External Quantum of Magnetic Moments in

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C H A P T E R 3

Quantum Mechanics of

Magnetic Moments in External Fields

1. The Equation of Motion

A. The Schrödinger Equation

In the Schrödinger representation of quantum mechanics, the time development of a quantum mechanical system is governed by the equa- tion

3Ψ

ί-8Γ = *Ψ> (1-1)

where Ψ is the state vector and the hamiltonian operator.¹ Relative to a complete orthonormal basis {u^}, the state vector may be expressed as

m

=

^{% c} , m , (1-2)

3

and the hamiltonian operator may be represented by an hermitian matrix:

Jtjk = (

^u

i >

^{^}^u^k)'y

= ^ki

^{· (1.3)}

T h e Uj are assumed to be independent of time, so that the time depend- ence of Ψ is entirely reflected by the time variation of the expansion coefficients C j .

T h e general theory of quantum mechanics deals with infinite-dimen- sional vector spaces, but the quantum mechanical analysis of magnetic

1 H e r e , a n d s u b s e q u e n t l y , d e n o t e s t h e e n e r g y o p e r a t o r d i v i d e d b y h.

75

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moments interacting with magnetic fields requires only finite-dimensional vector spaces. It will be assumed, therefore, that the basis contains η elements (n finite). T h e hamiltonian operator will be represented by an η χ η hermitian matrix and the state vector by an w-dimensional column vector. In specific problems, the Uj will be identified with the elements of a basis for an appropriate w-dimensional spin space. However, to ensure complete generality in the formulas to be developed in this section, the notation {ud) will be used to denote a basis whose elements will not be assumed to be eigenvectors of any operator unless an explicit statement is made to the contrary.

The time dependence of the expansion coefficients may be deduced from (1.1) upon noting that the effect of on any element of the basis can always be expressed as a linear combination of the :

^u

^s

= %^

^xi

u

^x ( ; = 1,2,...,«), (1.4)

1=1

with the

3tf

^xi given by (1.3). Substituting (1.2) and (1.4) in (1.1), and taking the scalar product of the resulting equation with u^k , one obtains

η

ià

^k

=^^

^k

jCj (k = 1, 2, ..., η). (1.5) 3=1

This set of equations can be written as the matrix equation

f

?

ⁿ

Μ

²

,

^(1.6)

Kin/ \ J

which is the matrix representation of (1.1) relative to the basis {uj}. It must be emphasized that (1.1) is a symbolic form for the equation of motion and is valid in all coordinate systems (bases); the matrix repre- sentation of the Schrödinger equation will be different relative to different bases.

For a given hamiltonian operator and a given basis, the time depend- ence of Ψ is obtained by solving η simultaneous first-order linear differential equations. T h e arbitrary constants of integration are given by the values of the components c^k(t) at some initial moment, say t = ' 0.

T h e system (1.5) always has one integral which can be easily deduced by introducing the dual of Ψ:

C l

\ w

^u

»

^V2

y t = (C l*

C*

^(1.7)

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1. T H E EQUATION OF M O T I O N 77

From (1.3) and (1.5) it is easy to show that satisfies

ί - - = - Ψ ^ . (1.8) Multiplying (1.1) from the left with Ψ\ and adding the result to the

equation obtained by multiplying (1.8) from the right with Ψ, one finds that

|(^) = o,

or

Ψ*Ψ = 2) I Φ)\2 = constant. (1.9) If the state vector is normalized, so that

3

= 1, (1.10).

the I Cj(t)\2 may be interpreted as the fractional contribution of Uj to the state Ψ at time t. Geometrically speaking, (1.10) requires the terminus of the state vector to lie on the surface of the unit hypersphere about the origin of an w-dimensional complex vector space.

B. Integration of the Schrödinger Equation

T h e integration of equations (1.5) is a relatively simple problem when is not an explicit function of time. Moreover, if the basis is such that each u^k is an eigenvector of J^, the integration is trivial. For if

J?u^k = Q^ku^ky (1.11)

then the matrix for Jf7 is diagonal,

(

^Ω¹ 0 ··· 0 \

° Ω* ? , (1.12)

0 0 .·· ü j and equations (1.5) reduce to

ick=Qkck. (1.13)

T h e condition dJfjdt = 0 requires the energy eigenvalues to be independent of time, so that equations (1.13) may be integrated to

c^k(t) = e-*°**c^k(0). (1.14)

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nt) = Xc^k(0)e-^u^ky (1.15) or, in matrix form,

ο-ιΩΛ Φ) \ /e~

^ißit

0

(1.16)

T h e solution (1.15) expresses the state vector as a linear superposition of the eigenvectors of Jf, but Ψ(ί) is not itself an eigenvector of unless all ck(0) = 0 except one, say £;(0). In this circumstance,

W(t) = φ)β-°^ .

