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Quantum Mechanics for Mathematicians: The

Spin 12

Particle in a Magnetic Field

Peter Woit

Department of Mathematics, Columbia University

woit@math.columbia.edu

October 15, 2012

The existence of a non-trivial double-cover Spin(3) of the three-dimensionalrotation group may seem to be a somewhat obscure mathematical fact but,remarkably, it is Spin(3) rather than SO(3) that is the symmetry group ofquantum systems describing elementary particles. Studies of atomic spectra inthe early days of quantum mechanics revealed twice as many states as expected,a phenomenon that is now understood to be due to the fact that the state spaceof an electron at a point is C2 rather than C . This state space is not a repre-sentation of the rotational symmetry group SO(3), but it is a representation ofthe double-cover Spin(3) = SU(2) (the standard representation of SU(2) ma-trices on C2). Particles with this degree of freedom are said to have spin 1/2,with the origin of this 1/2 precisely the same as the 1/2 we saw in the previousclass in the discussion of the Lie algebras of the two equivalent forms Sp(1) andSU(2) of the group Spin(3). This same C2 occurs for other matter particles(quarks, neutrinos, etc.) and appears to be a fundamental property of nature.Besides the doubling of the number of states, more complicated physical effectsoccur when such a particle is subjected to a magnetic field, and we will examinesome of this physics in this section.

1 The Spinor Representation

In the last section we examined in great detail various ways of looking at thethree-dimensional irreducible real representation of the groups SO(3), SU(2)and Sp(1). For SU(2) and Sp(1) however, there is a even simpler non-trivialirreducible representation: the representation of 2 by 2 complex matrices in

SU(2) on column vectors C2 by matrix multiplication or the representation ofunit quaternions in Sp(1) on H by scalar multiplication. Choosing an identifica-tion C2 = H these are isomorphic representations of isomorphic groups, and forconvenience we will generally stick to using SU(2) and complex numbers ratherthan quaternions. This irreducible representation is known as the spinor or

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spin representation of Spin(3) The homomorphism spinor defining the rep-resentation is just the identity map from SU(2) to itself.

The spin representation of Spin(3) is not a representation of SO(3). Thedouble cover map : Spin(3) SO(3) is a homomorphism, so given a rep-resentation (, V) of SO(3) one gets a representation ( , V) of Spin(3) bycomposition. But there is no homomorphism SO(3) SU(2) that would allowus to make the standard representation of SU(2) on C2 into an SO(3) repre-sentation. What is true is that we can try and define a representation of SO(3)by

: g SO(3) (g) = spinor(g) SU(2)

where g is some choice of one of the elements g SU(2) satisfying (g) = gThe problem with this is that we wont quite get a homomorphism. Changing

our choice of g will introduce a minus sign, so will only be a homomorphismup to sign

(g1)(g2) = (g1g2)

The nontrivial nature of the double-covering ensures that there is no way tocompletely eliminate all minus signs, no matter how we choose g. Something likethis which is not quite a representation, only one up to a sign ambiguity, is knownas a projective representation. So, the spinor representation of SU(2) =Spin(3) is only a projective representation of SO(3), not a true representationofSO(3). Quantum mechanics texts often deal with this phenomenon by notingthat physically there is an ambiguity in how one specifies the space of states H,with multiplication by an overall scalar not changing the eigenvalues of operatorsor the relative probabilities of observing these eigenvalues. As a result, the signambiguity has no physical effect. It seems more straightforward though to justwork from the beginning with the larger symmetry group Spin(3), accepting

that this is the correct symmetry group reflecting the action of rotations onthree-dimensional quantum systems.

2 The Spin 1/2 Particle in a Magnetic Field

Recall from our earlier discussion of the two-state quantum system, where thestate space is H = C2 that there is a four (real)-dimensional space of observableswith a general self-adjoint linear operator on H written as

M = c01+ c11 + c22 + c33

Multiplying by i, one gets skew-Hermitian operators, and thus elements of theLie algebra u(2). Exponentiating gives elements of the unitary group U(2), with

eic01 U(1) U(2)

andei(c11+c22+c33) SU(2) U(2)

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For an arbitrary two-state quantum system, neither the operators M nor thegroup SU(2) have any particular geometric significance. In some cases though,

the group SU(2) does have a geometric interpretation, reflecting its role as thedouble-cover Spin(3) and the fact that the group SO(3) acts on the physicalsystem by rotations of three-dimensional space. In these cases, the quantumsystem is said to carry spin, in particular spin one-half (we will later onencounter state spaces of higher spin values). We will from now on assume thatwe are dealing with a spin one-half state space.

