Handout 4: Implications of Relativistic Quantum Mechanics ... Until now, we have
skirted round the issue of the negative energy solutions of the Dirac equation.
November 2016
PX408: Relativistic Quantum Mechanics Tim Gershon
(
[email protected])
Handout 4: Implications of Relativistic Quantum Mechanics Interpretation of Negative Energy Solutions Until now, we have skirted round the issue of the negative energy solutions of the Dirac equation. Since we considered these to be one of the shortcomings of the Klein-Gordon equation, we should either apply the same logic here, or find a physically meaningful interpretation. In fact, it is impossible to ignore the negative energy solutions (although this was proposed by no less a physicist than Schr¨ odinger). The Dirac equation gives a non-zero probability of transition between a positive energy state and a negative energy state, so there must be some reason to explain why we do not observe such transitions. Moreover, we require a complete set of wavefunctions in order to be able to represent an arbitrary wavefunction as a (Fourier) expansion. If we exclude the negative energy states, we no longer have a complete set of wavefunctions.
10
E
10
E
8
8
6
6
4
4
+m
2 0
0
−2
-m
−2
−4
−4
−6
−6
−8 −10 −10 −8 −6 −4
+m
2
-m
−8 −2
0
2
4
6
8
10
|p|
−10 −10 −8 −6 −4
−2
0
2
4
6
8
10
|p|
Figure 1: (Left) Indicative positive and negative energy levels. (Right) An electron from the Dirac sea absorbs a photon and is promoted to an empty positive energy state, leaving behind a hole. Dirac was aware of these problems, and found a conceptual solution courtesy of the Pauli exclusion principle. He considered energy levels, as illustrated in Fig. 1, but assumed that all the negative energy states were filled. Therefore, the exclusion principle forbids transitions from positive energy states to the negative energy states. The existence of the “Dirac sea” of an infinite number of filled states may be philosophically unappealing, but it is at least mathematically consistent. This argument solves the problem of why positive energy electrons are not forever falling into lower and lower negative energy states, but it raises a question concerning the reverse process. Namely, it should be possible for a particle in a negative energy state to absorb a photon and be promoted into a positive energy state. Dirac suggested that when this occurs the particle leaves behind a “hole” in the sea of negative energy states (see Fig. 1(right)). The hole (i.e. the absence of a negative energy electron) behaves as a positive energy particle with opposite charge to the electron (i.e. as a positron).
This argument works well for the spin-1/2 electrons, and indeed a variant of the theory is still widely used in solid state physics today. However, there is still a problem if one considers spin0 particles described by the Klein-Gordon equation. Negative energies are also allowed for such particles, but the exclusion principle does not provide protection against transitions to the negative energy states. An alternative interpretation was presented by Feynman. The crux of the argument is that negative energy states represent particles moving backwards in time. That is, if E is positive, then µ µ e−ipµ x appears in the wavefunction of an electron with energy E = p0 . If E is negative, then e−ipµ x gives the wavefunction of a positron with energy E = −p0 (in fact, with four-momentum −pµ ). It µ is more convenient to write this as e+ipµ x , where the four-momentum is positive (E = p0 ). Note µ µ that e−ipµ x and e+ipµ x together provide the complete set of states required to write any arbitrary wavefunction as a Fourier series. Let us consider the implications of the possibility of having particles moving backwards in time. At first glance this appears to be more appropriate as a relativistic theory, since it puts spatial and temporal variables on the same footing. However, philosophical problems arise since in our day-to-day experience we have no concept of “backwards in time”. Indeed, the connection between microscopic and thermodynamic (or cosmological) time is a profound problem. However, we will not concern ourselves greatly with philosophical matters here. Recall that in non-relativistic quantum mechanics we could obtain the total amplitude for a transition from point A to point B by summing over all possible trajectories (appropriately weighted). In NRQM the complete set of