Lecture 11 { Spin, orbital, and total angular momentum 1 ...

[Pages:7]Lecture 11 ? Spin, orbital, and total angular momentum MATH-GA

2710.001 Mechanics

1 Very brief background

In 1922, a famous experiment conducted by Otto Stern and Walther Gerlach, involving particles subject to a nonuniform magnetic field, identified two surprising properties of the angular momentum: 1) the components of the angular momentum are quantized, i.e. they can only take certain discrete values; 2) there exists a type of angular momentum that is intrinsic to a particle, even a point particle, and that cannot be put in the form r ? p: this is the so-called "spin angular momentum". The total angular momentum J is the sum of the orbital angular momentum L and the spin angular momentum S: J = L + S. In this lecture, we will start from standard postulates for the angular momenta to derive the key characteristics highlighted by the Stern-Gerlach experiment.

2 General properties of angular momentum operators

2.1 Commutation relations between angular momentum operators

Let us first consider the orbital angular momentum L of a particle with position r and momentum p. In classical mechanics, L is given by

L=r?p so by the correspondence principle, the associated operator is

L= r? i

The operator for each components of the orbital angular momentum thus are

Lx

Ly

Lz

= y^p^z - z^p^y = i = z^p^x - x^p^z = i = x^p^y - y^p^x = i

y

z

-

z

y

z

x

-

x

z

x

y

-

y

x

We also define the operator

L2 = L2x + L2y + L2z

Let us start with the commutation relation between Lx and Ly:

[Lx, Ly] = [y^p^z - z^p^y, z^p^x - x^pz] = (y^p^z - z^p^y)(z^p^x - x^pz) - (z^p^x - x^pz)(y^p^z - z^p^y) = [y^p^z, z^p^x] - [z^p^y, z^p^x] - [y^p^z, x^p^z] + [z^p^y, x^p^z] = [y^p^z, z^p^x] + [z^p^y, x^p^z] = y^p^z[p^z, z^] + x^p^y[z^, p^z]

Now, we know that [z^, p^z] = i , so we conclude that

[Lx, Ly] = i (x^p^y - y^p^x) = i Lz By cyclical permutations one easily obtain the other relations:

[Ly, Lz] = i Lx [Lz, Lx] = i Ly

We note a very important results: the three components of the orbital angular momentum are not compatible with one another, and have associated uncertainty relations.

The operator L2, on the other hand, commutes with Lx, Ly and Lz. Indeed, [L2, Lz] = [L2x, Lz] + [L2y, Lz]

and [L2x, Lz] = LxLxLz - LzLxLx = LxLzLx - i LxLy - LxLzLx - i LyLx = -i (LxLy + LyLx)

1

Repeating the same calculation for [L2y, Lz], we would find

[L2y, Lz] = i (LxLy + LyLx)

so that finally and in exactly the same way, we would prove

[L2, Lz] = 0

[L2, Lx] = 0 [L2, Ly] = 0

A postulate of quantum mechanics is that all types of angular momentum operator J, orbital or spin, satisfy the following commutation relations:

[J 2, Jx] = 0 [J 2, Jy] = 0 [J 2, Jz] = 0

(1)

[Jx, Jy] = i Jz [Jy, Jz] = i Jx [Jz, Jx] = i Jy

(2)

We will now take this relations as a starting point, and derive general properties of any angular momentum operator J that satisfies these properties.

2.2 Eigenvalues of the operators J2 and Jz

We take the commutation relations given by Eq. (1) and Eq. (2) as our postulate, and show that alone they allow us to prove that the eigenvalues or J2 and Jz are quantized. Since J2 and Jz commute, there exists a basis of eigenvectors that are common to these two operators. Let us call |a, b an eigenstate of both J2, with eigenvalue 2a, and of Jz, with eigenvalue b. The factors 2 and appear because we have normalized the eigenvalues so that a and b are dimensionless numbers. We thus have

J 2|a, b = 2a|a, b

Jz|a, b = b|a, b

We also have the additional normalization condition

a, b|a, b = 1

Let us now construct the operators

J + = Jx + iJy J - = Jx - iJy

Note that J+ and J- are not Hermitian, but Hermitian conjugates of one another: (J+) = J-. We will now see what happens when one applies J+ and J- to the state |a, b .

