If δ = nπ, whatever the integer n, one is sure to find the neutrons in the same spin state as in the initial beam.. We shall analyse it in the specific case of a neutron beam, where it can
Trang 1Comparing the values bn of question 3.3 with this experimental result,
and recalling the result of a measurement of µ x for these values, explain why this proves that the state vector of a spin-1/2 particle changes sign under a
rotation by an odd multiple of 2π.
3.4 Solutions
Section 3.1: Neutron Interferences
3.1.1 The beams ABDC2 and ACDC2interfere Omitting the propagation
factors, one has, at C2 an amplitude
A2= α2β + β3= β(α2+ β2) Similarly, for ABDC3and ACDC3,
A3= 2αβ2.
The intensities at the two counters are
I2= R − 4R2T I3= 4R2T
3.1.2 When there is a phase shift δ in C, the above expressions get modified
as follows:
A2= α2βe iδ + β3= β(α2eiδ + β2) , A3= αβ2(1 + eiδ )
The intensities become
I2= R − 2R2T (1 + cos δ) I3= 2R2T (1 + cos δ)
The fact that I2+ I3 does not depend on the phase shift δ is a consequence
of the conservation of the total number of particles arriving at D.
Section 3.2: The Gravitational Effect
3.2.1 This results from elementary trigonometry.
(a) Since there is no recoil energy of the silicon atoms to be taken care of, the
neutron total energy (kinetic+potential) is a constant of the motion in all the
process This energy is given by E AC = p2/2M and E BD = (p − ∆p)2
/2M +
M gH sin φ, hence
∆p M2gH sin φ/p
(b) The velocity √
2gH is of the order of 0.5 m/s, and the neutron velocity
is v = h/M λ 2700 m/s The change in velocity ∆v is therefore very small:
∆v = gH/v 2 × 10 −4 m/s for φ = π/2.
Trang 23.4 Solutions 43
3.2.2 (a) The gravitational potential varies in exactly the same way along
AB and CD The neutron state in both cases is a plane wave with momen-tum p = h/λ just before A or C The same Schr¨odinger equation is used to determine the wave function at the end of the segments This implies that the
phases accumulated along the two segments AB and CD are equal.
(b) When comparing the segments AC and BD, the previous reasoning does
not apply, since the initial state of the neutron is not the same for the two
segments The initial state is exp(ipz/¯ h) for AC, and exp[i(p − ∆p)z/¯h] for
BD After travelling over a distance L = AC = BD, the phase difference
between the two paths is
δ = ∆p L
¯
M2gλd2
π¯ h2 tan θ sin φ
3.2.3 >From the previous result, one has δ2− δ1 = A g (sin φ2− sin φ1),
where A = M2λd2tan θ/(π¯ h2) Therefore,
g = δ2− δ1
A (sin φ2− sin φ1) .
There are 9 oscillations, i.e (δ2− δ1) = 18π, between φ1 =−32 ◦ and φ2 =
+24◦ , which gives g 9.8 ms −2 The relative precision of the experiment was
actually of the order of 10−3.
Section 3.3: Rotating a Spin 1/2 by 360 Degrees
3.3.1 SinceB is along the z axis, the magnetic Hamiltonian is:
ˆ
H M =−µ · B0=¯h ω
2
0 −1
.
At time t, the spin state is
|Σ(t) = √1
2
e−iωt/2
e+iωt/2
.
By a direct calculation of µ or by using Ehrenfest theorem (d
dt µ =
1
i¯h [ˆµ, ˆ H] ), we obtain:
dµ x
dt = ω µ y dµ dt y =−ωµ x dµ dt z = 0
Initiallyµ x = µ0 andµ y = µ z = 0; therefore,
µ = µ0 (cos ωt u x + sin ωt u y )
Trang 33.3.2 When the neutrons leave the field zone, the probability of finding
µ x = +µ0is
P x (+µ0) =|+x|Σ(T )|2= cos2ωT
2 = cos
2δ
with T = l/v = lM λ/h.