trajectories was limited to those which progressed monotonically forwards in time. In the new interpretation, additional possibilities are included. Let’s take a closer look at some of the new possibilities that are presented by the concept of particles travelling backwards in time. Figure 2 illustrates on the left a typical amplitude for an electron in non-relativistic quantum mechanics. In the middle there is a superficially similar trajectory that includes a leg in which the electron travels backwards in time, and is therefore only allowed in relativistic quantum mechanics. On the right, an alternative interpretation of the same diagram is shown, in which the electron travelling backwards in time is replaced with a positron travelling forwards in time. [Note that if absorption and emission of photons were added at the two “vertices”, these diagrams would represent Compton scattering amplitudes.] t
t
t
t = ta −
e
e−
e−
−
+
e e−
e
e−
t = tp
e−
e−
x
x
x
Figure 2: (Left) Typical non-relativistic quantum mechanical scattering; (middle) scattering possible only in relativistic quantum mechanics; (right) reinterpretation of the middle diagram. If we consider the progression in time shown in Figure 2(right), there are the following distinct stages: • t < tp : the initial e− is present; • t = tp : somewhere in space, independent of the initial e− , an e+ e− pair is produced;
• tp < t < ta : the initial e− and the produced e+ e− continue their independent trajectories; • t = ta : the initial e− meets the produced e+ and they annihilate; the produced e− continues its independent trajectory; • ta < t: the produced e− remains in the final state. It appears strange that the final state can contain an electron that is, in principle, completely different to that in the initial state. For example, it is not necessarily immediately obvious that the initial and final state particles must have the same momentum or spin. In fact, this requirement is enforced due to four-momentum (and spin) conservation at the production and annihilation points. (Also, the figure makes it appear as if the outgoing electron can have an arbitrary spatial shift relative to the incoming one – this is, of course, not the case.) In order to handle diagrams like those in Figure 2 correctly, we will need a theory that naturally handles pair production and annihilation. Quantum field theory provides this next step forward, as will be discussed in the Gauge Theories module. Nonetheless, we have seen that Dirac’s theory predicts the existence of an antiparticle partner to the electron, i.e. the positron. This particle is identical to the electron, but with all quantum numbers reversed (specifically, it has the same mass but opposite charge). The positron was discovered by Anderson in 1932, providing impressive vindication of Dirac’s theory. (Curiously, Dirac was not bold enough to explicitly predict the existence of the positron himself, and had spent some effort trying to understand how the only positively charged particle that was known at that time – the proton – could be the electron’s antiparticle but yet have a very different mass.) The Klein Paradox One interesting example of how pair production can explain results that are otherwise quite mysterious is the so-called “Klein paradox”. Consider a (spin-zero) particle travelling in the positive x direction and impinging on a potential barrier of height V and thickness a, as shown in Fig. 3. It’s wavefunction can be written as e−iEt+ipx . When it meets the barrier, it can either be reflected back 0 0 (Re−iEt−ipx ) or continue forwards with reduced energy and momentum (Ae−iE t+ip x ). At the end of 0 0 the barrier, it can either be reflected back (Be−iE t−ip x ), or continue forwards with the energy and momentum with which it started (T e−iEt+ipx ), in which case we can say it has tunnelled through (or over) the barrier.
V -iEt+ipx
e
Ae
-iE’t+ip’x -iEt+ipx
Te Re
-iEt-ipx
Be
x=0
-iE’t-ip’x
x=a
x
Figure 3: Particle impinging on a potential barrier. The relation between the energy outside and inside the barrier is simply E 0 = E − qV , and the standard relativistic energy-momentum relation gives us p p p = + E 2 − m2 and p0 = + (E − qV )2 − m2 , (1) since we have defined p and p0 to be in the positive x direction. After imposing boundary conditions so that the total wavefunction φ and its derivative dφ/dx are continuous at x = 0 and x = a, we can solve for R, A, B and T .