Since J2 commutes with Jx and Jy, we can write

J 2 J ?|a, b = J ? J 2|a, b = J ? 2a|a, b = 2aJ ?|a, b

where J? stands for either J+ or J-. We see that J?|a, b is an eigenvector of J2 with eigenvalue 2a. Likewise, using the commutation relations

[Jz, J +] = [Jz, Jx + iJy] = i Jy + i(-i )Jx = J +

we find

JzJ +|a, b = J +Jz|a, b + J +|a, b = (b + 1)|a, b

Following the same procedure, we could also show that

JzJ-|a, b = (b - 1)|a, b

We see that J+|a, b is an eigenvector of Jz with eigenvalue (b + 1), and J-|a, b is an eigenvector of Jz with eigenvalue (b - 1). For these reasons, J+ and J- are sometimes called ladder operators.

2

We just showed that J+|a, b is colinear with the normalized eigenstate |a, b + 1 , and that J-|a, b is colinear with the normalized eigenstate |a, b - 1 . There exist complex numbers c+ and c- such that

J +|a, b = c+|a, b + 1 J -|a, b = c-|a, b - 1

Since J- is the Hermitian conjugate of J+, the square of the norm of the ket of J+|a, b is

|c+|2 = a, b|J - J +|a, b = a, b|J -J +|a, b

We can easily compute

J -J + = Jx2 + i Jx, Jy + Jy2 = J 2 - Jz2 - Jz

Thus

|c+|2 = 2 [a - b(b + 1)]

In the same way, you can convince yourself that

|c-|2 = 2 [a - b(b - 1)]

Since |c+|2 and |c-|2 are positive quantities, we have restrictions on the allowable values for the eigenvalues a and b. Indeed, if for a given value of a, any value of b would be allowed, then by applying J+ or J- to |a, b multiple times, we would keep increasing the products b(b + 1) until |c+|2 becomes negative. The only way to avoid this contradiction is by saying that b must be restricted to a finite interval. Indeed, the only way of

stopping the iterative process is to find |c+| = 0 and c-| = 0 at some point. There must therefore exist:

1. A maximum value bmax of b such that a = bmax(bmax + 1)

2. A minimum value bmin of b such that a = bmin(bmin - 1)

The ket J +|a, bmax then is 0, as is the ket J -|a, bmin . We can write

bmax(bmax + 1) = bmin(bmin - 1)

The only solution of this equation with bmax > bmin is bmax = -bmin > 0. And since one can go from the eigenvalue bmin to bmax by steps of size 1 through the iterative application of J+, we have the additional condition

bmax = bmin + k

where k N, we conclude:

k bmax = 2 = -bmin

kN

Let us define j k/2, where j is either an integer or a half-integer. We then have a = j(j + 1), and b can vary from -j to j in steps of size 1. We can summarize this as the following important result:

? The eigenvalues of J2 are of the form 2j(j + 1) with j positive integer or half integer

? For a fixed j, the eigenvalues of Jz can be written in the form m, where m can take the following (2j + 1) values: -j, -j + 1, -j + 2, . . . , j - 2, j - 1, j

? If j is an integer, there is an odd number of eigenvalues of Jz for that j

? If j is a half-integer, there is an even number of eigenvalues of Jz for that j

Note that the eigenstates of J2 with eigenvalue 2j(j + 1) belong to a subspace of dimension at least 2j + 1. Indeed, we just found 2j + 1 of them that were also eigenstates of Jz with distinct eigenvalues; they are therefore orthogonal.

In order to motivate the next section, let us talk some more about the Stern-Gerlach experiment. It found that angular momentum was quantized, which can be seen as a consequence of the commutation relations (1) and (2). At this point, it could still be the case that angular momentum only consists of L, the orbital angular momentum, since L does satisfy (1) and (2). However, the Stern-Gerlach found a second important property of angular momentum: it can be such that j is half-integer. We will now show that this impossible if the angular momentum is only made of L, for reasons that have not yet been mentioned.

3

3 Orbital angular momentum

3.1 Orbital angular momentum in spherical coordinates

We use here the usual spherical coordinate system (, , ), with associated basis vectors e, e, and e, where is the colatitude and the azimuth. In these coordinates, the orbital angular momentum operator

is

L = e ?

