fulltext

Schrödinger-like formalism of relativistic quantum theory

for spin-zero particles (*)

YOUNG-SEA HUANG

Department of Physics, Soochow University - Shih-Lin, Taipei, Taiwan (ricevuto il 16 Aprile 1996; approvato il 30 Ottobre 1996)

Summary. — This paper presents a Schrödinger-like formalism of relativistic quantum theory alternative to the Klein-Gordon theory for spin-0 particles. The Schrödinger-like wave equation thus obtained is form-invariant, though its form is not manifestly Lorentz-covariant in the conventional sense. The alternative formalism resolves the great difficulties of negative probability density and Zitterbewegung in the Klein-Gordon theory. Ehrenfest’s theorem is preserved in the alternative formalism.

PACS 03.65 – Quantum mechanics. PACS 03.70 – Theory of quantized fields.

In relativistic quantum theory, the Dirac equation is used to describe spin-(1O2) particles, while the Klein-Gordon equation is used to describe spin-0 particles [1, 2]. The first section presents a Schrödinger-like formalism of relativistic quantum theory for spin-0 particles. The second section demonstrates that the Schrödinger-like formalism provides a simple scheme for solving relativistic quantum problems in a manner almost identical to what the Schrödinger wave equation does in non-relativistic quantum theory. Comparisons between the Schrödinger-like formalism and the Klein-Gordon theory are also given.

1. – Schrödinger-like formalism of relativistic quantum theory

According to special relativity, the total energy E and momentum p of a free material particle are E 4gmc2and p 4gmv, where m is the rest mass, c is the speed of light, and g 4 (12v2_/c2₎21 /2_{. The relativistic energy-momentum relation is}

E2 2 p2c2

4 m2c4_. (1)

(*) The author of this paper has agreed to not receive the proofs for correction.

From this equation, one has

Heff4 p2 2 m , (2)

if one defines the effective Hamiltonian Heff as

Hefff

E2

2 m2c4

2 mc2 .

(3)

Here, the effective Hamiltonian Heff for a free particle is 1 2mu

2

, where u 4gv. To obtain the wave equation of motion, one makes the replacement of the physical quantities in eq. (2) by the operators

p K2iˇ˜ (4) and HeffK iˇ ¯ ¯t , (5)

and lets the resulting operator equation act on a wave function c(r , t). Consequently, the wave equation of motion is obtained as

iˇ ¯ ¯t c(r , t) 42 ˇ2 2 m˜ 2 c(r , t) . (6)

Here, the time-like parameter t is the so-called proper time in Einstein’s special relativity; the time-like parameter t is related to the coordinate time t by dt 4g dt. The Schrödinger-like wave equation (6) has the plane-wave solution c(r , t) 4

A exp [2(iOˇ)(Hefft Z p Q r) ]. As an immediate consequence of the linear and homo-geneous properties of the Schrödinger-like wave equation, the wave packet c(r , t) 4 ( 2 pˇ)23 O 2

_s

_{A(p) exp}

_[

2(i O ˇ)

(

Heff(p) t Z p Q r

)]

d3p is a solution of this wave equation. Now, generalize the above formulation by considering a relativistic particle moving in a field of potential V(r). According to special relativity, the energy-momentum relation is given as

(E 2V)2 2 p2c2

4 m2c4_. (7)

Rewriting eq. (7) and using the definition equation (3), one obtains

Heff4 p2 2 m 1 Veff, (8) where Vefff 2 EV 2V 2 2 mc2 . (9)

Here, the quantity Heff can be thought of as an energy inclusive of the Newton-like relativistic kinetic energy 1

2mu 2

replacing the physical quantities by their associated operators as given by eqs. (4) and (5), and then operating on a wave function c(r , t), eq. (8) results in the Schrödinger-like wave equation

iˇ ¯ ¯tc(r , t) 4

g

2 ˇ2 2 m ˜2 1 Veff

h

c(r , t) . (10)

As in the nonrelativistic quantum theory, one defines the probability density by

r 4c*(r, t) c(r, t) .