Q1 Obtain the Klein-Gordon current for the system. Recall that µ JKG = i (φ∗ ∂ µ φ − φ∂ µ φ∗ ) − 2qAµ φ∗ φ ,
and note that Aµ = (V, 0, 0, 0) inside the barrier and is zero outside. Q2 Solve for R, A, B and T . The solution for T is −2 p0 i p 0 0 + sin(p a) . |T | = cos(p a) − 0 2 p p 2
(2)
Let us now consider the behaviour of T in various regimes: • qV < E − m Classically, the particle has enough momentum to get over the potential barrier (p0 is real), and quantum mechanically we find a corresponding result, with transmission probability |T | ≤ 1. • E − m < qV < E + m This is the territory of quantum mechanical tunnelling (p0 is imaginary), familiar from nonrelativistic quantum mechanics. A non-zero transmission probability is obtained, |T | < 1. • E + m < qV A strange thing happens: p0 is now real again. The system behaves much as it did for very low potential barriers: |T | ≤ 1. In spite of the large barrier, it is possible (for certain values of V and a) to have perfect transmission probability! Q3 Describe how |T | varies with V and a in each of the three regimes discussed above. What is the limiting value of |T | as V −→ ∞? This situation appears rather bizarre – that’s why it’s called a paradox, after all. However it can be resolved once we realise that the potential energy in the barrier is sufficient to create particleantiparticle pairs. We can then understand the transmission of particles through the barrier in terms of creation and annihilation events, as shown in Fig. 4.
t M
M
M
x=0
x=a
x
Figure 4: Resolution of the Klein paradox in terms of pair creation and annihilation. Particle and ¯ , respectively. antiparticle are labelled as M and M (In fact, the drawing in Fig. 3 is unphysical: the potential cannot behave as a step function, but must ramp up over some non-zero distance so that the electric field is finite. Extremely strong fields are necessary for pair production, and the Klein paradox has not yet been demonstrated experimentally.)
Q4 Calculate the Klein-Gordon density and 3-current inside the barrier for the case where E +m < qV . Interpret these results in terms of pair production. Q5 To set up an experimental test of the Klein paradox requires a potential change of V ≈ mc2 over the Compton length ~/mc. What is the required electric field? In what physical situations might such a strong field be realised? Dirac’s Explanation of g = 2 One of the most famous successes of relativistic quantum mechanics is Dirac’s explanation for the gyromagnetic ratio of the electron taking the value two. Let us first of all recap the problem as it arises in non-relativistic quantum mechanics. Consider the Schr¨ odinger equation for a spinless particle in an electromagnetic field i
∂ ψ = ∂t =
1 (−i∇ − qA)2 ψ 2m 1 −∇2 + iq(∇.A + A.∇) + q 2 A2 ψ . 2m
(3) (4)
It is important to note that the differential operators act on everything to their right, so that ∇.Aψ = ∇.(Aψ) = (∇.A)ψ + A.(∇ψ) ,
(5)
where we have used the fact that the dot product is commutative. Q6 By writing out explicitly ∇.Aψ =
∂ ∂x Ax ψ
+ . . ., prove Eq. (5).
We therefore obtain i
1 ∂ ψ= −∇2 + 2iqA.∇ + iq(∇.A) + q 2 A2 ψ . ∂t 2m
(6)
1 2 2 The last term ( 2m q A ψ) is often dropped as it is second order (i.e. it contains q 2 , which is a small correction due to the smallness of the fine-structure constant discussed below). However, we can include it for completeness in this discussion. Let us now choose A such that (i) (∇.A) = 0, and (ii) ∇ × A = B = (0 0 B). It is easily shown that these are satisfied with the choice A = 21 B(−y x 0). (The first requirement is called a choice of gauge, and is allowed due to the gauge invariance of the electromagnetic interaction – discussed in the Gauge Theories module. We can understand it by remembering that the electromagnetic field potential Aµ corresponds to photons, and these have only two degrees of freedom. The reduction of degrees of freedom from four – since Aµ is a four-vector – to two is due to the facts that (i) the photon is massless, and (ii) the photon has only two polarisation states. The third polarisation state is removed with the choice of gauge.) Then we find that iqB ∂ ∂ qB iq A.∇ψ = (−y + x )ψ = Lz ψ , (7) m 2m ∂x ∂y 2m
where the operator Lz gives the z-component of angular momentum. This expression generalises to iq q A.∇ψ = − L.Bψ , m 2m which we could alternatively write in terms of the magnetic moment µ =
(8) q 2m L.
