Spin One-Half, Bras, Kets, and Operators

SPIN ONE-HALF, BRAS, KETS, AND OPERATORS B. Zwiebach September 17, 2013 Contents 1 The Stern-Gerlach Experiment 1 2 Spin one-half states and operators 7 3 Properties of Pauli matrices and index notation 12 4 Spin states in arbitrary direction 16 1 The Stern-Gerlach Experiment In 1922, at the University of Frankfurt (Germany), Otto Stern and Walther Gerlach, did fundamental experiments in which beams of silver atoms were sent through inhomogeneous magnetic fields to observe their deflection. These experiments demonstrated that these atoms have quantized magnetic moments that can take two values. Although consistent with the idea that the electron had spin, this suggestion took a few more years to develop. Pauli introduced a “two-valued” degree of freedom for electrons, without suggesting a physical interpretation. Kronig suggested in 1925 that it this degree of freedom originated from the selfrotation of the electron. This idea was severely criticized by Pauli, and Kronig did not publish it. In the same year Uhlenbeck and Goudsmit had a similar idea, and Ehrenfest encouraged them to publish it. They are presently credited with the discovery that the electron has an intrinsic spin with value “one-half”. Much of the mathematics of spin one-half was developed by Pauli himself in 1927. It took in fact until 1927 before it was realized that the Stern-Gerlach experiment did measure the magnetic moment of the electron. A current on a closed loop induces a magnetic dipole moment. The magnetic moment vector j µ is proportional to the current I on the loop and the area A of the loop: j. j = IA µ (1.1) The vector area, for a planar loop is a vector normal to the loop and of length equal to the value of the area. The direction of the normal is determined from the direction of the current and the right-hand rule. The product µB of the magnetic moment times the magnetic field has units of energy, thus the units of µ are [µ] = erg or gauss 1 Joule Tesla (1.2) When we have a change distribution spinning we get a magnetic moment and, if the distribution has mass, an angular momentum. The magnetic moment and the angular momentum are proportional to each other, and the constant of proportionality is universal. To see this consider rotating radius R ring of charge with uniform charge distribution and total charge Q. Assume the ring is rotating about an axis perpendicular to the plane of the ring and going through its center. Let the tangential velocity at the ring be v. The current at the loop is equal to the linear charge density λ times the velocity: I = λv = Q v. 2πR (1.3) It follows that the magnitude µ of the dipole moment of the loop is µ = IA = Q Q v πR2 = Rv . 2πR 2 (1.4) Let the mass of the ring be M . The magnitude L of the angular momentum of the ring is then L = R(M v). As a result µ = Q Q RM v = L, 2M 2M (1.5) leading to the notable ratio µ Q = . L 2M (1.6) Note that the ratio does not depend on the radius of the ring, nor on its velocity. By superposition, any rotating distribution with uniform mass and charge density will have a ratio µ/L as above, with Q the total charge and M the total mass. The above is also written as µ = Q L. 2M (1.7) an classical electron going in a circular orbit around a nucleus will have both orbital angular momentum and a magnetic moment, related as above, with Q the electron charge and M the electron mass. In quantum mechanics the electron is not actually going in circles around the proton, but the right quantum analog exhibits both orbital angular momentum and magnetic moment. We can ask if the electron can have an intrinsic µ, as if it were, a tiny spinning ball. Well, it has an intrinsic µ but it cannot really be viewed as a rotating little ball of charge (this was part of Pauli’s objection to the original idea of spin). Moreover, we currently view the electron as an elementary particle with zero size, so the idea that it rotates is just not sensible. The classical relation, however, points to the correct result. Even if it has no size, the electron has an intrinsic spin S –intrinsic angular momentum. One could guess that e S? 2me Since angular momentum and spin have the same units we write this as µ = µ = e S ? 