(11)

It is immediately evident from the Schrödinger-like wave equation (10) that the probability density satisfies the equation of continuity

¯

¯tr 1˜QJ40 , (12)

where J is the probability current density as given by

J 4 ˇ

2 mi(c * ˜c 2c˜c*) .

(13)

The probability density defined herein is positive-definite. The probability is conserved in accordance with the equation of continuity.

With the wave function c(r , t) being normalized to unity, the expectation value of a dynamical variable O(r , p) representing a physical quantity can be calculated by the expression aOb 4

s

_{c * (r , t) O× c(r, t) d}3_{r, where O}×

4 O(r , 2iˇ˜) is the operator associated with the dynamical variable O. Consider the proper-time rate of change of the expectation value of the position vector, darb Odt. By using the Schrödinger-like wave equation (10) and its complex conjugate, one finds after a straightforward calculation that (14) darb dt 4



_{c * (r , t) r} ¯c(r , t) ¯t d 3 r 1



¯c * (r , t) ¯t rc(r , t) d 3 r 4 4 (iˇ)21

k



c * r

g

2ˇ 2 2 m ˜ 2 c 1Veffc

h

d3r 2



g

2ˇ2 2 m ˜ 2 c * 1Veffc *

h

rc d3r

l

4 4 2iˇ m



_{c * ˜c d}3 r 4 apb m .

Evidently, this result is the quantum counterpart of the equation p 4gmv4gmdr_dt 4

mdr

dt in relativistic classical mechanics; thus, it supports the conjecture HeffK iˇ ¯ ¯t

as given in the beginning of the formulation. On the contrary, according to the Klein-Gordon theory, for a free spin-0 particle the Hamiltonian in the Schrödinger form of the Klein-Gordon equation is given by H×_{4 (s}×

31 i s×2) p ×2 2 m1 mc 2 s × 3, where s×2

and s×

3 are the Pauli matrices [2-4]:

s ×₁₄

u

0 1 1 0

v

, s ×₂₄

u

0 i 2i 0

v

, and s ×₃₄

u

1 0 0 21

v

.

Using the equation of motion in the Heisenberg picture, one has the “velocity operator” for a free spin-0 particle,

d r× dt 4 1 iˇ[r×, H×] 4 (s×31 i s × 2) p × m .

Since the eigenvalues of s×

31 i s×2 are zero, the eigenvalues of v× 4dr× /dt are zero. This peculiar result means that the measured mean value of the speed of a free spin-0

particle should be always equal to zero. Moreover, one has the “position

operator” [4] r×(t) 4 r×(0) 1 p × Ht 1 e2 iHt 2 1 2 iH

g

v×( 0 ) 2 p × H

h

.

The last term on the right-hand side of this equation describes the so-called

Zitterbewegung—particles tremble rapidly even in the absence of external interaction [3-7]. The Klein-Gordon theory does not truly maintain Ehrenfest’s theorem, whereas the Schrödinger-like formalism presented herein does.

Consider, even more generally, a particle of charge e moving in external electro-magnetic field. According to special relativity, the relativistic energy-momentum relation is (E 2eF)22

g

P 2 e cA

h

2 c2_{4 m}2c4. (15)

Here, P 2 (eOc) A4pfgmv, as well as the scalar potential F(r, t) and the vector po-tential A(r , t) form a Lorentz covariant four-vector. This equation can be rewritten as

Heff4

g

P 2 e cA

h

2 2 m 1 Veff, (16)

where Vefff ( 2 EV 2V2) O2mc2, and V f eF . As in the above method, from eq. (16), one obtains the wave equation of motion

iˇ ¯ ¯t c(r , t) 4

C

`

`

`

D

g

2iˇ˜ 2 e cA

h

2 2 m 1 Veff

E

`

`

`

F

c(r , t) . (17)

It should be noted that the form of the Schrödinger-like wave equation (17) is invariant among inertial frames via the Lorentz transformation on the energy-momentum space. From the Lorentz-invariant relativistic energy-energy-momentum relation eq. (15), and by the formulations as described above, one can construct the

like wave equation in each inertial frame. Consequently, the obtained Schrödinger-like wave equations in all inertial frames have the same form, though their form is not

manifestly Lorentz-covariant in the conventional sense.

