4. Supplementary Notes on Time and Space Evolution of a Neutrino Beam


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1 Lecture Notes for Quantum Physics II & III 8.05 & Academic Year 1996/ Supplementary Notes on Time and Space Evolution of a Neutrino Beam c D. Stelitano 1996 As an example of a twostate system in quantum mechanics, we discuss the oscillations of neutrinos produced in the decay of π + ; such oscillations would occur only if neutrinos had nonvanishing masses, which still has to be experimentally determined. The energy of the neutrinos in this case is mainly kinetic, since their velocity is close to c, the speed of light. The dynamics of relativistic particles is a subject for a Quantum Field Theory course, and Quantum Mechanics is not able to analyze it properly; nevertheless, in this particular case, we can deduce a sensible result by introducing some reasonable approximations. We shall start from the time dependent Schrödinger equation i h ψ (r; t =Hψ(r; t (1 t and guess that the Hamiltonian for a relativistic free particle might be given by the following expression (resembling the classical relativistic energy so that in the Schrödinger picture we would have H = ˆp 2 c 2 + m 2 c 4 (2 i h t ψ (r; t = h 2 c m 2 c 4 ψ(r; t (3 A square root of an operator gives us quite a lot of troubles; we shall instead iterate the above equation and exploit the fact that the LHS commutes with the RHS, obtaining the following h 2 2 t 2 ψ (r; t =( h 2 c m 2 c 4 ψ (r; t (4 which is indeed one of the most famous (and early equation of relativistic quantum mechanics: the KleinGordon equation. Moving the LHS to the RHS 1
2 and dividing by h 2 c 2,weget ( 1 2 c 2 t m2 c 2 h 2 ψ (r; t = 0 (5 after defining the d Alambertian operator = 1 2 c 2 t 2 2 (6 we are able to rewrite the KleinGordon equation in a more compact and familiar way ( h = c =1 ( + m 2 ψ(x µ = 0 (7 where x µ =(ct, x, y, z. We shall give here an argument in onespace and onetime dimension (1 + 1 dimension, but it is clear that the threespace and onetime dimension (3 + 1 dim generalization is straightforward. In the dim counterpart of (4 h 2 2 ψ(x; t = ( h 2 c 2 2 t2 x 2 + m2 c 4 ψ(x; t (8 we separate the variables x and t by means of the usual ansatz (the RHS does not depend on t ψ(x; t =ϕ(xφ(t (9 After introducing a suitable constant E 2 (the square of the eigenenergy, we are led to two equations for the time and the spatial dependence; respectively h 2 2 t 2 φ(t =E2 φ(t (10 which is solved, for example, by 1 φ(t =e iet/ h (11 and ( h 2 c 2 2 x 2 + m2 c 4 ϕ(x =E 2 ϕ(x (12 which we shall recast as follows 2 x 2 ϕ(x = 1 h 2 ( E 2 m 2 c 4 c 2 ϕ(x (13 1 Note that the other choice of sign, φ(t = exp(+iet/ h corresponds to negative energy, which is problematic. This peculiar feature of the KleinGordon equation is only resolved in quantum field theories that include both particles and antiparticles. For the present, we ignore the negative energy solutions. 2
3 Upon defining ( E p 2 2 m 2 c 4 = c 2 (14 we are able to present it simply as ( 2 2 x p h 2 ϕ(x = ϕ(x (15 and solve it by means of wellknown eigenfunctions ϕ(x =e ipx/ h (16 Collecting the last expressions for φ and ϕ, we shall regard ψ(x; t =e ipx/ h e iet/ h (17 as the time evolution of the eigenfunction corresponding to the squared eigenenergy E 2. Another way of discussing the timeevolution of the wavefunction consists in formally introducing a timeevolution operator [ Û(t, t o = exp i ˆp ] 2 c 2 + m 2 c 4 (t t o / h (18 which in the Schrödinger representation reads [ ] Û S (t, t o = exp i h 2 c 2 2 / x 2 + m 2 c 4 (t t o / h (19 so that the timeevolution of a generic wavefunction is given by x ψ(t = x Û(t ψ(0 = ÛS(t x ψ(0 ÛS(t x ψ o (20 We shall now apply the previous arguments to the quantummechanical description of neutrino oscillations. Muon (ν µ and electron neutrinos (ν e are related to the neutrino mass eigenstates (ν 1 and ν 2 by means of the following linear superpositions x ν 1 o = cos ϑ x ν e o sin ϑ x ν µ o x ν 2 o = sin ϑ x ν e o + cos ϑ x ν µ o (21 where the subscript denotes the initial time t = 0. Inverting the last equations and applying the timeevolution operator to both sides of the resulting equations leads us to express the timedependent wavefunctions for the electron and muon neutrinos (eigenstates of the weak interaction in terms of those ones for the free relativistic Hamiltonian eigenstates: ν 1 and ν 2 x ν e (t = ÛS(t x ν e o = cos ϑ x ν 1 (t + sin ϑ x ν 2 (t x ν µ (t = ÛS(t x ν µ o = sin ϑ x ν 1 (t o + cos ϑ x ν 2 (t o (22 3