Physically, this means that an energy measurement would yield the value Qj with certainty, since the normalization condition requires

I^cj(^t)\^{2 =} 1· O^{n t ne} other hand, if all c^k are nonzero, an energy measure- ment will yield the values Ω¹ , Ω² , with probabilities | ^(ΐ)\², I^c2(t)1²> ··· · ^n a ny c a s e> t ne probability distribution does not change with time, since | ck(t)\² = \ ^ ( 0 ) |2 for all k.

T h e basis which reduces the matrix representative of a time- independent hamiltonian operator to diagonal form is called the energy representation or the Heisenberg coordinate system. This basis is seldom obvious in any given problem, whereas some other basis may be suggested by the specific nature of the problem. T h e integration of equations (1.5) can be reduced to the case just considered by transforming the initial basis into a basis for the Heisenberg coordinate system. Let C(t) and C\t) denote the column-vector representations of Ψ(ΐ) relative to the initial basis and the Heisenberg coordinate system, respectively. Let Τ denote the matrix of the transformation relating these bases. C(t) satisfies (1.6), and the substitution

C(t) = TC\t) (1.17) transforms (1.6) to

'Ί^Γ

⁼⁽^τ^~ ¹^{^} ^τ⁾^σ^·^(1.18)

When the hamiltonian matrix is subjected to a similarity transformation with Τy the latter reduces the former to diagonal form:

TtfT = T-WT = {Q^k o^kj)^y (1.19) and (1.5) is reduced to the form (1.13).

It follows that

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1. T H E EQUATION OF M O T I O N 79

C. Operational Solution of the Schrödinger Equation

T h e square matrix in (1.16) satisfies all the requirements of a unitary matrix. In fact, this matrix is the representative of the unitary operator

e

-ijet

jn the energy representation (cf. Appendix III). This can be verified by expanding the exponential operator and computing its matrix elements in the energy representation. T h e same result can also be derived by formal integration of (1.1) assuming that dJ^jdt = 0.

Alternatively, one can compute the successive derivatives of Ψ and expand the solution in a Taylor series. T h e latter computation proceeds as follows:

3Ψ 32Ψ 8Ψ dkW

**-a - -ejr - - « Ί τ » ( - « Τ * - · « ! - - < - ^**

^ψ

· k=0

^V⁰¹

't=0 k=0

^R

'

ψ{ΐ) = <τ***Ψ(0). (1.20) This equation provides an operational solution of the Schrödinger

equation which is valid in any basis, subject to the condition dJf?/dt = 0.

D. Constants of the Motion

T h e dynamical variables of a quantum mechanical system are represented by linear operators. If X denotes an arbitrary dynamical variable, its mean value at time t is defined (1) by the scalar product

<xy^t = (ψ, χψ) = ψ*χψ. (1.21)

In matrix notation, this equation is equivalent to

<X>t = XXc^k%X^kj, (1.22)

j k

where Xkj = (uk , XUJ). If the operator X is hermitian, it is easy to show that (X)^t is real.

T o illustrate the use of this definition, consider the computation of

< J f > , for the state (1.15). From (1.15) and (1.21), hence

or

(1.23)

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where the last form follows from the orthonormality of the uk . T h u s the mean value of the hamiltonian operator at time t is independent of the time and equal to the probable mean of the energy eigenvalues.

T h e mean value of an arbitrary operator will be a function of the time by virtue of the time dependence of the state vector Ψ. It is possible, however, to express the time dependence of an operator X in a purely operational form by defining the total time derivative of a quantum mechanical operator as

Evidently this definition requires the mean value of X at time t to be equal to the time derivative of the mean value of X at time t. In particular, the mean value of X is a constant independent of the time if X = 0.

When this is the case, X is said to be a constant of the motion.