3.3.3 The above probability is equal to 1 if δ = nπ (ωT = 2nπ), or B0= nb1
with
b1= 2π
2¯h2
µ0M lλ = 34.5 × 10 −4 T
For δ = nπ the magnetic moment has rotated by 2nπ around the z axis by
Larmor precession
3.3.4 The formulas are similar to those found in question 1.2 The phase of
the upper component of the spinor written in the{|+ z , |− z } basis, is shifted
by +δ, that of the lower component by −δ:
Amplitude at the counter C2: ei(p2·r−Et)/¯h √ β
2
β2+ α2eiδ
β2+ α2e−iδ
Amplitude at the counter C3: ei(p3·r−Et)/¯h αβ √2
2
1 + eiδ
1 + e−iδ
3.3.5 Since the measuring apparatus is insensitive to spin variables, we must
add the probabilities corresponding to S z =±1, each of which is the modulus
squared of a sum of amplitudes Altogether, we obtain the following intensities
of the total neutron flux in the two counters:
I2= R − 2R2T (1 + cos δ) , I3= 2R2T (1 + cos δ)
and
I2− I3= R − 4R2T (1 + cos δ)
3.3.6 There will be a minimum of I2−I3each time cos δ = +1, i.e δ = 2nπ.
This corresponds to a constructive interference in channel 3 On the other
hand, there appears a maximum if cos δ = −1, i.e δ = (2n + 1)π, and this corresponds to a destructive interference in channel 3 (I3= 0)
If δ = nπ, whatever the integer n, one is sure to find the neutrons in the same spin state as in the initial beam However, the interference pattern depends on the parity of n.
The experimental result ∆B = (64 ± 2) × 10 −4 T confirms that if the spin
has rotated by 4nπ, one recovers a constructive interference in channel 3 as in the absence of rotation, while if it has rotated by (4n + 2)π, the interference in
C3 is destructive The probability amplitude for the path ACD has changed
sign in this latter case, although a spin measurement in this path after the magnet will give exactly the same result as on the incoming beam
Trang 43.4 Solutions 45
References
A.W Overhauser, A.R Collela, and S.A Werner, Phys Rev Lett., 33, 1237 (1974); 34, 1472 (1975); 35, 1053 (1975) See also D Greenberger and A.W.
Overhauser, Scientific American, May 1980
Trang 5Spectroscopic Measurement
on a Neutron Beam
We present here a very precise method for spectroscopic measurements, due
to Norman Ramsey The method, using atomic or molecular beams, can be applied to a very large class of problems We shall analyse it in the specific case
of a neutron beam, where it can be used to determine the neutron magnetic moment with high accuracy, by measuring the Larmor precession frequency
in a magnetic field B0
A beam of neutrons is prepared with velocity v along the x axis The beam
is placed in a constant uniform magnetic field B0 directed along the z axis.
We write|+ and |− for the eigenstates of the z projection ˆ S zof the neutron
spin, and γ for the gyromagnetic ratio of the neutron: ˆ µ = γ ˆ S, ˆ µ being the
neutron magnetic moment operator, and ˆS its spin.
The neutrons are initially in the state|− When they approach the origin,
they cross a zone where an oscillating field B1(t) is applied in the (x, y) plane The components of B1 are
B 1x = B1e−r/a cos ω(t − z/c)
B 1y = B1e−r/a sin ω(t − z/c)
B 1z = 0 ,
(4.1)
where r =
x2+ y2 We assume that B1 is constant (strictly speaking it should vary in order to satisfy∇ · B = 0) and that B1 B0
In all parts of the chapter, the neutron motion in space is treated classically
as a linear uniform motion We are only interested in the quantum evolution
of the spin state
4.1 Ramsey Fringes
4.1.1 Consider a neutron whose motion in space is x = vt, y = 0, z = 0.
What is the Hamiltonian ˆH(t) describing the coupling of the neutron magnetic
moment with the fields B and B ?
Trang 648 4 Spectroscopic Measurement on a Neutron Beam
Setting ω0 =−γB0 and ω1 =−γB1, write the matrix representation of ˆ
H(t) in the basis {|+, |−}.
4.1.2 TreatingB1as a perturbation, calculate, in first order time-dependent perturbation theory, the probability of finding the neutron in the state |+
at time t = + ∞ (far from the interaction zone) if it was in the state |− at
t = −∞.
One measures the flux of neutrons which have flipped their spins, and are
in the state |+ when they leave the field zone This flux is proportional to the probability P −+that they have undergone the above transition.