4 We are discussing the case in which no electron neutrino is present at the initial time x ν e o = 0 (23 and the muon neutrino flux, originating from the decay of the π +, is described by a wave packet such as x ν µ o = Φ(pe ipx/ h (24 where Φ(p is a very welllocalized function centered in p o, the momentum transferred to ν µ during the decay process. By replacing the last two conditions in (21, we get the initial wavefunctions for the eigenstates of the free Hamiltonian x ν 1 o = sin ϑ Φ(pe ipx/ h x ν 2 o = cos ϑ Φ(pe ipx/ h (25 Let us discuss the propagation of ν 1 and omit the details for the case with ν 2 since they are very similar. By applying here either the timeevolution operator or the argument for the decomposition in timedependent eigenstates, we get x ν 1 (t = ÛS(t x ν 1 o = sin ϑ Φ(pe ipx/ h e ic p 2 +m 2 1 c2 t/ h (26 the expansion of the square root truncated to the first two terms reads p2 + m 2 c 2 p + m2 c 2 2p (27 so that x ν 1 (t sin ϑ Φ(pe ipx/ h e ic(p+ m 2 1 c2 2p t/ h (28 Notice at this point that Φ(p is peaked around p o (as if it were a narrow Gaussian function and causes the integrand to practically vanish outside a small neighbour of p o. In this interval the oscillations of the phase exp ( i m2 1 c3 t are [ ip(x ct h negligible with respect to exp phase as a constant and take it out of the integration x ν 1 (t sin ϑ exp ( i m2 1c 3 t ] ; we shall therefore regard the former Φ(pe ip(x ct/ h (29 4
5 we may also say that, from a physical point of view, it has just been assumed that the neutrinos produced in the decay process are moving at the light speed. Let us now introduce a more compact notation for the freely propagating wavepacket (with c velocity ψ f (x ct Φ(pe ip(x ct/ h (30 ψ f satisfies the following normalization condition dx ψ f (x ct 2 = 1 (31 and by means of it we can express the temporal evolution of the eigenstates of the free relativistic Hamiltonian as follows x ν 1 (t sin ϑ exp ( i m2 1 c3 t ψ f (x ct x ν 2 (t cos ϑ exp ( t ψ f (x ct (32 If we wish to evaluate the probability of finding an electron neutrino (revealed for example through another weak interaction process at a later time, we need the timedependent wavefunction for ν e as given in (22, that in the present case reads [ x ν e (t cos ϑ sin ϑ exp ( i m2 2c 3 t exp ( i m2 1c 3 ] t ψ f (x ct (33 We shall now simply assume that the probability density of finding an electron neutrino at the position x at time t is given by dp νe (x; t = x ν e (t x ν e (t = x ν e (t 2 (34 If we sit at a distance x o far from the neutrino source and wait for a ν e to pass by in the timeinterval dt about 2 t o = x o /c, then a length cdt of the wavefunction will cross the position x o in the interval dt; hence the probability of observing an electron neutrino in the finite timeinterval ɛ will be P νe (x o ;[t o ɛ, t o + ɛ] = t o+ɛ t cdt x o ɛ o ν e (t 2 = (cos ϑ sin ϑ ( 2 t o+ɛ exp t cdt o ɛ t exp ( i m2 1 2 c3 t ψf (x o ct 2 (cos ϑ sin ϑ 2 exp ( c x o exp ( i m2 1 c3 2 hp x 2 oc o to+ɛ t o ɛ cdt ψ f (x o ct 2 (35 2 t o is approximately the moment we expect the maximum of the neutrino wavefunction to be at x o. 5
6 Note that in the last approximation the slowoscillating phase has been moved out of the integration as we did above. Neglecting the tails of the very welllocalized wavepacket, which is essentially zero except when t = t o because of the form of ψ f,wehave to+ɛ t o ɛ cdt ψ f (x o ct 2 1 (36 and the familiar result for the probability of observing an electron neutrino at the distance x o is recovered by replacing E p o c ( P νe (x o = (cos ϑ sin ϑ 2 exp 2 hp x oc o exp ( i m2 1 c3 2 hp x 2 oc o ( (cos ϑ sin ϑ 2 exp 2 he x o exp ( i m2 1 c3 2 he x 2 o (37 6
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