A standard basis for the observables (besides the unit operator that generatesoverall U(1) transformations) is taken to be the operators sa, a = 1, 2, 3, where

sa = ia2

with the sa satisfying the commutation relations

[s1, s2] = s3, [s2, s3] = s1, [s3, s1] = s2

To make contact with the physics formalism, well define self-adjoint opera-tors

Sa = isa =a2

which will have real eigenvalues 12 .Note that the conventional definition of these operators in physics texts

includes a factor of

Sa = isa =a

2

This is because rotations of vectors are defined in physics texts using conjugationby the matrix

R(,w) = eiwS (compare to our R(,w) = eiwS = ews)

with the convention of dividing by a factor of appearing here for reasons thathave to do with the action of rotations on functions on R3 that we will encounterlater on. For now, one can either keep factors of out of the definitions of Saand the action of rotations, or just assume that we are working in units where= 1.

States in H = C2 that have a well-defined value of the observable Sa will bethe eigenvectors of Sa, with value for the observable the corresponding eigen-value, which will be 12 . Measurement theory postulates that if we perform themeasurement corresponding to Sa on an arbitrary state |, then

we will with probability c+ get a value of +12 and leave the state in an

eigenvector |a, +12 of Sa with eigenvalue +12

we will with probability c get a value of 12 and leave the state in an

eigenvector |a, 12 of Sa with eigenvalue 12

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where if

| = |a, +1

2 + |a,

1

2

we have

c+ =||2

||2 + ||2, c =

||2

||2 + ||2

After such a measurement, any attempt to measure another orthogonal compo-nent of S, say Sb, b = a will give

12 with equal probability and put the system

in a corresponding eigenvector of Sb.If a quantum system is in an arbitrary state | it may not have a well-

defined value for some observable A, but one can calculate the expected valueofA. This is the sum over a basis ofH consisting of eigenvectors (which will allbe orthogonal) of the corresponding eigenvalues, weighted by the probability oftheir occurrence. The calculation of this sum using expansion in eigenvectors of

Sa gives

|A|

|=

(a, +12 | + a, 12 |)A(|a, +

12 + |a,

12 )

(a, +12 | + a, 12 |)(|a, +

12 + |a,

12 )

=||2(+12) + ||

2(12)

||2 + ||2

=c+(+1

2) + c(

1

2)

One often chooses to simplify such calculations by normalizing states so thatthe denominator | is 1. Note that the same calculation works in general,as long as one has orthogonality and completeness of eigenvectors.

Recall that

R(, w) = ews

= cos(

2)1 i(w )sin(

2)

In the case of a spin one-half particle, the group Spin(3) = SU(2) acts onstates by the spinor representation with the element R(,w) SU(2) acting as

| R(,w)|

Taking adjoints and using unitarity, one has (thinking of vectors as columnvectors, elements of the dual space as row vectors) the following action on thedual state space

| |R(,w

)

1

The operators Sa transform under this same group according to the vectorrepresentation of SU(2), recall that an SO(3) rotation on vectors is given byconjugating by an SU(2) = Spin(3) group element according to

Sa R(,w)SaR(,w)1

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We see that if our observables transform as vectors, and states as spinors,the expectation values remain invariant:

|Sa| |R(,w)1R(,w)SaR(,w)

1R(,w)| = |Sa|

and all eigenvalues of observables remain invariant. One can also interpretthese joint transformations on states and observables as simply a change ofcoordinates, a rotation from the standard basis to a different one.

Next semester in this course we may get to the physics electromagnetic fieldsand how particles interact with them in quantum mechanics, but for now all weneed to know is that for a spin one-half particle, the spin degree of freedom thatwe are describing by H = C2 has a dynamics described by the Hamiltonian

H = B

Here B is the vector describing the magnetic field, and

= g(e)

2mcS

is an operator called the magnetic moment operator. The constants that appearare: e the electric charge, c the speed of light, m the mass of the particle, and g,a dimensionless number called the gyromagnetic ratio, which is approximately2 for an electron, about 5.6 for a proton.