Show that this probability has a resonant behavior as a function of the
applied angular frequency ω Plot P −+as a function of the distance from the
resonance ω − ω0 How does the width of the resonance curve vary with v and a?
The existence of this width puts a limit on the accuracy of the measurement
of ω0, and therefore of γ Is there an explanation of this on general grounds?
4.1.3 On the path of the beam, one adds a second zone with an oscillating
field B 1 This second zone is identical to the first but is translated along the
x axis by a distance b (b a):
B 1x = B1e−r /a cos ω(t − z/c)
B 1y = B1e−r /a sin ω(t − z/c)
B 1z = 0 ,
(4.2)
where r = ((x − b)2
+ y2)1/2
Show that the transition probability P −+ across the two zones can be
expressed in a simple way in terms of the transition probability calculated in the previous question
Why is it preferable to use a setup with two zones separated by a distance b
rather than a single zone, as in question 4.1.2, if one desires a good accuracy in
the measurement of the angular frequency ω0? What is the order of magnitude
of the improvement in the accuracy?
4.1.4 What would be the probability P −+ if one were to use N zones equally
spaced by the same distance b from one another? What optical system is this
reminiscent of?
4.1.5 Suppose now that the neutrons, still in the initial spin state |−, propagate along the z axis instead of the x axis Suppose that the length
of the interaction zone is b, i.e that the oscillating field is given by (4.1) for
−b/2 ≤ z ≤ +b/2 and is zero for |z| > b/2 Calculate the transition probability
P
−+in this new configuration.
For what value of ω is this probability maximum? Explain the difference
with the result obtained in question 4.1.2
Trang 74.1.6 In practice, the neutron beam has some velocity dispersion around the
value v Which of the two methods described in questions 4.1.3 and 4.1.5 is
preferable?
4.1.7 Numerical Application: The neutrons of the beam have a de Broglie
wavelength λn= 31 ˚A Calculate their velocity
In order to measure the neutron gyromagnetic ratio γn, one proceeds as in question 4.1.3 One can assume that the accuracy is given by
δω0=π 2
v
b .
The most accurate value of the neutron gyromagnetic ratio is currently
γn=−1.912 041 84 (±8.8 × 10 −7 ) q/M
p
where q is the unit charge and Mpthe proton mass In a field B0= 1 T, what
must be the length b in order to achieve this accuracy?
4.2 Solutions
4.1.1 The magnetic Hamiltonian is
ˆ
H(t) = −ˆµ.B = −γB0Sˆz + B 1x (t) ˆ S x + B 1y (t) ˆ S y
Since x = vt, y = z = 0,
ˆ
H(t) = −γB0Sˆz + B1 e−v|t|/a
ˆ
S x cos ωt + ˆ S y sin ωt
whose matrix representation is
ˆ
H(t) = ¯h
2
ω0 ω1exp(−v|t|/a − iωt)
ω1exp(−v|t|/a + iωt) −ω0
.
4.1.2 Let |ψ(t) = α(t)|+ + β(t)|− be the neutron state at time t The
Schr¨odinger equation gives the evolution of α and β:
i ˙α = ω0
2 α +
ω1
2 e
−iωt−v|t|/a β
i ˙β = ω1
2 e
iωt −v|t|/a α − ω0
2 β
We now introduce the variables ˜α and ˜ β:
˜
α(t) = α(t) e iω0t/2 β(t) = β(t) e˜ −iω0t/2 ,
whose evolution is given by
Trang 850 4 Spectroscopic Measurement on a Neutron Beam
i ˙˜α = ω1
2 e
i(ω0−ω)t−v|t|/a β˜
iβ =˙˜ ω1
2 e
i(ω −ω0)t −v|t|/a α ˜
The equation for ˜α can be formally integrated and it gives
˜
α(t) = ω1
2i
t
−∞
ei(ω0−ω)t −v|t |/a β(t˜ ) dt , (4.3)
where we have used the initial condition ˜α( −∞) = α(−∞) = 0 Now, since
we want the value of α(t) to first order in B1, we can replace ˜β(t ) by its
unperturbed value ˜β(t ) = 1 in the integral This gives
γ −+ ≡ ˜α(+∞) = ω1
2i
+∞
−∞
ei(ω0−ω)t −v|t |/a dt
= ω1v
ia
1
(ω − ω0)2+ (v/a)2 .