Q7 Show that the choice A = 12 B(−y x 0) satisfies (∇.A) = 0 and ∇ × A = B = (0 0 B).
Q8 By considering B fields aligned with x- and y- axes in turn, find equivalent expressions to Eq. (7) and show that Eq. (8) is an appropriate generalisation. Returning to the Schr¨ odinger equation as we left it at Eq. (6), we find i
∂ ∇2 q q2 2 ψ=− ψ− B.Lψ + A ψ. ∂t 2m 2m 2m
(9)
The B.L term causes splitting of states with definite values of l (the quantum number associated with the component of L aligned with the axis of the field B). This is called the Zeeman effect. Note that, with this theory, no splitting is expected for ground state orbits where L = 0 – the experimental observation of this splitting (by Stern-Gerlach) implied the existence of spin. q B.Sψ to the Hamiltonian, where We can na¨ıvely try to introduce spin by adding a term − 2m S = 12 σ and σ are the Pauli spin matrices. Then we would expect the splitting between states with qB sz = ± 12 to be 2m in magnitude. However, experimental observation gives a splitting that is twice this value. (In Dirac’s day the precision of the measurement was such that it was thought to be exactly twice; today we know that there are additional small corrections.) Dirac was able to explain the factor of two in the following way. We start from the momentum space Dirac equation Eψ = α.(p − qA)ψ + βmψ + qA0 ψ , (10) and write E = E0 + E1 = m + E1 , so that E1 ψ = α.(p − qA)ψ + (β − 1)mψ + qA0 ψ . Using the standard representation in block form: 1 0 0 σ β= α= 0 −1 σ 0
ψ=
(11)
ψ+ ψ−
(12)
we get E1 ψ+ = σ.(p − qA)ψ− + qA0 ψ+
(13) 0
E1 ψ− = σ.(p − qA)ψ+ − 2mψ− + qA ψ− . We can then write ψ− =
σ.(p − qA) ψ+ . 2m + E1 − qA0
(14)
(15)
In the non-relativistic limit E1 2m and taking also |qA0 | 2m we see that ψ− ∼ (velocity) × ψ+ . Therefore, in the non-relativistic limit we have E1 ψ+ =
(σ.(p − qA))2 ψ+ + qA0 ψ+ . 2m
(16)
This is the Pauli-Schr¨ odinger equation (written in a convenient form). The fact that we are able to obtain the Pauli-Schr¨odinger equation as a non-relativistic limit of the Dirac equation may appear trivial, but does have some importance. It means that we can claim that the successes of non-relativistic quantum mechanics are also successes of the relativistic theory. For example, it should now be apparant that the Dirac equation successfully describes the electronic energy levels in the hydrogen atom at least as well as the Schr¨odinger equation. It is worth briefly noting that we could have obtained the Pauli-Schr¨odinger equation directly from the generalised Schr¨ odinger equation by making the substitution appropriate for a spin- 12 particle in an electromagnetic field (instead of the minimal substitution for a spinless particle): p −→ p − qA −→ σ.(p − qA) .
(17)
Let us now expand the term of (σ.(p − qA))2 = (σ.(−i∇ − qA))2 to see how the Dirac g-factor of 2 arises. As before, we need to take care since the differential operators act on everything to their right. Writing it out explicitly (σ.(−i∇ − qA))2 ψ = (σ.(−i∇))2 ψ + iq ((σ.∇)(σ.A) + (σ.A)(σ.∇)) ψ + q 2 (σ.A)2 ψ (18) = −∇2 ψ + q 2 A2 ψ + iq (∇.A + iσ.(∇ × A) + A.∇ + iσ.(A × ∇)) ψ ,
(19)
where we have used the relation (σ.X)(σ.Y ) = X.Y + iσ.(X × Y ). Q9 Show that (σ.X)(σ.Y ) = X.Y + iσ.(X × Y ). Now, just as we applied the product rule for differentiation in the dot product in Eq. (5), we can do the same for the vector product ∇ × (Aψ) = (∇ × A)ψ − A × ∇ψ ,
(20)
where we have used the fact that the vector product is anticommutatitive. Q10 By writing ∇ × (Aψ) out explicitly, prove the result of Eq. (20). Inserting this result, we find (σ.(−i∇ − qA))2 ψ = −∇2 ψ + q 2 A2 ψ + 2iqA.∇ψ − qσ.(∇ × A)ψ 2
= (−i∇ − qA) ψ − qσ.Bψ .