2me 2 (1.8) (1.9) This is not exactly right. For electrons the magnetic moment is twice as large as the above relation suggests. One uses a constant “g-factor” to describe this µ = g e~ S , 2me ~ g = 2 for an electron. (1.10) This factor of two is in fact predicted by the Dirac equation for the electron, and has been verified experimentally. To describe the above more briefly, one introduces the canonical value µB of the dipole moment called the Bohr-magneton: µB ≡ e~ J = 9.27 × 10−24 . 2me Tesla (1.11) With this formula we get S , g = 2 for an electron. (1.12) ~ Both the magnetic moment and the angular momentum point in the same direction if the charge is µ = g µB positive. For the electron we thus get µ j = − g µB j S , ~ g = 2. (1.13) Another feature of magnetic dipoles is needed for our discussion. A dipole placed in a non-uniform magnetic field will experience a force. An illustration is given in Figure 1 below, where to the left we show a current ring whose associated dipole moment jµ points upward. The magnetic field lines diverge as we move up, so the magnetic field is stronger as we move down. This dipole will experience a force pointing down, as can be deduced by basic considerations. On a small piece of wire the force dFj is j . The vectors dFj are sketched in the right part of the figure. Their horizontal proportional to Ij × B components cancel out, but the result is a net force downwards. j is given by In general the equation for the force on a dipole jµ in a magnetic field B j . Fj = ∇(jµ · B) (1.14) j increases the fastest. Given that in our Note that the force points in the direction for which jµ · B j are parallel, this direction is the direction in which the magnitude of B j increases situation jµ and B the fastest. The Stern-Gerlach experiment uses atoms of silver. Silver atoms have 47 electrons. Forty-six of them fill completely the n = 1, 2, 3, and 4 levels. The last electron is an n = 5 electron with zero orbital angular momentum (a 5s state). The only possible angular momentum is the intrinsic angular momentum of the last electron. Thus a magnetic dipole moment is also that of the last electron (the nucleus has much smaller dipole moment and can be ignored). The silver is vaporized in an oven and with a help of a collimating slit a narrow beam of silver atoms is send down to a magnet configuration. 3 Figure 1: A magnetic dipole in a non-uniform magnetic field will experience a force. The force points in the j grows the fastest. In this case the force is downward. direction for which j µ·B In the situation described by Figure 2 the magnetic field points mostly in the positive z direction, and the gradient is also in the positive z-direction. As a result, the above equation gives ∂Bz Fj ≃ ∇(µz Bz ) = µz ∇Bz ≃ µz jez , ∂z (1.15) and the atoms experience a force in the z-direction proportional to the z-component of their magnetic moment. Undeflected atoms would hit the detector screen at the point P . Atoms with positive µz should be deflected upwards and atoms with negative µz should be deflected downwards. Figure 2: A sketch of the Stern-Gerlach apparatus. An oven and a collimating slit produces a narrow beam of silver atoms. The beam goes through a region with a strong magnetic field and a strong gradient, both in the z-direction. A screen, to the right, acts as a detector. The oven source produces atoms with magnetic moments pointing in random directions and thus 4 the expectation was that the z-component of the magnetic moment would define a smooth probability distribution leading to a detection that would be roughly like the one indicated on the left side of Figure 3. Surprisingly, the observed result was two separate peaks as if all atoms had either a fixed positive µz or a fixed negative µz . This is shown on the right side of the figure. The fact that the peaks are spatially separated led to the original cumbersome name of “space quantization.” The Stern Gerlach experiment demonstrates the quantization of the dipole moment, and by theoretical inference from (1.13), the quantization of the spin (or intrinsic) angular momentum. Figure 3: Left: the pattern on the detector screen that would be expected from classical physics. Right: the observed pattern, showing two separated peaks corresponding to up and down magnetic moments. It follows from (1.13) that Sz . (1.16) ~ The deflections calculated using the details of the magnetic field configuration are consistent with µz = − 2 µB Sz 1 ~ Sz = ± , or = ± . 