The Schrodinger equation is

d

dt|(t) = (

i

)( B)|(t)

with solution|(t) = U(t)|(0)

where

U(t) = eitB = eit

ge2mc

SB = eitge2mc

isB = etge2mc

sB = etge|B|2mc

s B|B|

The time evolution of a state is thus given at time t by a rotation about theaxis w = B|B| by an angle

ge|B|t

2mc

a rotation taking place with angular velocityge|B|2mc .

The amount of non-trivial physics that is described by this simple system isimpressive, including:

The Zeeman effect: this is the splitting of atomic energy levels that occurswhen an atom is put in a constant magnetic field. With respect to theenergy levels for no magnetic field, where both states in H = C2 have thesame energy, the term in the Hamiltonian given above adds

ge|B|

4mc

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to the two energy levels, giving a splitting between them proportional tothe size of the magnetic field.

The Stern-Gerlach experiment: here one passes a beam of spin one-halfquantum systems through an inhomogeneous magnetic field. One canarrange this in such a way as to pick out a specific direction w, and splitthe beam into two components, of eigenvalue +12 and

12 for the operator

w S.

Nuclear magnetic resonance spectroscopy: one can subject a spin one-half system to a time-varying magnetic field B(t), which will be describedby the same Schrodinger equation, although now the solution cannot befound just by exponentiating a matrix. Nuclei of atoms provide spin one-half systems that can be probed by with time and space-varying magneticfields, allowing imaging of the material that they make up.

Quantum computing: attempts to build a quantum computer involve try-ing to put together multiple systems of this kind (qubits), keeping themisolated from perturbations by the environment, but still allowing inter-action with the system in a way that preserves its quantum behavior.The 2012 Physics Nobel prize was awarded for experimental work makingprogress in this direction.

3 The Heisenberg Picture

So far in this course weve been describing what is known as the Schr odingerpicture of quantum mechanics. States in H are functions of time, obeyingthe Schrodinger equation determined by a Hamiltonian observable H, while

observable self-adjoint operators A are time-independent. Time evolution isgiven by a unitary transformation

U(t) = eitH, |(t) = U(t)|(0)

One can instead use U(t) to make a unitary transformation that puts thetime-dependence in the observables, removing it from the states, as follows:

|(t) |(t)H = U1(t)|(t) = |(0), A AH(t) = U

1(t)AU(t)

where the H subscripts for Heisenberg indicate that we are dealing withHeisenberg picture observables and states. One can easily see that the physi-cally observable quantities given by eigenvalues and expectations values remain

the same:

H(t)|AH|(t)H = (t)|U(t)(U1(t)AU(t))U1(t)|(t) = (t)|A|(t)

In the Heisenberg picture the dynamics is given by a differential equationnot for the states but for the operators. Recall from our discussion of the adjoint

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representation the formula

ddt

(etXY etX) = ( ddt

(etXY))etX + etXY( ddt

etX)

= XetXY etX etXY etXX

Using this with

Y = A, X = iH

we findd

dtAH(t) = [i

H

, AH(t)] =

i

[H, AH(t)]

and this equation determines the time evolution of the observables in the Heisen-berg picture.

Applying this to the case of the spin one-half system in a magnetic field, and

taking for our observable S we find

d

dtSH(t) =

i

[H,SH(t)] = i

eg

2mc[SH(t) B,SH(t)]

We know from earlier that the solution will be

SH(t) = U(t)SH(0)U(t)1

for

U(t) = etge|B|2mc

s B|B|

and thus the spin vector observable evolves in the Heisenberg picture by rotating

about the magnetic field vector with angular velocityge|B|2mc .

4 The Bloch Sphere and Complex ProjectiveSpace

There is a different approach one can take to characterizing states of a quantumsystem with H = C2. Multiplication of vectors in H by a non-zero complexnumber does not change eigenvectors, eigenvalues or expectation values, so ar-guably has no physical effect. Multiplication by a real scalar just correspondsto a change in normalization of the state, and we will often use this freedomto work with normalized states, those satisfying | = 1. With normalizedstates, one still has the freedom to multiply states by a phase ei without chang-ing eigenvectors, eigenvalues or expectation values. In terms of group theory, theoverall U(1) in the unitary group U(2) acts on H acts on H by a representationof U(1), which can be characterized by an integer, the corresponding charge,but this decouples from the rest of the observables and is not of much interest.One is mainly interested in the SU(2) part of the U(2), and the observablesthat correspond to its Lie algebra.