= i

i

e

-

e sin

The Cartesian components of L are

Lx = i

- sin - cot cos

Ly = i

cos - cot sin

Lz = i

For the sake of completeness, one may also calculate the expression for L2. This is most conveniently done by applying L2 to an arbitrary function (r):

L2 = - 2

e

-

e sin

e

-

e sin

=- 2

e

e

- e

e sin

- e sin

e

+ e sin

e sin

Now, since e/ = -e, e/ = e cos , e/ = 0, e/ = -(e sin + e cos ), this becomes

L2 = - 2

2

1 2

2

+ cot

+

sin2

2

In other words,

L2 = - 2

2

1 2

2

+

cot

+

sin2

2

(3)

Lastly, it can be convenient to use the variable u cos instead of . It is easy to show that in the u,

variables, L2 is

L2 = - 2

(1

-

u2

)

2 u2

-

2u

u

+

1

1 - u2

2 2

(4)

3.2 Eigenvalues and eigenfunctions of the orbital angular momenta

Eigenfunctions of Lz

Let us solve for the eigenfunction of the operator Lz associated with the eigenvalue b, which we call Yb(, , ). By definition, we have

LzYb = bYb

Yb = i

bYb

1 Yb = ib Yb

We solve this equation by separation of the variables: we write Yb(, , ) = P (, )eib, where P is an arbitrary function of and (at this point). Now, here is a crucial point that we did not highlight enough in the previous lectures: wavefunctions in r representation must be single-valued function of the space coordinates, so that we can define their Fourier transforms. For Yb to be a single-valued function, we must have

Yb(, , ) = Yb(, , + 2)

which means that b is an integer. We write b = m N, and we can say that the eigenvalues of Lz are m with m N. This result does not contradict the general results of Section 2.2, but adds an additional constraint regarding orbital angular momenta: half-integer values of m and therefore j are not allowed.

4

In conclusion, the eigenvalues of L2 are of the form 2l(l + 1) with l N, and the eigenvalues Lz for a given l take all the integer values from -l to +l.

Common eigenfunctions of L2 and Lz

We call Ylm the eigenfunction that is common to L2 and Lz with respective eigenvalues 2l(l + 1) and m. We already know that l N and that m can take integer values between -l and +l. We also saw that since Ylm is an eigenfunction of Lz, we can write

Ylm = Plm(, cos )eim

Note that in principle Plm is a function of and , but it is more convenient to use cos as a variable instead of , as we will now see.

Since the variable does not appear in L2, let us ignore it in the notation. Following Eq.(4), the eigenvalue equation for L2 is

(1

-

u2

)

d2Plm(u) du2

-

2u dPlm(u) du

-

m2 l(l + 1) - 1 - u2

Plm(u) = 0

All we have to do is solve the equation above for m = 0, because we can then use the ladder operators L? = Lx ? iLy to go from Yl0 to Ylm for all desired m values. Writing Pl(u) = Pl0(u), Pl(u) satisfies

(1 - u2)Pl (u) - 2uPl (u) + l(l + 1)Pl(u) = 0

(5)

Eq. (5) is the Legendre differential equation. The only solutions that do not have singularities at u = ?1

are polynomials called Legendre polynomials, which we already encountered in this course. Note that the eigenfunctions Ylm presented here, to within a normalization that we will not discuss here,

are called spherical harmonics.

4 Spin angular momentum

The Stern-Gerlach experiments showed that there are atoms and particles for which the angular momentum Jz can only take two values. We therefore are in the case for which j = 1/2, which is the smallest allowable angular momentum. We saw that this angular momentum could not be put in the usual form r ? p, and does not have an equivalent in classical mechanics. It is an intrinsic angular momentum that physicists have decided to call spin. At this point, we know plenty of its mathematical properties to investigate it in detail: it satisfies the commutation relations (1) and (2), and is such that j = 1/2.

4.1 Representation of the states of a particle with spin 1/2

Electrons, protons, and neutrons, and quarks inside the protons and neutrons, are called fermions, meaning that their spin angular momentum is a half-integer. More precisely, let S be the spin angular momentum; fermions are such that they are always in an eigenstate of S2 with eigenvalue 2 ? 1/2 ? (1/2 + 1) = 3/4 2. We saw that the state of a particle without spin was given by the wavefunction (r). For a particle with spin, one adds S2 and Sz to x, y, and z to form to complete set of observables. In fact, we know that for fermions the measurement of S2 always leads to 3/4 2, so only Sz provides new information. A complete measurement leads to simultaneous knowledge of the coordinates x, y, and z, as well as Sz, which can only take the two values sz = ? 2 . We thus define a new probability amplitude (r, sz) such that the probability of finding the particle in the volume dr around r0 with the value sz for the component Sz is given by |(r0, sz)|2.

