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Spin

Introduction

The Stern Gerlach experiment for the simplest possible atom, hydrogen in its ground state, demonstrated unambiguously that the component of the magnetic moment of the atom along the z-axis could only have two values.  It had been well established by this time that the magnetic moment vector was along the same axis as the angular momentum.  This is obviously true for the Bohr model of hydrogen, where the circulating electron is equivalent to a ring current, generating a magnetic dipole.  The problem is, though, that a magnetic moment generated in this way by orbital angular momentum will have a minimum of  three possible values of its z-component: the lowest nonzero orbital angular momentum is  with allowed values of the z-component   

 

Recall, however, that in our derivation of allowed angular momentum eigenvalues from very general properties of rotation operators, we found that although for any system the allowed values of m form a ladder with spacing  we could not rule out half-integral m values.  The lowest such case,  would in fact have just two allowed m values:   However, this cannot be any kind of orbital angular momentum because the z-component of the orbital wave function  has a factor  and therefore picks up a factor -1 on rotating through  meaning  is not single-valued, which doesn’t make sense for a Schrödinger wave function. 

 

Yet the experimental result is clear.  Therefore, this must be a new kind of non-orbital angular momentum.  It is called “spin”, the simple picture being that just as the Earth has orbital angular momentum in its yearly circle around the sun, and also spin angular momentum from its daily turning, the electron has an  analogous spin.  But the analogy has obvious limitations: the Earth’s spin is after all made up of material orbiting around the axis through the poles, the electron’s spin cannot similarly be imagined as arising from a rotating body, since orbital angular momenta always come in integral multiples of

 

Fortunately, this lack of a simple quasi-mechanical picture underlying electron spin doesn’t prevent us from using the general angular momentum machinery previously developed, which followed just from analyzing the effect of spatial rotation on a quantum mechanical system.  Recall this led to the spacing of the ladder of eigenvalues, and to values of the matrix elements of angular momentum components Ji between the eigenkets  enough information to construct matrix representations of the rotation operators for a system of given angular momentum.  As an example, for the orbital angular momentum  state, we constructed the  matrix representation of an arbitrary rotation operator  in the space with orthonormal basis   (in the  notation).  The spin  case can be handled in exactly the same way.

Spinors, Spin Operators, Pauli Matrices

The Hilbert space of angular momentum states for spin one-half is two dimensional.  Various notations are used:  or even, more graphically,

 

 

Any state of the spin can be written

 

 

and this two-dimensional ket is called a spinor.  

 

Operators on spinors are necessarily matrices.  We shall follow the usual practice of denoting the angular momentum components Ji  by  Si for spins.

 

From our definition of the spinor,

 

The general formulas for raising and lowering operators

 

 

become for  simply

 

 

so

 

It follows immediately that an appropriate matrix representation for spin one-half is

 

 

These three matrices representing the (x, y, z) spin components are called the Pauli spin matrices.  They are hermitian, traceless, and obey

 This can be written

 

The total spin

 

Any  matrix can be written in the form

 

 

Exercise:  prove the above statements, then use your results to show that

 

(a)  

 

(b)  

Relating the Spinor to the Spin Direction

But how do  in  relate to which way the spin’s pointing?   To find out, let’s assume that it’s pointing up along the unit vector  that is, in the direction    In other words, it’s in the eigenstate of the operator  having eigenvalue unity:

 

 

Evaluating, , using elementary trigonometric identities

 

 

where we have multiplied by an overall phase factor  to make it look nicer.  Note that the spinor is also correctly normalized.

 

The physically significant parameter for spin direction is just the ratio   Note that any complex number can be represented as , with  so for any possible spinor, there’s a direction along which the spin points up with probability one.