2. – Applications of the Schrödinger-like formalism of relativistic quantum theory for spin-zero particles

2.1. The one-dimensional square step potential. – Consider a particle impinging on a potential step which has the form

V(x) 4./ ´ 0 , _{x E0 ,} V 4const , x D0 . (18) Case 1: V EE2mc2.

The kinetic energy of the particle in the region x D0 is larger than zero. From eq. (10), the time-independent wave equation of motion is

d2c(x) dx2 1 k 2 c(x) 40 , k 4

g

2 mHeff ˇ2

h

1 /2 , for x E0 , (19a) and d2 c(x) dx2 1 k 8 2 c(x) 40 , k 84

g

2 m(Heff2 Veff) ˇ2

h

1 /2 , for x D0 . (19b)

The general solution is

c(x) 4./ ´ A eikx 1 B e2ikx, _{x E0 ,} C eik 8 x, _{x D0 ,} (20)

where A, B, and C are constants. Since the particle is incident from the region x E0, the term e2ik 8 x _{is not included in the solution in the region x D0. By matching c(x) and} dc(x) Odx across the step at x40, one obtains that A1B4C, and k(A2B) 4k 8 C. Consequently, one has

B A 4 k 2k 8 k 1k 8 , (21) and C A 4 2 k k 1k 8 . (22)

Now consider the probability current density J. Substituting the wave function eq. (20) into the probability current density J equation (13), one finds that

J 4./ ´

(ˇk Om)[NAN22 NBN2] , _{x E0 ,}

(ˇk 8Om)NCN2, _{x D0 .}

(23)

From eqs. (21) and (22), one has NBN2 NAN2 1 k 8 k NCN2 NAN2 4 1 . (24) Hence (ˇk Om)[NAN2

2NBN2] 4 (ˇk 8Om)NCN2_{; as it should be for the stationary state in} this case, the probability current density is constant everywhere. Consequently, the reflection coefficient is R 4 NBN 2 NAN2 4 (k 2k 8)2 (k 1k 8)2 4

g

k

E2_{2 m}2c4₂

k

_{(E 2V)}2_{2 m}2c4

h

2

g

k

E2 2 m2c4 1

k

(E 2V)2 2 m2c4

h

2 , (25)

and the transmission coefficient is

T 4 k 8 k NCN2 NAN2 4 4 kk 8 (k 1k 8)2 4 4

k

(E2 2 m2c4 )[ (E 2V)2 2 m2c4_]

g

k

E2 2 m2c4 1

k

(E 2V)2 2 m2c4

h

2 . (26)

From eqs. (24), (25), and (26), one obtains, as expected, the conservation of probability flux, R 1T41.

Case 2: E 2mc2

E V E E 1 mc2.

The kinetic energy of the particle in the region x D0 is less than zero, but larger than 22mc2_{. Then, the equation of motion is}

d2 c(x) dx2 1 k 2 c(x) 40 , k 4

g

2 mHeff ˇ2

h

1 O2 , for x E0 , (27a) and d2 c(x) dx2 2 k 2 c(x) 40 , k 4

g

2 m(Veff2 Heff) ˇ2

h

1 O2 , for x D0 . (27b)

The general solution is

c(x) 4./ ´ A eikx 1 B e2ikx, _{x E0 ,} D e2kx_{, x D0 ,} (28)

where A, B and D are constants. The term ekx is not included in the solution in the region x D0, because this term diverges in the limit xK1Q. By applying the boundary conditions that c(x) and dc(x) Odx are continuous at x40, one obtains that

A 1B4D, and ik(A2B) 42kD. Then, one has that BOA4 (1 2ikOk)O (1 1ikOk) 4 eia_{, and D OA42O (11ikOk), where a42 tg}21_{(2kOk). Consequently, the reflection} co-efficient is R4NBN2

Case 3: V DE1mc2_.