While acceptable, this notation treats continuous variables (the coordinates) on an equal footing with the discrete variable sz. For this reason, one will preferably adopt the following notation:

+(r) (r, + 2 ) -(r) (r, - 2 ) A general wavefunction thus has two components, which we write in the form of a column vector:

(r) =

+(r) -(r)

= +(r)

1 0

+ -(r)

0 1

(6)

5

In the sum above, each term is the product of an ordinary wavefunction (+(r) or -(r)), and a column vector (+ = (1, 0)T or - = (0, 1)T ) called a spinor. Spin operators such as Sx, Sy or Sz only act on

spinors and in this representation can be seen as 2 ? 2 matrices. Since our spinors are eigenstates of Sz, we

write

Sz = 2

1 0

0 -1

(7)

Note that spinors have played an important role in geometry, and were discovered and studied by the French mathematician Elie Cartan much before their usefulness in quantum mechanics was understood.

4.2 Pauli's spin matrices

We now work in the subspace S of the eigenstates of S2. The dimension of S is 2. We choose a representation that uses the basis of eigenstates of Sz:

|z+ =

1 0

|z- =

0 1

We already know the expression for Sz in this representation, given by Eq. (7). What is the expression for

Sx and Sy? This is what we derive now. Note first that with a rotation, one can make the z-axis coincide with the x-axis or with the y-axis. This

implies that the matrices Sx, Sy and Sz have the same trace and the same determinant. From Sz, we thus know that the trace of the three matrices is 0 and the determinant is 2/4.

For the simplicity of the notation, let us define the matrices x, y and z such that

Sx 2 x

Sy 2 y

Sz 2 z

x, y and z are called Pauli's matrices, and are subject, as we have seen, to the following 3 constraints:

1. They are Hermitian matrices. This is because of the components of the spin angular momentum are physical observables. The most general form for Pauli's matrices must therefore be

a c - id c + id b

(8)

where a, b, c, and d are real numbers.

2. The trace of Pauli's matrices is zero, so that a = -b. Any Pauli matrix can therefore be written as the sum a3 + c1 + d2

with

1 =

0 1

1 0

2 =

0 i

-i 0

3 =

1 0

0 -1

Note that 3 = z 3. The determinant of Pauli's matrices is -1, so that a2 + c2 + d2 = 1

We need one more ingredient to fully characterize Sx and Sy, namely the commutation relations

[1, 2] = 2i3 [2, 3] = 2i1 [3, 1] = 2i2

Let us now write the most general form for Sx:

Sx = 2 (c1 + d2 + a3)

with a2 + c2 + d2 = 1

We know from Eq. (2) that [Sx, Sz] = -i Sy. Plugging our expressions for Sx and Sz in terms of 1, 2, and 3 in this commutation relation, we find

Sy = 2 (c2 - d1)

6

We then use this form for Sy into [Sy, Sz] = i Sx to find

Sx = 2 (c1 + d2) Hence, a = 0, so c2 + d2 = 1 and we can write

Sx = 2 (cos 1 - sin 2) for some real number . We then also have

Sy = 2 (sin 1 + cos 2)

For any angle , the matrices above satisfy the appropriate commutation relation. At this point, we have fixed the z-axis, but are still free to rotate the x and y axes by an angle - so that x = 1 and y = 2. With this choice of axes, we have

Sx = 2 x = 2

0 1

1 0

Sy = 2 y = 2

0 i

-i 0

Sz = 2 3 = 2

1 0

0 -1

The eigenvectors of Sx and Sy in this representation are

|x+ = 1 2

1 1

|x- = 1 2

1 -1

|y+ = 1 2

1 i

|y- = 1 2

1 -i

and they can be written as follows in the basis of eigenvectors of Sz:

|x+

1 =

|z+ + |z-

2

|x-

1 =

|z+ - |z-

2

|y+

1 =

|z+ + i|z-

2

|y-

1 =

|z+ - i|z-

2

7

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download