The Spin Rotation Operator

The rotation operator for rotation through an angle about an axis in the direction of the unit vector  is, using

 

 

(Warning: we’re following standard notation here, but don’t confuse this --angle turned through—with the  in writing  in terms of )

 

Expanding the exponential,

 

 

and using

 

 

Writing this in the same D-notation we used for orbital angular momentum earlier (the superscript refers to the j-value)

 

 

The rotation operator is a  matrix operating on the ket space

 

 

Explicitly, it is

 

 

 

Notice that this matrix has the form

 

with

 

The inverse of this rotation operator is clearly given by replacing  with  that is,

 

 

These  matrices have determinant  and so are unitary.  They clearly form a group, since they represent operations of rotation on a spin.  This group is called SU(2), the 2 refers to the dimensionality, the U to their being unitary, and the S signifying determinant +1.

 

Note that for rotation about the z-axis,  it is more natural to replace  and the rotation operator becomes

 

 

In particular, the wave function is multiplied by -1 for a rotation of   Since this is true for any initial wave function, it is clearly also true for rotation throughabout any axis.

 

Exercise: write down the infinitesimal version of the rotation operator  for spin ½ , and prove that  that is,  is rotated in the same way as an ordinary three-vector—note particularly that the change depends on the angle rotated through, as opposed to the half-angle, so, reassuringly, there is no -1 for a complete rotation (as there cannot be—the direction of the spin is a physical observable, and cannot be changed on rotating the measuring frame through ).

Spin Precession in a Magnetic Field

As a warm up exercise, consider a magnetized classical object spinning about its center of mass, with angular momentum  and parallel magnetic moment.  The constant  is called the gyromagnetic ratio.  Now add a magnetic field , say in the z-direction. This will exert a torque , easily solved to find the angular momentum vector  precessing about the magnetic field direction with angular velocity of precession .

 

(Proof: from , take  Of course, dLz/dt = 0, since  is perpendicular to , which is in the z-direction.)

 

The exact same result comes from the quantum mechanics of an electron spin in a magnetic field.  The electron has magnetic dipole moment , where  and g (known as the Landé g-factor) is very close to 2.  (This g-factor terminology is used more widely: the magnetic moment of an atom is writtenis the Bohr magneton, and g depends on the total orbital angular momentum and total spin of the particular atom.)

 

The Hamiltonian for the interaction of the electron’s dipole moment with the magnetic field is , hence the time development is

 

with the propagator

 

but this is exactly the rotation operator (as shown earlier) through an angle   about !

 

For an arbitrary initial spin orientation

 

 

the propagator for a magnetic field in the z-direction

 

so the time-dependent spinor is

 

 

The angle  between the spin and the field stays constant, the azimuthal angle around the field increases as  exactly as in the classical case.

 

Exercise: for a spin initially pointing along the x-axis, prove that

Paramagnetic Resonance

We have shown that the spin precession frequency is independent of the angle of the spin to the field.  Consider how all this looks in a frame of reference which is itself rotating with angular velocity  about the z-axis. Let’s call the magnetic field , because we’ll soon be adding another one. 

 

In the rotating frame, the observed precession frequency is , so there is a different effective field  in the rotating frame. Obviously, if the frame rotates exactly at the precession frequency,  spins pointing in any direction will remain at rest in that frame—there’s no effective field at all.

 

Suppose now we add a small rotating magnetic field with angular frequency  in the x,y plane, so the total magnetic field

 

 

The effective magnetic field in the frame rotating with the same frequency  as the small added field is

 

 

Now, if we tune the angular frequency of the small rotating field so that it exactly matches the precession frequency in the original static magnetic field,  all the magnetic moment will see in the rotating frame is the small field in the x-direction!  It will therefore precess about the x-direction at the slow angular speed   This matching of the small field rotation frequency with the large field spin precession frequency is the “resonance”.

 

If the spins are lined up preferentially in the z-direction by the static field, and the small resonant oscillating field is switched on for a time such that  the spins will be preferentially in the y-direction in the rotating frame, so in the lab they will be rotating in the x,y plane, and a coil will pick up an ac signal from the induced emf.