2 2 ~ (1.17) A particle with such possible values of Sz /~ is called a spin one-half particle. The magnitude of the magnetic moments is one Bohr magneton. With the magnetic field and its gradient along the z-direction, the Stern-Gerlach apparatus mea­ j in the z direction. To streamline our pictures we will denote such sures the component of the spin S apparatus as a box with a ẑ label, as in Figure 4. The box lets the input beam come in from the left and lets out two beams from the right side. If we placed a detector to the right, the top beam would be identified as having atoms with Sz = ~/2 and the bottom having atoms with Sz = −~/2.1 Let us now consider thought experiments in which we put a few SG apparatus in series. In the first configuration, shown at the top of Figure 5, the first box is a ẑ SG machine, where we block the Sz = −~/2 output beam and let only the Sz = ~/2 beam go into the next machine. This machine acts as a filter. The second SG apparatus is also a ẑ machine. Since all ingoing particles have Sz = ~/2 the second machine lets those out the top output and nothing comes out in the bottom output. The quantum mechanical lesson here is that Sz = ~/2 states have no component or amplitude along Sz = −~/2. These are said to be orthogonal states. 1 In the quantum mechanical view of the experiment, a single atom can be in both beams, with different amplitudes. Only the act of measurement, which corresponds to the act of placing the detector screen, forces the atom to decide in which beam it is. 5 Figure 4: Left: A schematic representation of the SG apparatus, minus the screen. Figure 5: Left: Three configurations of SG boxes. The second configuration in the figure shows the outgoing Sz = ~/2 beam from the first machine going into an x̂-machine. The outputs of this machine are –in analogy to the ẑ machine– Sx = ~/2 and Sx = −~/2. Classically an object with angular momentum along the z axis has no component of angular momentum along the x axis, these are orthogonal directions. But the result of the experiment indicates that quantum mechanically this is not true for spins. About half of the Sz = ~/2 atoms exit through the top Sx = ~/2 output, and the other half exit through the bottom Sx = −~/2 output. Quantum mechanically, a state with a definite value of Sz has an amplitude along the state Sx = ~/2 as well as an amplitude along the state Sx = −~/2. In the third and bottom configuration the Sz = ~/2 beam from the first machine goes into the x̂ machine and the top output is blocked so that we only have an Sx = −~/2 output. That beam is 6 fed into a ẑ type machine. One could speculate that the beam entering the third machine has both Sx = −~/2 and Sz = ~/2, as it is composed of silver atoms that made it through both machines. If that were the case the third machine would let all atoms out the top output. This speculation is falsified by the result. There is no memory of the first filter: the particles out of the second machine do not anymore have Sz = ~/2. We find half of the particles make it out of the third machine with Sz = ~/2 and the other half with Sz = −~/2. In the following section we discuss a mathematical framework consistent with the the results of the above thought experiments. 2 Spin one-half states and operators The SG experiment suggests that the spin states of the electron can be described using two basis vectors (or kets): |z; +) and |z; −) . (2.1) The first corresponds to an electron with Sz = ~2 . The z label indicates the component of the spin, and the + the fact that the component of spin is positive. This state is also called ‘spin up’ along z. The second state corresponds to an electron with Sz = − ~2 , that is a ‘spin down’ along z. Mathematically, we have an operator Ŝz for which the above states are eigenstates, with opposite eigenvalues: ~ |z; +) 2 ~ Ŝz |z; −) = − |z; −) . 