If the differentiation on the right side of (1.24) is carried out, one obtains, since Ψ and W^f are nonzero,

§ = f ⁺ ^t^ X l (1.25) ^[

T h e first term on the right gives the contribution to the total time derivative from the explicit time dependence of X> and the second term refers to an implicit time dependence resulting from the lack of com- mutivity of X with J f \ It follows that an operator X is a constant of the motion if it commutes with and is not an explicit function of the time.

In particular, if X is taken to be the hamiltonian operator, the energy of the system will be a constant of the motion, provided dJ^jdt = 0. This condition was explicitly used in the derivation of (1.23).

T h e integration of (1.25) is easily carried out in the energy representa- tion for an operator that is not an explicit function of the time. Upon resolving both sides of the equation into matrix elements, one obtains

X^kS{t)=i{Q^k-Q^s)X^k${t), (1.26)

which integrates to

X^kj(t) = Χ^Μ(0)β«°*-°>»⁹ (1.27) where Xkj(0) = X ^{K J} . Equation (1.27) shows that the diagonal elements

of X are constants, while the off-diagonal elements of X oscillate with the difference frequency ( Q^K — Ω³) .

In the energy representation, the matrices of e± i y et are diagonal with matrix elements

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2. M A G N E T I C M O M E N T S I N S T A T I O N A R Y F I E L D S 81

2 . Magnetic Moments in Stationary Fields

A. The Hamiltonian Operator

T h e hamiltonian operator for a nuclear magnetic moment in a magnetic field H is obtained by equating

hJif

to the classical energy — μ · H and replacing μ with the quantum mechanical vector operator yh\. T h u s

= —γΚΆ · I, or

= - y H - I . (2.1) It will be assumed that Η does not depend upon time, so that

d^jdt = 0. N o restriction will be imposed upon the direction of H , but it will be convenient to express the magnetic field in terms of its polar angles:

Η = Hn = //(sin θ cos φ, sin θ sin φ, cos 0), (2.2) where η is a unit vector in the direction of H , and Η = | Η |. In this

notation, the hamiltonian operator becomes

3tf

=

—yHvi '

I = —γΗ(Ιχ sin θ cos φ + Iy sin θ sin φ + Iz cos Θ). (2.3) Thus (1.27) can be written

L s

or

X(t) = e^iJtrtX(0)e-^iJirt. (1.28)

This equation can be obtained without reference to any basis by a direct integration of (1.25) with dXjdt = 0. For this purpose, rewrite (1.25) in the form

X - iJeX = e^iJirt ^ (e~^iJtrtX) = -iXJT, and multiply from the left with e~~im to obtain

—-— = —iSffi, at

where S(t) = e~iJirX(t). This equation integrates to S(t) = S(0)e~iJirt = X(py-i<*rt^{t T} ^{h e} definition of S now yields (1.28).

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B. The Spin-/ Particle

Since

dtffjdt =

0, the hamiltonian operator is a constant of the motion.

Moreover, I² commutes with each term in Jf7, so that the square of the spin angular momentum is also a constant of the motion. T h u s the determination of the eigenvalues and eigenvectors of is equivalent to the determination of the simultaneous eigenvectors of I² and η · I.

T h e eigenvalues of η · I are m = —I, —I + 1, and the corre- sponding eigenvectors are

**I /, m) = %e-™'*D**

^m

>

^m

{d)\/, m'), (2.4)

m'

where the expansion on the right is in terms of the basis which diago- nalizes Iz, and where the Dm>^m(9) are given by (3.45) or (3.46) of Chapter 2. T h e eigenvalues of 2tf are

Qm = -yHm, (2.5)

so that

**V(t) = X cJPY"*™ I /,**

^m

). (2.6)

m

Equations (2.4) and (2.6) can now be used to compute the mean value of any spin operator. However, the expressions for Dm'm(6) for arbitrary / are so complicated that they are not suitable for use in an illustrative example, whereas / = \ contains all the essential features of the general problem, and provides a convenient illustration of the results established in Section 1.

C. The Spin-^ Particle

T h e hamiltonian matrix, relative to the basis which diagonalizes I^z, may be obtained at once by writing I = ^ σ, and inserting the Pauli spin matrices in (2.3):

_ _ yH_

^/cos θ^e~ⁱ⁽^f s i n

θ\

T h e eigenvalues of ^f⁷ are given by the roots of d e t( J f — XI) = 0, namely λ = i \yti = Ω± , in agreement with (2.5).

