Calculation of the expectation values of the spin and the magnetic moment of the gamma photons created as a result of the electronpositron annihilation Mesude Saglam * and Ziya Saglam **
*Ankara University, Department of Physics, 06100 Tandogan, Ankara, Turkey ** Department of Physics, Faculty of Science, Aksaray University 68100, Aksaray, Turkey
Abstract We have calculated the expectation values of the spin and the intrinsic magnetic moment of the gamma photons created as a result of the electron-positron annihilation. We show that, depending on its helicity a gamma photon propagating in z direction with an angular frequency carries a magnetic moment of z (ec 2 ) along the propagation direction. Here the (+) and (-) signs stand for the right hand and left circular helicity respectively. We also show that whatever the helicity is, the spin of each gamma photon is equal to zero (but not 1 !). We argue that in a Stern-Gerlach experiment (SGE) the magnetic moment is an important property but not the spin of the particles. Because of these two symmetric values of the magnetic moment, we expect a splitting of the gamma photon beam into two symmetric subbeams in a (SGE). We believe that the present result will be helpful for understanding the recent attempts on the (SGE) with slow light and the behavior of the dark polaritons and also the atomic spinor polaritons.
Keywords: Magnetic moment, quantum flux, gamma photons, canonical angular momentum, electron-positron annihilation, right (left) hand circular helicity.
Corresponding Author: Mesude Saglam Address: Ankara University, Department of Physics, 06100-Tandogan, Ankara, Turkey e-mail:
[email protected] Fax: + 90 312 223 2395
1 Introduction In an earlier study [1] we calculated the intrinsic quantum flux of gamma photons created as a result of the electron-positron ( e - e ) annihilation. By using the flux conservation rule, which is an element of the conservation of the canonical angular momentum, we found that each gamma photon carries an intrinsic quantum flux of 0 h e along the propagation direction. Here the (+) and (-) signs stand for the right and left hand circular helicity respectively. The aim of the present work is to calculate expectation values of the spin and the intrinsic magnetic moment of the gamma photons created as a result of the electronpositron annihilation. We show that, depending on its helicity a gamma photon propagating in z direction with an angular frequency , carries a magnetic moment of z (ec 2 ) along the propagation direction. Here the (+) and (-) signs stand for the right hand (rh) and left hand (lh ) circular helicity respectively. But on the other hand the expectation values of the spin of each gamma photon is found to be equal to zero (but not 1 !). We believe that the present result is valid for any photon with an angular frequency . Therefore we hope that, the present result will be helpful for understanding the recent attempts on the Stern-Gerlach experiment (SGE) with slow light and the behavior of the dark polaritons and also the atomic spinor polaritons [2-4]. The first experiment about the magnetic moment of a photon was done in 1896 by Zeeman [5] who discovered the spectral lines from sodium, when it was put in a flame, were split up into several components in a strong magnetic field. A long period after Zeeman, experimental attempts about photon’s magnetic moment through the (SGE) with photons have started recently by Karpa et al [2-4]. SGE was devised in 1922 to show that electron has a non-zero magnetic moment [6] and later it was also extended to the nuclei [7]. In general, in the SGE the magnetic field gradient serves as a detector for the particle’s magnetic moment vector: If the magnetic field is a non uniform field along the z direction such as B Bz ( z ) zˆ , then because of the torque ( B ) on the particles, the magnetic moment vector can have z-component only: z zˆ ( where z 0 or z 0 ). On the other hand the potential energy U B produces a force of F z (Bz z ) zˆ which has quantized values as z . Depending on