2 Ŝz |z; +) = + (2.2) If we have two basis states, then the state space of electron spin is a two-dimensional complex vector space. Each vector in this vector space represents a possible state of the electron spin. We are not discussing other degrees of freedom of the electron, such as its position, momentum, or energy. The general vector in the two-dimensional space is an arbitrary linear combination of the basis states and thus takes the form |Ψ) = c1 |z; +) + c2 |z; −) , with c1 , c2 ∈ C (2.3) It is customary to call the state |z; +) the first basis state and it denote by |1). The state |z; −) is called the second basis state and is denoted by |2). States are vectors in some vector space. In a two- dimensional vector space a vector is explicitly represented as a column vector with two components. ( ) ( ) 1 0 The first basis vector is represented as and the second basis vector is represented as . Thus 0 1 we have the following names for states and their concrete representation as column vectors ( ) 1 |z : +) = |1) ←→ , 0 (2.4) ( ) 0 |z : −) = |2) ←→ . 1 7 Using these options the state in (2.3) takes the possible forms ( ) ( ) ( ) 1 0 c1 |Ψ) = c1 |z; +) + c2 |z; −) = c1 |1) + c2 |2) ←→ c1 + c2 = . 0 1 c2 (2.5) As we mentioned before, the top experiment in Figure 5 suggests that we have an orthonormal basis. The state |z; +) entering the second machine must have zero overlap with |z, −) since no such down spins emerge. Moreover the overlap of |z; +) with itself must be one, as all states emerge from the second machine top output. We thus write (z; −|z; +) = 0 , (z; +|z; +) = 1 . (2.6) (z; +|z; −) = 0 , (z; −|z; −) = 1 . (2.7) and similarly, we expect Using the notation where the basis states are labeled as |1) and |2) we have the simpler form that summarizes the four equations above: (i|j) = δij , i, j = 1, 2. (2.8) We have not yet made precise what we mean by the ‘bras’ so let us do so briefly. We define the basis ‘bras’ as the row vectors obtained by transposition and complex conjugation: (1| ←→ (1, 0) , (2| ←→ (0, 1) . (2.9) Given states |α) and |β) ( ) α1 |α) = α1 |1) + α2 |2) ←→ α2 ( ) β1 |β) = β1 |1) + β2 |2) ←→ β2 (2.10) (α| ≡ α∗1 (1| + α∗2 (2| ←→ (α∗1 , α∗2 ) (2.11) we associate and the ‘bra-ket’ inner product is defined as the ‘obvious’ matrix product of the row vector and column vector representatives: (α|β) ≡ (α∗1 , α∗2 ) · ( ) β1 = α∗1 β1 + α∗2 β2 . β2 (2.12) Note that this definition is consistent with (2.8). When we represent the states as two-component column vectors the operators that act on the states to give new states can be represented as two-by-two matrices. We can thus represent the operator Ŝz as a 2 × 2 matrix which we claim takes the form Ŝz ~ = 2 ( 8 ) 1 0 . 0 −1 (2.13) To test this, it suffices to verify that the matrix Ŝz acts on the column vectors that represent the basis states as expected from (2.2). Indeed, ( )( ) ( ) ~ 1 ~ 1 ~ 0 1 = + = + |z; +) Ŝz |z; +) = + 0 2 0 −1 2 0 2 ( )( ) ( ) ~ 1 ~ 0 ~ 0 0 Ŝz |z; −) = + = − = − |z; −) . 1 2 0 −1 2 1 2 (2.14) In fact, the states |1) and |2), viewed as column vectors are the eigenstates of matrix Ŝz . There is nothing particular about the z axis. We could have started with a SG apparatus that measures spin along the x axis and we would have been led to an operator Ŝx . Had we used the y axis we would have been led to the operator Ŝy . Since spin represents angular momentum (albeit of intrinsic type), it is expected to have three components, just like orbital angular momentum has three components: L̂x , L̂y , and L̂z . These are all hermitian operators, written as products of coordinates and momenta in three-dimensional space. Writing L̂x = L̂1 , L̂y = L̂2 , and L̂z = L̂3 , their commutation relations can be briefly stated as L̂i , L̂j = i~ǫijk L̂k (2.15) This is the famous algebra of angular momentum, repeated indices are summed over the values 1,2,3, and ǫijk is the totally antisymmetric symbol with ǫ123 = +1. Make sure that you understand this notation clearly, and can use it to see that it implies the relations [L̂x , L̂y ] = i~L̂z , [L̂y , L̂z ] = i~L̂x , (2.16) [L̂z , L̂x ] = i~L̂y . While, for example, L̂z = x̂p̂y − ŷp̂x is a hermitian operator written in terms of coordinates and momenta, we have no