T h e eigenvectors of 3tf satisfy the matrix equation

γΗ /cos θ e-

^iq>

sin θ

2 \ei(P s i n θ ^{- c o s} W W * W '

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c^x sin Θ 2'

Combining this result with the normalization condition, one obtains , ,· 1 — cos2 - ,

1 + tan2 ((9/2) 2 so that

Θ

c

^x

= e

^l7

i cos 2 β

c2 = ^(<ρ+η) sin ^

where η is an arbitrary (real) phase angle.

An analogous calculation with Ω+ yields . , . θ

c

^x

= e

^lr

i sin -

( f l+ = £ y / / ) . c2 = _β*<<ρ+η') cos -

The phases, for reasons to be indicated subsequently, are chosen as follows:

or, in expanded form, vH

— •^L^- (ε^λ cos θ + c2e~^lçp sin θ) = Q±cx ,

— (c^** sin Θ — c² cos 0) = i2±c² ·

These equations are not independent, but either one can be used to determine the ratio c²\c^x. T h e values of c¹ and c² are then determined, except for an arbitrary phase factor, by the normalization condition

k i l2+ l'a l2 = l.

For Ω_ = — J γΗ> the first of equations (2.8) gives c9 1 — cos θ Θ — = ^ t a n -

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T h e eigenvectors of Jf7 can now be expressed as:

| + ) = exp(— \icp) c o s - | + > + e x p( ^ V )^{s i n} ^ ~ >⁼

exp( — ψφ) cos 2 exp(^99) sin ^

I —)= — exp(— &<p) sin - | + > + e x p( ^ V ) cos - | =

where | ± ) = | £ , ± i) and | ± > = I £ , ± £>· [Note that J T | ± ) =

•^τ! ± ) · ] Compounding these eigenvectors into a 2 X 2 matrix, one obtains the unitary matrix which diagonalizes

( θ exp(— \ίψ) cos ^ —exp(— \ίψ) sin

Γ = t/«/«(ç,,e,0) = | /

^{θ λ}

\ . (2.11)

e x p ( s i n ^ εχρ^ΐφ) cos -

Equations (2.9) through (2.11) should be compared with the analysis in Section 3.D of Chapter 2, where identical results were obtained by expanding the operator e^im'°l². T h e choice of the phase angles η and η was in fact motivated by the desire to secure agreement with the previous calculation. One could also obtain | ± ) and Τ from the general trans- formation formula derived in Chapter 2.

The state vector and its dual are given by

W{t) = €⁺(0)β-*^Ω-* I + ) + cjfi)e-^ia^ I - ) ,

(2.12)

W\t) = c⁺* ( 0 y^O- * ( + I + c_*(P)e^ia+\- |.

From these equations, it follows that the mean value of a generic spin operator X is

ΨΧΨ = I r+( 0 ) |2 ( + I X I + ) + I U 0 ) l2( - | * | - )

+ c+* ( 0 ) £ _ ( 0 ) é > - ^ ( + ! X I - ) + c+( 0 ) £ _ * ( 0 ) e > ^ ( - ι X I + ) .

(2.13)

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If the system is initially in the state | + ) , c_(0) = 0, and the mean value of Iz is

(+ IAI +) =Icos*| — isin»|

1 A

= - COS

Ό.

T h e first line may be interpreted as stating that if one measures Iz for a system in the state | + ), one observes the values m = + \ and m = — \ with the probabilities cos2(#/2) and sin2(0/2), respectively;

the second line shows that the mean value of Iz may be obtained by projecting the eigenvalue \ along the ζ direction. Similar remarks apply if the system is initially in the state | — ), so tl*at c+(0) = 0. T h e mean value of Iz is then

(- I h I - ) = \ sin² \ ~ \^{c o s2} \ = - \^{c os θ}'

T h e results should be compared with the elementary discussion of the Stern-Gerlach experiment in Chapter 1.

When c⁺(0) and c_(0) are both nonzero, the mean values of and I^z are

</+>. = ^ji(k

⁺

(0 )l

²

- k-(0)l

²

) sin θ

+ c⁺*(0)c_(0)e-*y^Ht cos²1 -

*

⁺

**(0)c_*(0)é^ sin

²

1|,

</_>, = e~^(\ c⁺(0)\* - I £_(0)|²) sin θ

-ε⁺*(0)ε_(0)β-^^Ηί sin21 + ε⁺(0)ε_*(0)β^^Η* cos21|,

</«>« = ^K(k

⁺

(O)|

²

-k_(O)l

²

)cos0

-(c

⁺

*(0 )<r_(0 )erM« +

^£⁺( 0 ) ' - * ( 0 ) ^Η ί) sin 0}.

