Scientific & Technical Information Technical Reports and was provided to UNT Digital Library by the UNT Libraries Government Documents Department.
Extracted Text
The following text was automatically extracted from the image on this page using optical character recognition software:
with
I C~=2 [( mP all j j+3 (8E+ 2Th~j
spinj charged particle[2]. Let us introduce the Dirac Hamiltonian
H=ck+cS.Y- e)yomc2 (12)
having made use of the Dirac's matrices
_,(_2O g) 'Y=( ' 5=m_ T (_ cl' (13)
7 - 0 ' Z 0 0I , () - 7= -6 0
where 6 is a vector whose components are the Pauli's matrices
0 -i _1 0 0 1
(X (i 0) 0 -1), '- z=(1 0) (14)
I is the 2 x 2 identity matrix, 0 the null matrix and having chosen the y-axis parallel to
the main magnetic field. A standard derivation leads to the non relativistic expression
of the Hamiltonian exhibiting the SG interaction with the "normal" magnetic moment
H = e + -1(- eA)2 _ (d - B) (15)
which coincides with the Pauli equation and is valid in the PRF.
To complete the derivation we must add the contribution from the anomalous
magnetic moment to the SG energy term in the previous equation, with a factor
1 + a = , yielding
- a 2 - "B = - t* B with * = g40. (16)
In order to obtain the z-component of the SG force in the Laboratory frame along
the direction of motion of the particle, we must boost the whole Pauli term of Eq.(15)
by using the unitary operator U in the Hilbert space[4], which expresses the Lorentz
transformation
U-1 [g (Yom . ')s U = g 4 (yod f r') [S-1(4oa,)S + S-' (ioa)S + S-' (yoo)S]
(17)
that can be written in terms of the equivalent transformation in the 4 x 4 spinor space
S = exp 170(l - 5)2} = cosh 2 + ( 2 )sinh (18)
with
t= , coshu = = y = Lorentz factor, sinhu = VVW2 (_ 3 . (19)3
Upcoming Pages
Here’s what’s next.
Search Inside
This report can be searched. Note: Results may vary based on the legibility of text within the document.
Tools / Downloads
Get a copy of this page or view the extracted text.
Citing and Sharing
Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.
Reference the current page of this Report.
Conte,M.; Luccio, A. U. & Pusterla, M. Relativistic Stern-Gerlach Interaction in an RF Cavity, report, May 1, 2009; [Upton, New York]. (https://digital.library.unt.edu/ark:/67531/metadc932801/m1/4/: accessed March 29, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.