such construction for Ŝz . The latter is a more abstract operator, it does not act on wavefunctions ψ(jx) but rather on the 2-component column vectors introduced above. The operator Ŝz is just a two-by-two hermitian2 matrix with constant entries! If spin is a quantum mechanical angular momentum, we must have that the triplet of operators Ŝz , Ŝx , and Ŝy satisfy [Ŝx , Ŝy ] = i~Ŝz , [Ŝy , Ŝz ] = i~Ŝx , (2.17) [Ŝz , Ŝx ] = i~Ŝy , or, again using numerical subscripts for the components (Ŝ1 = Ŝx , · · · ) we must have Ŝi , Ŝj 2 = i~ǫijk Ŝk . (2.18) Hermitian means that the matrix is preserved by taking the operations of transposition and complex conjugation. 9 We can now try to figure out how the matrices for Ŝx and Ŝy must look, given that we know the matrix for Ŝz . We have a few constraints. First the matrices must be hermitian, just like the angular momentum operators are. Two-by-two hermitian matrices take the form ( ) 2c a − ib , with a, b, c, d ∈ R a + ib 2d (2.19) Indeed, you can easily see that transposing and complex conjugating gives exactly the same matrix. Since the two-by-two identity matrix commutes with every matrix, we can subtract from the above matrix any multiple of the identity without any loss of generality. Subtracting the identity matrix times (c + d) we find the still hermitian matrix ( ) c − d a − ib , with a, b, c, d ∈ R a + ib d − c (2.20) Since we are on the lookout for Ŝx and Ŝy we can subtract a matrix proportional to Ŝz . Since Ŝz is diagonal with entries of same value but opposite signs, we can cancel the diagonal terms above and are left over with ( ) 0 a − ib , with a, b ∈ R a + ib 0 (2.21) Thinking of the space of two-by-two hermitian matrices as a real vector space, the hermitian matrices given above can be associated to two basis “vectors” that are the matrices ( ) ( ) 0 1 0 −i , 1 0 i 0 (2.22) since multiplying the first by the real constant a and the second by the real constant b and adding gives us the matrix above. In fact, together with the identity matrix and the Ŝz matrix, with the ~/2 deleted, ( ) ( ) 1 0 1 0 , , 0 1 0 −1 (2.23) we got the complete set of four two-by-two matrices that viewed as basis vectors in a real vector space, can be used to build the most general hermitian two-by-two matrix by using real linear combinations. Back to our problem, we are supposed to find Ŝx and Ŝy among the matrices in (2.22). The overall scale of the matrices can be fixed by the constraint that their eigenvalues be ±~/2, just like they are for Ŝz . Let us give the eigenvalues (denoted by λ) and the associated normalized eigenvectors for these two matrices. Short computations (can you do them?) give ( ) ( ) ( ) 1 1 1 0 1 1 : λ = 1, for √ , λ = −1, for √ , 1 0 2 1 2 −1 for the first matrix and ( ) 0 −i : i 0 1 λ = 1, for √ 2 ( ) 1 , i 10 1 λ = −1, for √ 2 ( ) 1 , −i (2.24) (2.25) ( ) c1 for the second matrix. In case you are puzzled by the normalizations, note that a vector is c2 normalized if |c1 |2 + |c2 |2 = 1. Since the eigenvalues of both matrices are ±1, we tentatively identify ( ) ( ) ~ 0 1 ~ 0 −i Ŝx = , (2.26) , Ŝy = 2 1 0 2 i 0 which have, at least, the correct eigenvalues. But in fact, these also satisfy the commutation relations! Indeed, we check that, as desired, ~2 [Sˆx , Sˆy ] = 4 ( 0 1 1 0 ~2 = 4 ( i 0 ~2 = 4 ( 2i 0 )( ) ( )( ) 0 −i 0 −i 0 1 − i 0 i 0 1 0 ) ( ) 0 −i 0 − −i 0 i ) ( ) ~ 1 0 0 = i~ = i~Sˆz . −2i 2 0 −1 (2.27) All in all we have Ŝx ~ = 2 ( ) 0 1 , 1 0 Ŝy ~ = 2 ( ) 0 −i , i 0 Ŝz ~ = 2 ( 1 0 0 −1 ) . (2.28) Exercise. Verify that the above matrices satisfy the other two commutation relations in (2.17). You could ask if we got the unique solution for Ŝx and Ŝy given the choice of Ŝz ? The answer is no, but it is not our fault, as illustrated by the following check: Exercise. Check that the set of commutation relations of the spin operators are in fact preserved when we replace Ŝx → −Ŝy and Ŝy → Ŝx . The solution we gave is the one conventionally used by all physicists. Any other solution is physically equivalent to the one we gave (as will be explained in more detail after we develop more results). The solution defines the Pauli matrices σi by writing Ŝi = ~ σi . 