These equations show that (I^)t,

<Cy)/ ,

and (J^i are sinusoidal functions of time, and that the evaluation of the mean value of a component of the spin in any direction other than that of Η requires a knowledge of

I^{C +}( 0 ) |2, I cjf))\\ and c+(0)c_*(Q).

When the applied field is along the ζ axis (Θ = 0), I^z is a constant of the motion, and the preceding equations yield, taking φ = 0,

<4>

^t

= 4{

^c+

*(0)c_(0)e-"'

^ir<

+ c

⁺

(0)c_*(oyv

^A

'},

= -c⁺*(0)c_(0)e-'y>" +^{C +}( 0 )^C_ * ( 0 y " < J, (2.14)

<0,

= i d

c

⁺

(0) ι

²

- 1 mo) I

²

}·

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These equations show that the mean values of Ix and Iy generate a rotating component of angular momentum in the xy plane. Since μ — yÄI, similar remarks hold for the mean values of the components of μ. This appears to contradict the calculation of Section 4. A in Chapter 1 , where the only nonvanishing component of the nuclear magnetization was Mz = M0 . One must recognize, however, that the mean values com- puted above refer to a single particle, whereas the calculation of Chapter 1 was based on statistical considerations. T h e components of the macroscopic magnetization are actually the statistical averages of the components of <μ>, taken over the magnetic moments in a sufficiently large volume. If it is assumed that

where N⁰ is the number of nuclear magnetic moments per unit volume, then the statistical averages of (I^xy^t and (Iy}^{t a r}^e zero. This could come about, for example, if the relative phases η^+ί — of the products c^+jc*j = r^+jr_j exp[i(y^+j — η-j)] are distributed at random (2, 3).

D. The Heisenberg Equations of Motion

T h e results derived in the preceding section may be regarded from a different point of view by using (1.25) to obtain the equations of motion for the components of the magnetic moment operator μ = yh\. Upon evaluating the required commutators, one finds that

These operator equations, which are valid for arbitrary fields, are called the Heisenberg equations of motion for the components of μ. An examina- tion of the right-hand members of (2.15) shows that the Heisenberg equations can be condensed to

γ{Η

^ζ

μ

^υ

— Η

^ν

μ

^ζ

\ Ύ(Η

^χ

μ

^ζ

— Η

^ζ

μ

^χ

), γ{Η

^υ

μ

^χ

— Η

^χ

μ

^υ

).

(2.15)

άμ

~dt γΗ χ μ. (2.16)

Since the components of Η are scalar quantities, there is no problem with commutivity in the cross product Η Χ μ.

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2. M A G N E T I C M O M E N T S I N S T A T I O N A R Y F I E L D S 87 T h e equation of motion satisfied by the expectation value of μ follows at once from (1.24) and (2.16):

^<μ>, = -7Η Χ< μ > , . (2.17) Thus the expectation value of the quantum mechanical operator for μ

satisfies the same equation of motion as a classical magnetic moment in a magnetic field H. In particular, if Η is independent of time, <μ>^

precesses about Η with angular velocity — y H .

Equation (2.17) provides the quantum mechanical justification for the classical description of the resonance phenomenon presented in Chapter 1 . Indeed, all the classical calculations carried out in Chapter 1 can be transcribed into the language of quantum mechanics by inter- preting the classical variables as quantum mechanical expectation values.

T h e relation between the classical and quantum mechanical descrip- tions of the motion of a magnetic moment in an applied field admits an interesting geometric interpretation in the case of a particle with spin \ . This interpretation is based on a simple geometric representation of the quantization of a particle with spin \ .

T h e most general state of a spin- \ particle is of the form

^

^{= c}

\ I \ )~ky

"4"^c

2 I \ >

2~X

and is essentially determined by the ratio

ζ = c

²

lc

¹ . When

ζ

is given, c^x and c2 are determined, except for an arbitrary phase factor, by the normalization condition.