(21) (22)
Q11 Work carefully through the steps required to reach Eq. (22), stating any assumptions necessary. So, finally, E1 ψ+ =
1 (−i∇ − qA)2 − qσ.B ψ+ + qA0 ψ+ . 2m
(23)
q q The term we are interested in is, of course, − 2m σ.B = −g 2m S.B (recall that S = 12 σ), and thus we have shown that the Dirac equation predicts g = 2. It is curious to note that while the Dirac prediction of g = 2 is a great triumph of relativistic quantum mechanics, the calculation of the anomalous magnetic moment (i.e. corrections that lead to g 6= 2), originally done by Schwinger, is a similarly important success of Q.E.D. This point will be discussed further in the Gauge Theories module.
Fine Structure of the Hydrogen Atom We now consider other relativistic corrections. One straightforward approach is to use the freefield Schr¨odinger equation and expand T =E−m=
p p4 p2 m2 + p2 − m = − ... 2m 8m3
From this it is easy to see that the lowest order correction is 1 ∆Erel = − p4 . 8m3
(24)
(25)
To calculate the size of this effect on the electronic orbits of the hydrogen atom, we need to calculate p2 the expectation value of p4 , hp4 i. This can be done using the non-relativistic approximation 2m = En − V , so p4 h 2 i = En2 − 2En hV i + hV 2 i , (26) 4m and since in the hydrogen atom V ∝ 1r , we need only to calculate h 1r i and h r12 i. We will not work through the calculation, but simply state the solutions 1 1 h i= 2 r n a
h
1 1 i= , 1 2 r l + 2 n3 a2
(27)
where a is the Bohr radius. In natural units a = (mα)−1 , where α is the “fine structure constant” – 1 a dimensionless number that is found empirically to take the value α ≈ 137 (and in natural units is equal to the charge of the electron squared). These then give the result ! 2n 3 α4 m − . (28) ∆Erel = − 4 4n 2 l + 12 Since the relativistic correction goes as α4 , while the energy levels themselves go as α2 , it is clear that the relativistic corrections are small effects. 2
2
Q12 Using V = − er , En = − α2nm 2 and the results of Eq. (27), prove the result of Eq. (28). Q13 Rewrite ∆Erel in terms of En . A more complete treatment, coming from the non-relativistic expansion of (σ.(p − qA))2 can be found in Feynman (QED, p.50–53). Using E1 = E − m, the result is E1 ψ = V ψ +
q 1 4 q2 1 (p − qA)2 ψ − σ.Bψ − p ψ + (∇.E + 2σ.(p − qA) × E) ψ , 2m 2m 8m3 8m2
(29)
where E and B are electric and magnetic field vectors, respectively. The first and second terms are those that appear in the conventional non-relativistic Schr¨odinger equation for a spinless particle, the third is the Stern-Gerlach term (with the Dirac g-factor of 2 implicit), and the fourth is the relativistic correction we discussed above. The fifth term is due to “spin-orbit coupling”. If we put A = 0 for the moment, consider the σ.(p × E) term, and write E ∝ rr3 , we see that this term gives something that looks like σ.( p×r ) = σ.L where L is the orbital r3 r3 angular momentum. It follows that the correction due to this effect is ∆Eso =
q2 (L.S) , 2m2 r3
(30)
and the reason to call this spin-orbit coupling should be obvious. [It is interesting to note that this correction can be calculated non-relativistically. Just as for the g-factor, such a calculation gives the wrong answer by a factor of 2. A more advanced calculation incorporating an effect known as “Thomas precession” gives the correct result.] It is useful to use the total angular momentum J = L + S to find L.S = =
1 2 J − L2 − S 2 2 1 (j(j + 1) − l(l + 1) − s(s + 1)) , 2
(31) (32)
and this time we need the expectation value of h
1 , r3
which is
1 1 i= , 1 3 r l l + 2 (l + 1)n3 a3
(33)
and so it can be shown that ∆Eso
α4 m j(j + 1) − l(l + 1) − = 4n3 l l + 12 (l + 1)
3 4
.