2 We then have that the Pauli matrices are ( ) ( ) 0 1 0 −i σ1 = , σ2 = , 1 0 i 0 (2.29) σ3 = ( ) 1 0 . 0 −1 (2.30) Let us describe the eigenstates of Ŝx , which given (2.26) can be read from (2.24): Ŝx |x; ±) = ± |x; ±) . (2.31) with 1 |x; +) = √ |z; +) + 2 1 |x; −) = √ |z; +) − 2 ( ) 1 1 1 √ |z; −) ←→ √ , 2 2 1 ( ) 1 1 1 √ |z; −) ←→ √ , −1 2 2 11 (2.32) Note that these states are orthogonal to each other. The above equations can be inverted to find 1 |z; +) = √ |x; +) + 2 1 |z; −) = √ |x; +) − 2 1 √ |x; −) 2 1 √ |x; −) 2 (2.33) These relations are consistent with the second experiment shown in Figure 5. The state |z; +) entering the second, x̂-type SG apparatus, has equal probability to be found in |x; +) as it has probability to be found in |x; −). This is reflected in the first of the above relations, since we have the amplitudes 1 (x; +|z; +) = √ , 2 1 (x; −|z; +) = √ . 2 (2.34) These probabilities, being equal to the norm squared of the amplitudes, are 1/2 in both cases. The relative minus sign on the second equation above is needed to make it orthogonal to the state on the first equation. We can finally consider the eigenstates of Ŝy axis. We have ~ Ŝy |y; ±) = ± |y; ±) . 2 (2.35) and using (2.25) we read 1 |y; +) = √ |z; +) + 2 1 |y; −) = √ |z; +) − 2 ( ) i 1 1 √ |z; −) ←→ √ , 2 2 i ( ) i 1 1 √ |z; −) ←→ √ . 2 2 −i (2.36) Note that this time the superposition of |z; ±) states involves complex numbers (there would be no way to find y type states without them). 3 Properties of Pauli matrices and index notation Since we know the commutation relations for the spin operators and we have Si = ~2 σi , it follows that   Ŝi , Ŝj = i~ ǫijk Ŝk ,  ~ ~ ~ σi , σj = i~ ǫijk σk . 22 2 (3.37) (3.38) Cancelling the ~’s and some factors of two, we find [σi , σj ] = 2i ǫijk σk . 12 (3.39) Another important property of the Pauli matrices is that they square to the identity matrix. This is best checked explicitly (do it!): (σ1 )2 = (σ2 )2 = (σ3 )2 = 1 . (3.40) This property “explains” that the eigenvalues of each of the Pauli matrices could only be plus or minus one. Indeed, the eigenvalues of a matrix satisfy the algebraic equation that the matrix satisfies. Take for example a matrix M that satisfies the matrix equation M 2 + αM + β1 = 0 (3.41) Let v be an eigenvector of M with eigenvalue λ: M v = λv. Let the above equation act on v M 2 v + αM v + β1v = 0 → λ2 v + αλv + βv = 0 → (λ2 + αλ + β)v = 0 , (3.42) and since v = 0 (by definition an eigenvector cannot be zero!) we conclude that λ2 + αλ + β = 0, as claimed. For the case of the Pauli matrices we have (σi )2 = 1 and therefore the eigenvalues must satisfy λ2 = 1. As a result, λ = ±1 are the only options. We also note, by inspection, that the Pauli matrices have zero trace, namely, the sum of entries on the diagonal is zero: tr(σi ) = 0 , i = 1, 2, 3. (3.43) A fact from linear algebra is that the trace of a matrix is equal to the sum of its eigenvalues. So each Pauli matrix must have two eigenvalues that add up to zero. Since the eigenvalues can only be plus or minus one, we must have one of each. This shows that each of the Pauli matrices has a plus one and a minus one eigenvalue. If you compute a commutator of Pauli matrices by hand you might notice a curious property. Take the commutator of σ1 and σ2 : [σ1 , σ2 ] = σ1 σ2 − σ2 σ1 . The two contributions on the right hand side give ( )( ) ( ) 0 1 0 −i i 0 σ1 σ2 = = , 1 0 i 0 0 −i ( )( ) ( ) 0 −i 0 1 −i 0 = . σ2 σ1 = i 0 1 0 0 i (3.44) (3.45) The second contribution is minus the first, so that both terms contribute equally to the commutator! In other words, σ1 σ2 = − σ2 σ1 . (3.46) This equation is taken to mean that σ1 and σ2 anticommute. Just like we define the commutator of two operators X, Y by [X, Y ] ≡ XY − Y X, we define the anticommutator, denoted by curly brackets, by Anticommutator: {X , Y } ≡ XY + Y X . 