Majorana Spin-Flip Transition of Atom in Varying Magnetic Field(1932)

Oriented atoms in a variable magnetic field(∗)
Ettore Majorana
Summary. — The author calculates the probability of non-adiabatic processes when an oriented atomic beam passes close to a point where the magnetic field vanishes.
ψ = C1/2ψ1/2 + C−1/2ψ−1/2 .
The
state
is
therefore
essentially
defined
by
theቤተ መጻሕፍቲ ባይዱ
ratio
. C−1/2
C1/2
If the phases of ψ1/2 and ψ−1/2(∗) are chosen so as to obtain the normal representation
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direction. The probability of agreement for two states represented by the points P and P is given by
W (P, P ) = cos2 1 α , 2
where α is the angle P OP . The probability vanishes, i.e. the two states are orthogonal, when P and P are diametrically opposite. For j > 1/2 there is in general no direction along which the atom is oriented, i.e. with a value determined by angular momentum. In spite of this, an intrinsic geometric representation similar to the previous one is still possible. The only difference is that every state, rather than being represented by a single point, is represented by 2j points on the unit sphere. Indeed let us consider a generic state:
tang ϑ eiϕ = C−1/2 .
2
C1/2
If O is the center of the sphere, the vector radius OP defines the direction along
which the momentum in the state ψ has value 1/2. In the case j = 1/2 the most generic
rotational
state
then
corresponds
to
oriented
atoms
with
momentum
1h 2 2π
in
an
arbitrary
(2) P. Gu¨ttinger, “Z. Physik”, 73, 169 (1932).
(∗) In “Il Nuovo Cimento” it is erroneusly printed ψ1/2. (Note of the Editor, see also E. Amaldi,
On the other hand, we cannot obtain a zero field in a point on the trajectory of the molecular beam except by several trials, for example by using two orthogonal auxiliary fields that can be regulated independently; this makes it difficult to perform the experiment until fast means to detect the beam are available. Nevertheless we anticipate the discussion to better clarify the nature of the dynamical problem arising from the rotation of a magnetic atom in an arbitrarily varying magnetic field. Our calculations will show that both the classical and the quantum-mechanical treatment require the integration of the same differential equations. It follows that when the classic solution is known, as in the case of a uniformly rotating field with constant angular velocity —which is the problem treated by Gu¨ttinger—, the quantum solution can be immediately derived. For the problem that we will study later, namely the passing close to a point of vanishing field with a slowly varying gradient, the quantum treatment is mathematically more convenient. Also in this case to obtain the general solution it is enough to solve the simplest case j = 1/2.
Societ`a Italiana di Fisica
125
126
Scientific Paper no. 6
The problem has been discussed theoretically by Gu¨ttinger(2) in the case of a uniformly rotating field with constant intensity. In this paper we will suppose instead that the molecular beam passes close to a point where the magnetic field vanishes. This case is of special importance because, if the beam were to pass exactly by a point of zero field, all the atoms would invert their orientation.
(∗) Translated from “Il Nuovo Cimento”, vol. 9, 1932, pp. 43-50, by P. Radicati di Brozolo. (1) T. E. Phipps and O. Stern “Z. Physik”, 73, 185 (1932).
of angular momenta, the state ψ can be represented, as is known, in an invariant way by
a point P on a unit sphere whose spherical coordinates ϑ and ϕ are defined by(∗∗)
As is well known, an oriented atom in a slowly varying magnetic field follows adiabatically the direction, assumed to be variable, of the field. This is the cause of a phenomenon that has recently been observed: if a molecular beam emerging from a Stern-Gerlach experiment is passed through another Stern-Gerlach experiment, one does not observe any further splitting of the beam. The reason is the following: all the atoms have the same orientation since all of them have followed exactly the stray field inevitably existing in the region between the pole-pieces which are meant to produce the orientation of the beam and those which should test this orientation after a certain distance. However, Phipps has undertaken some experiments to observe a non-adiabatic variation of the field in this region; this requires that the field be sufficiently weak and the variation of its direction sufficiently fast so that its rotational frequency be comparable with the Larmor frequency. Since it is difficult to reduce the intensity of the field below a few gauss(1), it is necessary, with a velocity of the beam of the order of 105 cm/sec, that the direction of the field changes appreciably over a distance of a fraction of a millimeter. These are, therefore, very delicate experiments that have not yet produced any decisive result.