whether ( z 0 or z 0 ), the particles are deflected upward or downward. In the SGE with electron, the electron beam was deflected into two sub-beams which mean that z takes only two possible quantized values which are equal to z B (e 2m0 ) where B is the Bohr magneton. Because of the negative charge of the electron, the relation between spin and the magnetic moment was set as g B S where g is the Lande-g factor which is equal to 2 and S is the spin angular momentum vector. So in the SGE with electron, the z component of spin ( S z ) takes only two possible values which are: S z 1 2 . Therefore if (Bz z ) 0 then the spin-down electrons will be deflected upward while the spin-up ones will be deflected downward. Similarly if we had SGE with a positron beam, because of the positive charge, the spin-down positrons will be deflected downward while the spin-up ones will be deflected upward. But when it comes to the gamma photons, which are the composite particles made up of an electron and a positron, the direction of the deflection is determined by the sign of the z . Therefore the magnetic moment of the gamma photon is an important property (not spin) to determine the direction of the
deflection. We have calculated the magnetic moment of a gamma photon and found that the z -component of the magnetic moment is equal to z (ec 2 ) where is the angular frequency of photon. Here the (+) and (-) signs correspond to (rh) and (lh ) circular helicity respectively. On the other hand the spin calculations show that whatever the helicity is, the spin of the gamma photon is equal to zero (but not 1 !). Electron-positron ( e - e ) annihilation process has been known for almost about 75 years [1, 8]. After the collision we have two gamma photons with the same energy but with different helicities. In this collision we have the conservation of charge, energy, linear momentum and the conservation of the canonical angular momentum which requires the conservation of the total magnetic moments and the total quantum flux as well[1]. Magnetic flux quantization has been known for more than 60 years [9]. Recently Saglam and Boyacioglu [10], with a semiclassical method, calculated quantized magnetic fluxes through the Landau orbits of an electron including the effect of the spinning motion. They showed that depending on the spin orientations, the spin contribution to the quantized magnetic flux of electron is equal to ( 0 2) . Here ( 0 h e) is the flux unit and equal to 4.14 x1015T .m 2 . Similarly it was shown by Saglam and Sahin [1] that, depending on the spin orientations, the spin contribution to the quantized magnetic flux of positron is equal to ( 0 2) . The aim of the present work is to calculate the the expectation values of spin and the intrinsic magnetic moment of the gamma photons created as a result of the electronpositron annihilation. We believe that the present work will be helpful to understand the recent Stern-Gerlach experiment of photons [2-4] in detail. In the theoretical side, the present work will bring a new insight to the quantum mechanics as well.
2 The intrinsic magnetic moments of an electron and a positron The definition of the magnetic moment vector ( ) for a free electron ( e ) and a free positron ( e ) are given by [1]
e g B S
(1)
e g B S
(2) Here g is the Lande-g factor which is equal to 2 for both particles and S is the spin angular momentum. Depending on the spin orientation, the z- components of the magnetic moments for spin-up and down electron are:
z ( e )
z ( e )
e B 2m0
(3)
e B 2m0
(4)
Similarly the related z- components of the magnetic moments for spin-up and down positron are:
z ( e ) z ( e )
e B 2m0
(5)
e B 2m0
(6)
3 Calculation of the expectation values of spin of gamma photons To calculate the expectation values of the spin and the magnetic moment of a gamma photon, our starting point will be the electron-positron annihilation process ending with the creation of two gamma photons with right and left hand circular helicities. After the collision we will have two photons with the same energy E m0 c 2 (kc hc / ) but with different helicities. Just before the collision the relative spin orientation of controlled by the Heisenberg exchange Hamiltonian [11]: U exc 2 JS1 S2