N o w the ratio c²/c¹ can be represented by a point ζ = ξ + ίη in the complex plane, and the latter can be stereographically projected onto the unit sphere about the origin, as described in Chapter 1 . This pro- jection sets up a one-to-one correspondence between the spin states of a spin- \ particle (points in the complex ζ plane) and the points on the unit sphere about the origin of a cartesian-coordinate system.

Let the point Ρ(φ, θ) on the unit sphere be the image of ζ, and let ζ denote the spin state corresponding to m — + \ for the direction (φ, θ).

From (2.14) of Chapter 1 , it follows that

Ρ{Ψ,θ)^ζ=^- = β^^ηΘ-, m = + 1 .

T h e point on the unit sphere corresponding to m — — \ is the point diametrically opposite to Ρ(φ, θ):

Ρ(φ + τ τ , π - 0)<->ζ' = - ~ = ?± = - ^ c o t ^ , m = - \ .

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These conclusions are verified by the ratios of the components of the eigenvectors | ± ) given in (2.9) and (2.10).

T h e stereographic projection thus provides a geometric representation of the quantization of a spin-^ particle. T h e axis of quantization is represented by a line L passing through the center of the unit sphere S.

The direction of L is defined by the polar angles ψ and Θ, and the spin states corresponding to m = ± \ are represented by the two diametri- cally opposite points determined by the intersection of L with S. T h e generalization of this representation leads to the description of a particle with spin I as a composite system of 21 spin- J particles (4).

Consider now the classical and quantum mechanical equations of motion:

T h e Schrödinger equation is equivalent to the matrix equation

— I

^e

A = ^(

^{Hz H}

-\(

^Cl

\

dt \cj 2 \H⁺ -Hj\c²r

where H^± — H^x ± iH^y , and thus to the system of linear differential equations

c\ = l\ (Hzci

+

^H

-

^c

2)>

^é*^{= l}~2 (^H+Cl ~^Hz°^'

U p o n introducing the parameter ζ = c²/c¹ , one finds that ζ is a solution of

ζ = 1-ξ(Η+-2ΗΖζ-Η_ζ%

which is identical with (2.11) of Chapter 1. Hence the solutions of the classical and quantum mechanical equations of motion may be obtained from the solution of the Riccati equation common to both problems.

Geometrically speaking, the point ζ traces out a path in the complex plane whose time rate of change is governed by the equation Ψ = \ίγΆ * οψ. Except for normalization, this path describes the change of c2jc¹ with time. Each point of the path in the complex plane can be stereographically projected onto the unit sphere S; the projected path on S is identical to that generated by the motion of μ/μ, which satisfies the equation of motion μ/μ, = —yH Χ (μ/μ).

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3. M A G N E T I C M O M E N T S I N R O T A T I N G F I E L D S 89 3. Magnetic Moments in Rotating Fields

A. Transformation of the Hamiltonian Operator

When the magnetic field is an explicit function of the time, d-Jtifjdt Φ 0, and the Schrödinger equation for a particle with spin / is equivalent to a set of 21 + 1 simultaneous first order linear differential equations with variable coefficients. T h e integration of these equations is often a difficult problem, but the special case of a rotating magnetic field can be treated by a procedure that is analogous to the rotating coordinate transformation used in the corresponding classical problem.

For γ > 0, the rotating field is given by

Hx = Ηλ cos œt, Hy = —H1 sin œt, Hz = H0 , and the hamiltonian operator by

jT(f) = yH · I = —{ω^χ(Ιχ cos œt — I^y sin ωί) + ω⁰Ιζ}, (3.1) where ω1 = γΗ1 and ω0 = γΗ0 .

T h e analogous classical problem was shown to be equivalent to that of a magnetic moment in a stationary field when observed from a coordinate system rotating about the ζ axis with frequency ω and a sense corresponding to the sign of —γ. It follows, from the transformation theory of Chapter 2, that the time dependence of the hamiltonian operator can be removed by introducing the transformation:

Ψ(ί) = *^<ω"*Φ(ί). (3.2)

Differentiating and substituting the resulting expression for Ψ in (1.1) yields

8Φ

i ^- = {e-^Wit) e^'z + ωΙζ}Φ. (3.3) T h e indicated transformation of J^(t) is easily carried out with the help

of equations (3.7) of Chapter 2. T h e final result is

i ~ = ~{ωλΙχ + ΔΙΖ}Φ = -Ωη · ΙΦ, (3.4) where the notational abbreviations are identical with those introduced

in Chapter 1 [equations (3.8)].