13 (3.47) In this language we have checked that {σ1 , σ2 } = 0 , (3.48) and the property σ12 = 1, for example, can be rewritten as {σ1 , σ1 } = 2 · 1 . (3.49) In fact, as you can check (two cases to examine) that any two different Pauli matrices anticommute: {σi , σj } = 0 , for i 6= j . (3.50) We can easily improve on this equation to make it work also when i is equal to j. We claim that {σi , σj } = 2δij 1 . (3.51) 6 j the right-hand side vanishes, as needed, and when i is equal to j, the right-hand Indeed, when i = side gives 2 · 1, also as needed in view of (3.49) and its analogs for the other Pauli matrices. Both the commutator and anti-commutator identities for the Pauli matrices can be summarized in a single equation. This is possible because, for any two operators X, Y we have XY = 1 2 {X, Y } + 1 2 [X, Y ] , (3.52) as you should confirm by expansion. Applied to the product of two Pauli matrices and using our expressions for the commutator and anticommutator we get σi σj = δij 1 + i ǫijk σk . (3.53) This equation can be recast in vector notation. Denote by bold symbols three-component vectors, for example, a = (a1 , a2 , a3 ) and b = (b1 , b2 , b3 ). Then the dot product a · b = a1 b1 + a2 b2 + a3 b3 = ai bi = ai bj δij . (3.54) Note the use of the sum convention: repeated indices are summed over. Moreover, note that bj δij = bi (can you see why?). We also have that a · a = |a|2 . (3.55) Cross products use the epsilon symbol. Make sure you understand why (a × b)k = ai bj ǫijk . (3.56) We can also have triplets of operators, or matrices. For the Pauli matrices we denote σ ≡ (σ1 , σ2 , σ3 ) . (3.57) We can construct a matrix by dot product of a vector a with the ‘vector’ σ. We define a · σ ≡ a1 σ1 + a2 σ2 + a3 σ3 = ai σi . 14 (3.58) Note that a·σ is just a single two-by-two matrix. Since the components of a are numbers, and numbers commute with matrices, this dot product is commutative: a · σ = σ · a. We are now ready to rewrite (3.53). Multiply this equation by ai bj to get ai σi bj σj = ai bj δij 1 + i (ai bj ǫijk ) σk = (a · b) 1 + i (a × b)k σk , (3.59) so that, finally, we get the matrix equation (a · σ)(b · σ) = (a · b) 1 + i (a × b) · σ . (3.60) As a simple application we take b = a. We then have a · a = |a|2 and a × a = 0, so that the above equation gives (a · σ)2 = |a|2 1 . (3.61) When a is a unit vector this becomes (n · σ)2 = 1 , n a unit vector. (3.62) The epsilon symbol satisfies useful identities. One can show that the product of two epsilons with one index contracted is a sum of products of Kronecker deltas: ǫijk ǫipq = δjp δkq − δjq δkp . (3.63) Its contraction (setting p = j) is also useful: ǫijk ǫijq = 2δkq . (3.64) The first of these two allows one to prove the familiar vector identity a × (b × c) = b (a · c) − (a · b) c . (3.65) It will be useful later on to consider the dot and cross products of operator triplets. Given the operators X = (X̂1 , X̂2 , X̂3 ) and Y = (Ŷ1 , Ŷ2 , Ŷ3 ) we define ˆ i Yˆi , X·Y ≡ X (X × Y)i ≡ ǫijk X̂j Ŷk . (3.66) In these definitions the order of the operators on the right hand side is as in the left-hand side. This is important to keep track of, since the X̂i and Ŷj operators may not commute. The dot product of two operator triplets is not necessarily commutative, nor is the cross product necessarily antisymmetric. 15 4 Spin states in arbitrary direction We consider here the description and analysis of spin states that point in arbitrary directions, as specified by a unit vector n: n = (nx , ny , nz ) = (sin θ cos φ, sin θ sin φ, cos θ) . (4.67) Here θ and φ are the familiar polar and azimuthal angles. We view the spatial vector n as a triplet of numbers. Just like we did for σ, we can define S as the triplet of operators S = (Ŝx , Ŝy , Ŝz ) . (4.68) Note that, in fact, S = ~ σ. 2 (4.69) We can use S to obtain, by a dot product with n a spin operator Ŝn that has a simple interpretation: Ŝn ≡ n · S ≡ nx Ŝx + ny Ŝy + nz Ŝz = ~ n·σ. 2 (4.70) Note that Ŝn is just an operator, or a hermitian matrix. We view Ŝn as the spin operator in the direction of the unit vector n. To convince you that this makes sense note that, for example, when n points along z, we have (nx , ny , nz ) = (0, 0, 1) and Ŝn becomes Ŝz . The same holds, of course, for the x and y directions. Moreover, just like all the Ŝi , the eigenvalues of Ŝn are ±~/2. This is needed physically, since all directions are physically equivalent and those two values for spin must be the only allowed values for all directions. To see that this is true we first compute the square of the matrix Ŝn :  ~ 2  ~ 2 , (4.71) (Ŝn )2 = (n · σ)2 = 2 2 using (3.62). Moreover, since the Pauli matrices are traceless so is Ŝn : tr(Ŝn ) = ni tr(Ŝi ) = ni ~ tr(σi ) = 0 . 