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Working with normalized states in this case corresponds to working withunit-length vectors in C2, which are given by points on the unit sphere S3. If

we dont care about the overall U(1) action, we can imagine identifying all statesthat are related by a phase transformation. Using this equivalence relation wecan define a new set, whose elements are the cosets, elements of S3 C2,with elements that differ just by multiplication by ei identified. The set ofthese elements forms a new geometrical space, called the coset space, oftenwritten S3/U(1). This structure is called a fibering of S3 by circles, and isknown as the Hopf fibration. Try an internet search for various visualizationsof the geometrical structure involved, a surprising way decomposition of three-dimensional space into non-intersecting curves.

The same space can be represented in a different way, as C2/C, by takingall elements ofC2 and identifying those related by muliplication by a non-zerocomplex number. If we were just using real numbers, R2/R can be thought ofas the space of all lines in the plane going through the origin.

One sees that each such line hits the unit circle in two opposite points, sothis set could be parametrized by a semi-circle, identifying the points at thetwo ends. This space is given the name RP1, the real projective line, and

the analog space of lines through the origin in Rn is called RPn1. What weare interested in is the complex analog CP1, which is often called the complexprojective line.

To better understand CP1, one would like to put coordinates on it. A

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standard way to choose such a coordinate is to associate to the vector

z1z2

C2

the complex number z1/z2. Overall multiplication by a complex number willdrop out in this ratio, so one gets different values for each of the of the coordinatez1/z2 for each different coset element, and it appears that elements of CP

1

correspond to points on the complex plane. There is however one problem withthis coordinate: the coset of

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does not have a well-defined value: as one approaches this point one moves offto infinity in the complex plane. In some sense the space CP1 is the complexplane, but with a point at infinity added.

It turns out that CP1 is best thought of not as plane together with a point,but as a sphere, with the relation to the plane and the point at infinity given bystereographic projection. Here one creates a one-to-one mapping by consideringthe lines that go from a point on the sphere to the north pole of the sphere.Such lines will intersect the plane in a point, and give a one-to-one mappingbetween points on the plane and points on the sphere, except for the north pole.Now, one can identify the north pole with the point at infinity, and thus thespace CP1 can be identified with the space S2. The picture looks like this

and the equations relating coordinates (X1, X2, X3) on the sphere and the

complex coordinate z1/z2 = z = x + iy on the plane are given by

x =X1

1 X3, y =

X21 X3

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and

X1 =2x

x2 + y2 + 1, X2 =

2y

x2 + y2 + 1, X3 =

x2 + y2 1

x2 + y2 + 1

The action of SU(2) on H by

z1

z2

z1

z2

takes

z =z1z2

z +

z +

Such transformations of the complex plane are conformal (angle-preserving)transformations known as Mobius transformations. One can check that thecorresponding transformation on the sphere is the rotation of the sphere in R3

corresponding to this SU(2) = Spin(3) transformation.In physics language, this sphere CP1 is known as the Bloch sphere. It

provides a useful parametrization of the states of the qubit system, up to scalarmultiplication, which is supposed to be physically irrelevant. The North poleis the spin-up state, the South pole is the spin-down state, and along the

equator one finds the two states that have definite values for S1, as well as thetwo that have definite values for S2.

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Notice that the inner product on vectors in H does not correspond at all

to the inner product of unit vectors in R

3

. The North and South poles of theBloch sphere correspond to orthogonal vectors in H, but they are not at allorthogonal thinking of the corresponding points on the Bloch sphere as vectorsintR3. Similarly, eigenvectors for S1 and S2 are orthogonal on the Bloch sphere,but not at all orthogonal int H.

Many of the properties of the Bloch sphere parametrization of states in H arespecial to the fact that H = C2. In the next class we will study systems of spinn2 , where H = C

n. In these cases there is still a two-dimensional Bloch sphere,but only certain states in H are parametrized by it. We will see other examplesof systems with coherent states analogous to the states parametrized by theBloch sphere, but the case H has the special property that all states (up toscalar multiplication) are such coherent states.

5 For Further Reading

Just about every quantum mechanics textbook works out this example of a spin1/2 particle in a magnetic field. For one arbitrarily chosen example, see Chapter14 of [1] or

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References

[1] Shankar, R., Principles of Quantum Mechanics, 2nd Ed., Springer, 1994.

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