e and
e will be (7)
Where S1 , S 2 are the spin vectors and J is the exchange integral. If we had two electrons instead, the exchange energy together with the Pauli exclusion principle requires that the spins of the electrons must be antiparallel. But for colliding ( e , e ) system, the minimum energy is obtained for parallel spins. Therefore just before the collision the total z-component of the spin of the colliding ( e , e ) system will be 1 . That means there are two possibilities:
a) The electron is in spin-down state and the positron is in spin-down state. b) The electron is in spin-up state and the positron is in spin-up state. In Dirac notation the states (a) and (b) can be defined as:
a e e
(8a)
b e e
(8b)
which build an orthonormalized set. Namely
dV dV
2
e e e e 1
(8c)
dV dV
2
e e e e 1
(8d)
dV dV
2
e e e e 0
(8e)
e e e e 0
(8f)
* a a
* b b
* a b
1
1
1
dV dV * b a
1
2
where dV1 and dV2 are the volume elements for electron and positron respectively. The expectation values of the total z-components of the spin eigenstates (a) and (b) given in (8a, 8b) are:
S
z
[( S1 ) z ( S2 ) z ] for the
( S z ) a a* S za dV1dV2 a* [( S1 ) z ( S 2 ) z ]a dV1dV2 e e [( S1 ) z ( S 2 ) z ] e e
(9a)
e ( S1 ) z e e e e e e ( S 2 ) z e
1 1 1 2 2
( S z ) b b* S zb dV1dV2 b* [( S1 ) z ( S 2 ) z ]b dV1dV2 e e [( S1 ) z ( S 2 ) z ] e e
(9b)
e ( S1 ) z e e e e e e ( S 2 ) z e
1 1 1 2 2
where S1 and S 2 are the spins of electron and positron respectively. Since the states (a) and (b) are equally probable, the total wave function i before the collision will be given by [1]:
i
1 1 (a b ) ( e e e e ) 2 2
(10)
which corresponds to the quantum entanglement of the states a e e
and
a e e . In passing as far as the calculations of the expectation values are concerned, because of the orthonormalization conditions given by (8c-8f), any combinations of a and b such as (1 2 ) (a b ) will give us the same result. The expectation value of
S
z
for the initial state i in (10) is:
1 (a* b* ) S z (a b )dV1dV2 2 1 1 { a* S za dV1dV2 b* S zb dV1dV2 } ( 1 1) 0 2 2
S
z initial
i* S z i dV1dV2
(11)
where we used the results of (9a, 9b). As was shown by Saglam and Sahin[1], the total wave function, f of the system after the collision,
will
correspond to
another
entanglement
of the
states e , e ,
e and e provided that the conservation of charge, energy, linear momentum and
the canonical angular momentum are met. Concerning the spin, the total z-component of the
spin of the final state must be zero again. The possible entanglements that fulfill the above conditions will be [1]:
f
1 ( e e e e ) 2
(12)
which corresponds to two different gamma photons created with opposite circular helicities. After the collision the eigenstates corresponding to the right hand and left circular helicities are:
Here a' and b' respectively.
a' e e
(13a)
b' e e
(13b)
corresponds to the gamma photons with (rh) and (lh ) circular helicity
As we did in (9a, 9b), the expectation values of the total z-components of the spin S z [( S1 ) z (S2 ) z ] for the eigenstates a' and b' of (13) are:
( S z ) 'a ' a S z ' a dV1dV2 ' a [( S1 ) z ( S 2 ) z ] ' a dV1dV2 *
*
e e [( S1 ) z ( S 2 ) z ] e e
(14a)
1 1 e ( S1 ) z e e e e e e ( S 2 ) z e 0 2 2 ( S z ) 'b ' b S z ' b dV1dV2 ' b [( S1 ) z ( S 2 ) z ] ' b dV1dV2 *
*
e e [( S1 ) z ( S 2 ) z ] e e
(14b)
e ( S1 ) z e e e e e e ( S 2 ) z e
1 1 0 2 2
(14a) and (14b) state that, whatever the helicity is, the spin of the gamma photon is equal to zero (but not 1 !). Finally we calculate expectation value of
S
z
for the final state f :
* * 1 ( ' a ' b ) S z ( ' a ' b )dV1dV2 2 * * 1 1 1 { ' a S z ' a dV1dV2 ' b S z ' b dV1dV2 } [( S z ) 'a ( S z ) 'b ] (0 0) 0 2 2 2
S
z
final
*f S z f dV1dV2
(15)
where we used (14a, 14b). If we compare the expectation value of S z before and after the collision [(11) and (15)], we see that the conservation of the spin angular momentum is fulfilled.