(34)
We can now put together the relativistic correction and the spin-orbit coupling correction to obtain the total fine-structure correction, which can be shown to be ! α4 m 3 2n − ∆Efs = − 4 , (35) 4n 2 j + 12 where the derivation uses the fact that j can only take the values l ± 21 . This formula is identical to that for the relativistic correction alone given in Eq. (28), except that l has been replaced by j. There are several points worth noting • all corrections are negative; • the nth energy level En splits into n sublevels (corresponding to j = 12 , 32 , . . . 2n−1 2 ); • states with different values of l (but the same j) can occupy the same energy level. A schematic of the fine structure corrections is shown in Fig. 5. 3D5/2 3P3/2 3D3/2 3S1/2 3P1/2
2P3/2 2S1/2 2P1/2
3D5/2 3P3/2 3D3/2 3S1/2 3P1/2
2P3/2 2S1/2 2P1/2
1S1/2 1S1/2
unperturbed
with fine structure
Figure 5: Fine structure of the electronic energy levels in the hydrogen atom. Not to scale. The Necessity of Quantum Field Theory As we have seen, relativistic quantum mechanics has enjoyed tremendous success. However, we know that it is still not a completely satisfactory theory. The reasons for this all ultimately boil
down to the fact that there is no mechanism to handle the creation and annihilation of particle– antiparticle pairs in RQM. Consequently, several empirical observations cannot be explained without a more advanced theory. These experimental observations include • high energy experiments, in which particle–antiparticle creation/annihilation is directly observed; • cosmic ray collisions in the Earth’s upper atmosphere, detailed studies of which show effects due to particle–antiparticle creation; • nuclear screening effects, as discussed briefly in an earlier handout; • precise measurements of the electron g-factor reveal that its value is slightly different from the RQM prediction of 2; • precision studies of the hydrogen spectrum reveal additional structure (so-called “hyperfine structure”), in particular the “Lamb shift” in which states with different values of the quantum number l (but the same j) are indeed split into different levels. To be able to explain these effects, we need to move beyond the limitations of the current theory. This limitation arises from the fact that the RQM we have discussed is a “semi-classical” theory, in the sense that, although we have treated the electron quantum mechanically, the electromagnetic field has been treated in an essentially classical matter. We need a theory in which the field is also quantised (the quantisation of the field is sometimes called “second quantisation”). That is to say, we need a quantum field theory. This will be introduced in the Gauge Theories module. Eulogy for the Dirac Equation To conclude our discussion, let us recall some of the shortcomings of non-relativistic quantum mechanics, and see how our relativistic treatment, in particular the Dirac equation, has improved the situation. • a non-relativistic theory is intrinsically unsatisfactory ,→ the relativistic treatment obvious remedies this problem; • the concept of spin in an ad-hoc addition to NRQM ,→ spin arises naturally as a consequence of Dirac’s theory – solutions of the Dirac equation can be distinguished by the helicity operator; • there is no physical reasoning behind Pauli’s exclusion principle ,→ this remains the case – we understand the origin of the spin quantum number, but not why two states cannot have the same set of quantum numbers; • the g-factor of the electron and the fine structure of the hydrogen atom cannot be explained ,→ the situation is much improved, but we can see that there are still some discrepancies between the theory and precise measurements; • particle–antiparticle creation/annihilation cannot be explained ,→ we still have this problem. In some sense, the theory of relativistic quantum mechanics can be considered as an intermediate step towards a more complete theory. (In the Gauge Theories module, the remaining problems will be resolved.) Nonetheless, the step we have taken is a very significant one indeed. Crucially, we have gained some understanding of the purely quantum mechanical concept of spin. Without the Dirac equation, spin does not make sense. With Dirac’s theory, things begin to fall into place.