2 (4.72) By the same argument we used for Pauli matrices, we conclude that the eigenvalues of Ŝn are indeed ±~/2. For an arbitrary direction we can write the matrix Ŝn explicitly: ( ) ( ) ( ) ~� 0 1 0 −i 1 0 � nx Ŝn = + ny + nz 1 0 i 0 0 −1 2 ( ) ~ nx − iny nz = −nz 2 nx + iny � � cos θ sin θe−iφ ~ . = 2 sin θeiφ − cos θ (4.73) Since the eigenvalues of Ŝn are ±~/2 the associated spin eigenstates, denoted as |n; ±), satisfy ~ Ŝn |n; ±) = ± |n; ±) . 2 16 (4.74) The states |n; +) and |n; −) represent, respectively, a spin state that points up along n, and a spin state that points down along n. We can also find the eigenvalues of the matrix Ŝn by direct computation. The eigenvalues are the roots of the equation det(Ŝn − λ1) = 0: � ~ � ! ~ −iφ cos θ − λ sin θe ~2 ~2 2 2 2 2 2 det 2~ = λ − (cos θ + sin θ) = λ − = 0. iφ 4 4 − 2~ cos θ − λ 2 sin θe (4.75) The eigenvalues are thus λ = ±~/2, as claimed. To find the eigenvector v associated with the eigenvalue λ we must solve the linear equation (Ŝn − λ1)v = 0. We denote by |n; +) the eigenvector associated with the eigenvalue ~/2. For this eigenvector we write the ansatz ( ) c1 |n; +) = c1 |+) + c2 |−) = , c2 (4.76) where for notational simplicity |±) refer to the states |z; ±). The eigenvector equation becomes (Ŝn − 2~ 1)|n; +) = 0 and explicitly reads � ~ cos θ − 1 2 sin θeiφ sin θe−iφ � !� � ! c1 − cos θ − 1 c2 = 0. (4.77) Either equation gives the same relation between c1 and c2 . The top equation, for example gives c2 = eiφ sin 2θ 1 − cos θ c1 . c1 = eiφ sin θ cos 2θ (4.78) (Check that the second equation gives the same relation.) We want normalized states, and therefore � � 2 θ sin θ 2 (4.79) |c1 |2 + |c2 |2 = 1 → |c1 |2 1 + = 1 → |c1 |2 = cos2 . θ 2 2 cos 2 Since the overall phase of the eigenstate is not observable we take the simplest option for c1 : c1 = cos θ2 , c2 = sin θ2 exp(iφ) , (4.80) that is |n; +) = cos 2θ |+) + sin 2θ eiφ |−) . (4.81) As a quick check we see that for θ = 0, which corresponds to a unit vector n = e3 along the plus z direction we get |e3 ; +) = |+). Note that even though φ is ambiguous when θ = 0, this does not affect our answer, since the term with φ dependence vanishes. In the same way one can obtain the normalized eigenstate corresponding to −~/2. A simple phase choice gives |n; −) = sin 2θ |+) − cos 2θ eiφ |−) . (4.82) If we again consider the θ = 0 direction, this time the ambiguity of φ remains in the term that contains the |z; −) state. It is convenient to multiply this state by the phase −e−iφ . Doing this, the pair of 17 eigenstates read3 |n; +) = cos 2θ |+) + sin θ2 eiφ |−) , |n; −) = − sin θ2 e−iφ |+) + cos 2θ |−) . (4.83) The vectors are normalized. Furthermore, they are orthogonal (n; −|n; +) = − sin θ2 eiφ cos θ2 + cos θ2 sin 2θ eiφ = 0 . (4.84) Therefore, |n; +) and |n; −) are an orthonormal pair of states. Let us verify that the |n; ±) reduce to the known results as n points along the z, x, and y axes. Again, if n = (0, 0, 1) = e3 , we have θ = 0, and hence |e3 ; +) = |+) , |e3 ; −) = |−) , (4.85) which are, as expected, the familiar eigenstates of Ŝz . If we point along the x axis, n = (1, 0, 0) = e1 which corresponds to θ = π/2, φ = 0. Hence 1 |e1 ; +) = √ (|+) + |−)) = |x; +) , 2 1 |e1 ; −) = √ (−|+) + |−)) = −|x; −) , 2 (4.86) where we compared with (2.32). Note that the second state came out with an overall minus sign. Since overall phases (or signs) are physically irrelevant, this is the expected answer: we got the eigenvectors of Ŝx . Finally, if n = (0, 1, 0) = e2 , we have θ = π/2, φ = π/2 and hence, with e±iφ = ±i, we have 1 |e2 ; +) = √ (|+) + i|−)) = |y; +) , 2 1 1 |e2 ; −) = √ (i|+) + |−)) = i √ (|+) − i|−)) = i|y; −) (4.87) 2 2 which are, up to a phase for the second one, the eigenvectors of Ŝy . 3 The formula (4.83) works nicely at the north pole (θ = 0), but at the south pole (θ = π) the φ ambiguity shows up again. If one works near the south pole multiplying the results in (4.83) by suitable phases will do the job. The fact that no formula works well unambiguously through the full the sphere is not an accident. 18