T h e effective hamiltonian operator in (3.4) is — Ωη · I, and, since

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this operator does not depend explicitly upon the time, the operational solution of (3.4) is

Φ(ί) = e^iQtn'^l<P(0). (3.5)

From (3.2), Φ(0) = Ψ(0), and

Ψ(ί) = β*ω»ζ€ίΩΙη'ιΨ(0). (3.6)

Β. The Transition Probability

T h e state vector can be expressed as a linear combination of the eigenvectors of Iz :

^(0 = Xc

^m

(0|/,m>. (3.7)

m

T h e probability amplitudes Cm(t) are no longer complex exponential functions of the time, so that the | Cm(t)\2 change with time. These changes can be given a more precise physical interpretation by assuming that the rf-field amplitude is zero for t < 0, but equal to H± for t ^ 0.

It will be assumed that when t — 0 the state vector is defined by

I

^c

m (0 )l

²

=

^{1> C}^m'(0) = °>^{f or}^m

' Φ

^m

-

A t s o m e t i m e t

> °>

t n e s t a t e

vector is given by (3.7), and the absolute square of Cm'(t) represents the probability that the spin is in the state | / , mts) at time t. This probability2

will be denoted Pm^^m'{t):

i> w ( 0 = I C^m>{t)\* = |</, rri ι ψ(φ\\ (3.8)

From (3.6), (3.8), and the initial condition Ψ(0) = CJ0)\ I, m), it follows that

Pm^mit) = l</,

m

I

^ * V^M' »

I /,

rri)\\

N o w <J, m'\ e^iœtI* = e-^{i w}'^{w <}< 7 ,^m' |, so that

Pm^rnit) =

| < / , I»' | | / , lff>|a. (3.9)

C. The Spin~| Particle

Equation (3.9) can be applied at once to the case / = \ (4, 5), for the exponential operator expands to

,-O/n Ï ι t • · ·Q ί β

£«»tn.i _ yc os __ 1_ m . σ s m _ _ _ ^

2 T h e probability defined by (3.8) is a conditional probability; Pm^m'{t) gives t h e p r o b a b i l - ity t h a t t h e spin is in t h e state | / , m'> at t i m e t on t h e condition t h a t it was in t h e state

I / , w> at t = 0.

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3. M A G N E T I C M O M E N T S I N R O T A T I N G F I E L D S 91 so that

/ ! JL Ι „iQtn-l I 1 JL

\2 > 2 I

^e

I 2 > 2

•> = f sin —

<-|, — -J I

η · σ | \ , •!•>

T h u s

A^-i/siO

= sin²0sirr ^Ts i n2^ f . (3.10) 2

(ω

⁰

— ω)

²

+

^ω^χ

Equation (3.10) shows that

^1/2^-1/2(0

*^{s a n} oscillatory function of the time with amplitude sin²0 = ω^/Ω² and angular oscillation frequency

\ Ω. T h e maximum amplitude of

^1/2-^-1/2(0

*^{s u r n t}y , which is achieved at resonance (Δ — 0) or when | ω^χ | ^> | Δ | ; in either case, the oscillation frequency is a^/2. These properties are shown in Fig. 3.1, where

^ι/2->-ι/2(0

i^s plotted for several values of the parameter χ = ω^χ\Ω.

F I G . 3 . 1 . T h e transition p r o b a b i l i t y

-Pi/

²

->-i/2 (0

f °^r several values of t h e p a r a m e t e r χ = ω1/Ω.

D. Majorana's Formula for Spin /

T h e determination of P^m^^m'(t) for a particle with spin / can be carried out by using the Euler decomposition

X- I 1.0

(3.11)

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where Qt and η are given parameters and the Euler angles φ, χ, φ are to be determined. These quantities³ satisfy the following relations:

cos a. sin \Qt = sin sin ^(φ — φ), cos β sin \Qt = sin \χ cos ^(φ — φ).

cos γ sin ^Qt = cos \χ sin ^(φ + φ), cos \Qt = cos Ι^χ cos \(φ + φ), η = (cos α, cos β, cos γ) = (œJQ, 0, ^Ι/Ώ).