4 Calculation of the expectation values of the magnetic moment of gamma photons: To calculate the expectation values of the magnetic moment of the gamma photons, we first consider the expectation values of the magnetic moment before the collision. The expectation values of the z-components of the magnetic moment vectors for the eigenstates ( a , b ) of the initial entangled state ( i ) can be calculated from (3-6) and (8a,b) : ( z ) a
e e 0 2m0 2m0
( z ) b
e e 0 2m0 2m0
(16a)
(16b)
Similarly by using (3-6) and (13a, 13b), the expectation values of the z-components of the magnetic moment vectors for the eigenstates ( a' , b' ) of the final entangled state f : ( z ) 'a
e e e 2 B 2m0 2m0 m0
( z ) 'b
e e e 2 B 2m0 2m0 m0
(16c)
(16d)
(16c) and (16d) give us the expectation values of the z-components of the magnetic moment vectors of the gamma photons with (rh) and (lh ) circular helicity respectively. Because in the eigenstate a' e e
we have (rh) circular helicity while in the eigenstate
b' e e we have (lh ) circular helicity [1]. Substituting the relation ( m0c 2 ) in (16c) and (16d), we can write z-components of the magnetic moments of the gamma photons in terms of the angular frequency, :
z ( rh)
ec 2
z (lh )
ec 2
(17a)
(17b)
Next by using (16a-16d) we can calculate the expectation value of
z
(the total z-
component of the magnetic moment) for the initial entangled state, i and the final entangled state, f given in (10,12) :
1 1 z initial [( z ) a ( z ) b ] (0 0) 0 2 2
(18a)
1 1 z final [( z ) 'a ( z ) b' ] (2 B 2 B ) 0 2 2
(18b)
If we compare the expectation value of z for the initial state and the final state (18a,b), we see that conservation of the total magnetic moment is also fulfilled. Therefore, the electron-positron annihilation resulting with the creation of two gamma photons simply corresponds to the transition from the initial entangled states i to the final entangled states f ; Namely:
1 1 e e e e e e e e . 2 2
(19)
At present we are not able to decide about the correct replacement of ( ) in both wave functions in (19). This is a well known problem of quantum mechanics. The answer can be given only in the future after the discovery of the so called hidden variables [12-15].
5 Conclusions We have calculated the expectation values of the spin and the intrinsic magnetic moment of the gamma photons created as a result of the electron-positron annihilation. We show that, whatever the helicity is, the spin of the gamma photon is equal to zero (but not 1 !). We also show that, depending on its helicity a gamma photon propagating in z direction with an angular frequency carries a magnetic moment of z (ec 2 ) along the propagation direction. Here the (+) and (-) signs stand for the right hand and left circular helicity respectively. Because of these two symmetric values of the magnetic moment, we expect a splitting of the photon beam into two symmetric subbeams in a Stern-Gerlach experiment. The splitting is expected to be more prominent for low energy photons. We believe that the present result will be helpful for understanding the recent attempts on the Stern-Gerlach experiment with slow light and the behavior of the dark polaritons and also the atomic spinor polaritons.
References 1. Saglam, M. and Sahin, G.: Photon in the frame of the current loop model. Int. J. Mod. Phys. B.23 (24) 4977-4985 (2009). 2. Karpa, L. and Weitz, M.: Slow light in inhomogeneous and transverse fields. New J. Phys. 10 045015-045026 (2008). 3. Karpa, L. and Weitz, M.: Deflection of slow light in an a Stern-Gerlach magnetic field. IEEE, 17 11 (2007). 4. Karpa, L. and Weitz, M.: Slow light in inhomogeneous and transverse fields. Nature Physics, 2 332-333 (2006). 5. Zeeman, P.: The effect of magnetization on the nature of light emitted by asubstance. Nature 347347(1897). 6. Gerlach, W. and Stern, O.: Das magnetische moment des silberatoms. Zeitschrift für Physics, 9 353-355 (1922). 7. Sidles, J.A.: Folded Stern-Gerlach experiment as a means for detecting nuclear magnetic resonance in individual nuclei.Physical Review Letters, 68 1124-1127 (1992). 8. Griffiths, D.: Introduction to Elementary Particles. John Wiley and Sons, Canada.(1987). 9. London, F.: Superfluids. John Wiley and Sons(1950). 10.Saglam, M. and Boyacioglu, B.: Calculation of the flux associated with the electron’s spin on the basis of the magnetic top model. Int. J. of Mod. Phys. B, 16, 607-614 (2002). 11.Slater, J.C .: Atomic shielding constants. Phys.Rev. 36, 57-64 (1930). 12.Einstein, A., Podolsky, B. and Rosen, N.: Can quantum mechanical description of physical reality be considered complete? Physical Review, 47, 777-780 (1935). 13. Bohr, N.: Can quantum mechanical description of physical reality be considered complete? Physical Review, 48, 696-702 (1935). 14. Schrödinger, E.: Discussion of Probability Relations between Separated Systems. Proceeding of the Cambridge Philosophical Society, 31, 555-562 (1935). 15. Fine, A.: Hidden Variables, Joint Probability and the Bell Inequalities.Physical Review Letters 48, 291-295 (1982).