The transition probability is therefore
P −+=ω
2v2
a2
1
[(ω0− ω)2+ (v/a)2]2 .
Fig 4.1 Transition probability in one zone
The width of the resonance curve is of the order of v/a This quantity is the inverse of the time τ = a/v a neutron spends in the oscillating field From the uncertainty relation δE.τ ∼ ¯h, when an interaction lasts a finite time τ the accuracy of the energy measurement δE is bounded by δE ≥ ¯h/τ Therefore,
from first principles, one expects that the resonance curve will have a width
of the order of ¯h/τ in energy, or 1/τ in angular frequency.
Trang 94.1.3 In the two-zone case, the transition amplitude (in first order
pertur-bation theory) becomes
γ −+= ω2i1
+∞
−∞
ei(ω0−ω)t−v|t|/a dt + +∞
−∞
ei(ω0−ω)t−|vt−b|/a dt
.
If we make the change of variables t = t − b/v in the second integral, we
obtain
γ −+= ω2i1
1 + ei(ω0−ω)b/v +∞
−∞
ei(ω0−ω)t−v|t|/a dt ,
which is the same formula as previously but multiplied by 1 + ei(ω0−ω)b/v If
we square this expression, in order to find the probability, we obtain
P −+=4ω
2v2
a2
1
[(ω0− ω)2+ v2/a2]2 cos
2 (ω0− ω)b 2v
.
Fig 4.2 Ramsey fringes in a two-zone setup
The envelope of this curve is, up to a factor of 4, the same as the previous curve However, owing to the extra oscillating factor, the width at
half-maximum of the central peak is now of order πv/(2b) The parameter which now governs the accuracy is the total time b/v that the neutron spends in the
apparatus, going from one zone to the other
In spectroscopic measurements, it is important to locate the exact position
of the maximum of the peak Multiplying the width of the peak by a factor
a/b ( 1 since a b) results in a major improvement of the measurement
accuracy Of course one could in principle build a single interaction zone of large size∼ b, but it would be difficult to maintain a well controlled oscillating
field over such a large region From a practical point of view, it is much simpler
to use small interaction zones of size a and to separate them by a large distance b.
Trang 1052 4 Spectroscopic Measurement on a Neutron Beam
4.1.4 It is quite straightforward to generalize the previous results to an
arbitrary number of zones:
γ −+=ω2i1
1 + ei(ω0−ω)b/v+· · · + e i(N −1)(ω0−ω)b/v
×
+∞
−∞
ei(ω0−ω)t−v|t|/a dt
P −+=
ω2v2
a2
1
[(ω0− ω)2+ v2/a2]2
sin2[N (ω0− ω)b/2v]
sin2[(ω0− ω)b/2v] .
As far as amplitudes are concerned, there is a complete analogy with a dif-fraction grating in optics
The neutron (more generally, the particle or the atom) has some transition
amplitude t for undergoing a spin flip in a given interaction zone The total amplitude T is the sum
T = t + t e iφ + t e 2iφ + ,
where eiφ is the phase shift between two zones
4.1.5 We now set z = vt, and x = y = 0 for the neutron trajectory This
will modify the phase of the field (Doppler effect)
ω(t − z/c) → ω(1 − v/c)t = ˜ωt with ω = ω (1˜ − v/c)
and we must integrate the evolution of ˜α:
i ˙˜α = ω1
2 e
i(ω0−˜ ω)t β˜
(with ˜β 1) between ti =−b/(2v) and tf= b/(2v) The transition probability
is then
P
−+ = ω12
sin2[(ω0− ˜ω)b/(2v)]
(ω0− ˜ω)2 , which has a width of the order of b/v but is centered at
˜
ω = ω0⇒ ω = ω0
1− v/c ω0(1 + v/c)
Comparing with question 4.1.2, we find that the resonance frequency is dis-placed: The neutron moves in the propagation direction of the field, and there
is a first order Doppler shift of the resonance frequency
4.1.6 If the neutron beam has some velocity dispersion, the experimental
result will be the same as calculated above, but smeared over the velocity distribution
In the method of question 4.1.3, the position of lateral fringes, and the
width of the central peak, vary with v A velocity distribution will lead to