Since Pm^mr(t) is the absolute square of a matrix element taken with respect to the eigenvectors of Iz, the decomposition (3.11) gives

p m ^ ) = Κ Λ « ' I 1h^m>\² = 3 L . ( x ) . (3-12) where the last equality follows from (3.48) of Chapter 2. Thus P^m^^mf(t) is determined when χ is expressed in terms of known parameters. From cos β = 0, it follows that φ — φ = π, so that

sin \χ = -~- sin \Ωί — sin θ sin (3.13) Therefore,

w( 0 - ( / + m)! ( / - m)\ (7 + m')! (/ - m')\ (cos | ) "

v j V ( - l )f c[ t a n (x/ 2 ) ] — + » * }*

) ^ ( / - « ' - £)! (m' - m + k)\(I + m — k)\k\ \ ' { V

Equation (3.14), which, by virtue of (3.10) and (3.13), relates P^m^^m'{t) for arbitrary values of 7 to ^ 1 / 2 ^ - 1 / 2 ( 0 » *s known as Majorana!s formula (3-4). The Majorana formula possesses the following symmetry prop- erties:

Pm->m'(t) Pm'-*m(t) P—m->—m'(j)i (3.15) which are direct consequences of the symmetry properties of the D^mm>(6) established in Chapter 2.

T h e Majorana formula shows that if a spin is initially in the state I 7, my, transitions to all other states | 7, m^fs) are possible. T o illustrate this point and the application of (3.14) in a specific instance, suppose that the spin is in the state | 7, —7> at t = 0. For the assumed value

3 See A p p e n d i x I I .

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3. M A G N E T I C M O M E N T S I N R O T A T I N G F I E L D S 93

of tny the only value of k allowed by the condition that no factorial argument be negative is k = 0, for all values of triThus the Majorana formula reduces to

P - ^ - ( 0

= ⁽

⁷

^!

⁷

^JHf ⁾

^{2 i}

^{(tan^)'}

^{+ m}

^'-

^(3-16)

This equation reduces to (3.10) for / = rri = \ . It is easily verified that the P_^m\t) satisfy the normalization condition

m'

For upon changing the summation index from rri to r = / — rri = 0, 1, 27, and using the binomial expansion for (1 + sf \ one finds that

= ( « >^{8 2}| )^{2 Ι}( ϊ + t a n²| )^{2 7} = 1.

T h e probabilities of the transitions, 11, —1} —> \ I, m'}, rri = —J,

—I + 1, are easily determined. For example, the probability that the spin is still in the state | / , —/> at time t is

P-M{t) =

(cos

²

!)

²

',

while the probability that the single quantum transition | / , — Is) —•

I / , — / + 1> has occurred in the time interval t is

P-M+1{t) =

2/(cos

²

g "

tan* | .

The probability of the 2/-tuple quantum transition | / , — J> —• | / , + / >

is

P _7_+ /( 0 =

(sin

²

g

^{2 /}

.

T h e transition probabilities Ρ P _ ! _ , 0( i ) , and P^^t) are sketched in Fig. 3.2 for / = 1 and χ = ω^Ω = 1 (Δ = 0, or | ωλ | >

From the figure it is clear that the probability of the single quantum transition | 1, — 1 > - > | 1, 0> never exceeds \ , whereas the probability

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F I G. 3.2. T h e transition probabilities P _1_>_1( 0 , P _ i ^o( 0 > a n d P-i^{t) for / = 1

and χ = œ

^l

jQ = 1.

of the double quantum transition | 1 , — 1 > — * | 1 , 1 > is unity for Qt = π, 377, . . . . It should be noted, however, that the rate of increase of is not comparable to that of P^^t) until t & π/5.

R E F E R E N C E S

1. L . I. Schiff, " Q u a n t u m M e c h a n i c s , " 2 n d ed., C h a p . V I . M c G r a w - H i l l , N e w York 1955.

2. A . A b r a g a m , " T h e Principles of N u c l e a r M a g n e t i s m , " C h a p . I I . Oxford U n i v . Press, L o n d o n a n d N e w York, 1961.

3. R. C. T o l m a n , " T h e Principles of Statistical M e c h a n i c s , " C h a p . I X . Oxford U n i v . Press, L o n d o n a n d N e w York, 1938.

4. E. Majorana, Nuovo Cimento 9, 43 (1932).

5. (a) J. Schwinger, Phys. Rev. 5 1 , 648 (1937). (b) I. I. R a b i , ibid. 5 1 , 652 (1937).