In this case, the kinetic energy of the particle in the region x D0 is less than 22 mc2. Since Heff2 VeffD 0 for this case, the equation of motion is given, again, by eq. (19) in case 1. Consequently, the reflection coefficient R and the transmission coefficient T are given, respectively, by eq. (25) and eq. (26) in case 1. Thus, one has 0 G

R G1, 0 GTG1, and R1T41.

These predictions are the same as the predictions of the Klein-Gordon theory,

provided that in case 3 the sign of the momentum p in the equation

p2_c2

4 (E 2 V)22 m2c4 _{is chosen to be positive [4, 8-11]. But, if p is chosen to be} negative, then the Klein-Gordon theory predicts that the reflection coefficient R D1

and the transmission coefficient _{T E0 in the case 3—the so-called Klein}

paradox [12, 13]. The paradoxical predictions in the regime of strong fields have never been tested experimentally [14, 15].

2.2. The pionic atom. – Consider a pionic atom consisting of a nucleus of charge 1Ze and a pion of mass m and charge 2e interacting by means of the Coulomb potential, V(r) 42Ze2O 4 pe0r , where r is the distance between the nucleus and the pion. Then, the time-independent wave equation H×effc(r) 4

g

2

ˇ2

2 m˜ 2

1 Veff

h

c(r) 4

Heffc(r) in polar spherical coordinates is given as

(29)

y

₂ ˇ 2 2 m

g

d2 dr2 1 2 r d dr

h

1 L ×2 2 mr2 2 E mc2

g

Ze2 4 pe0

h

1 r 2 2 1 2 mc2

g

Ze2 4 pe0

h

2 ₁ r2

z

c(r , u , f) 4Heffc(r , u , f) ,

where L× f2iˇ(r3˜). Since the operators H×eff, L×2 and L×z commute, and the

spherical harmonics Ylm(u , f) are simultaneous eigenfunctions of L×2 and L×z, solutions

of eq. (29) are expected as cE , l , m(r) 4RE , l(r) Ylm(u , f), where l 40, 1, 2, R, and m 42l, 2l11, R, l. By substituting cE , l , m(r) into eq. (29), the radial differential

equation follows as (30)

y

₂ ˇ 2 2 m

g

d2 dr2 1 2 r d dr

h

1 l(l 11) ˇ2 2 mr2 2 E mc2

g

Ze2 4 pe0

h

1 r 2 2 1 2 mc2

g

Ze2 4 pe0

h

2 ₁ r2

z

R(r) 4HeffR(r) . Here, for simplicity, R(r) f RE , l(r). Let u(r) 4rR(r). In terms of the dimensionless

quantities a f e 2 4 pe0ˇc , l f Ze 2 4 pe0ˇc

g

2E2 2 mc2Heff

h

1 O2 4 Za

g

2E 2 2 mc2Heff

h

1 O2 ,

and

r f

g

28 mHeff

ˇ2

h

1 O2

r ,

eq. (30) then becomes

y

d2 dr2 2 1 4 1 l r 2 b(b 11) r2

z

u(r) 40 , (31) where b(b 11) 4l(l11)2 (Za)2

. By using u(r) 4e2r O 2_rb 11_{f (r), one has the}

differential equation for f (r)

y

r d 2 dr2 1 ( 2 b 1 2 2 r) d dr 1 (l 2 b 2 1 )

z

f (r) 40 . (32)

This equation has the generalized Laguerre polynomial L2 b 11

l 2b21 as its solution,

provided that n 8fl2b21 is a positive integer or zero [16]. Thus, the solutions of eq. (31) are

u(r) Ae2r O 2

rb 11L_{l 2b21}2 b 11 (r) . (33)

By defining the principal quantum number as nfn 81l11, then one has l4n1b2l. Consequently, the normalized solutions of eq. (30) are

Rnl(r) 4

y

Z3 a03 4 l4

g

1 1 (Za)2 l2

h

23 O 2 _{G(l 2b)} G(l 1b11)

z

1 O2 e2r O 2_rb_L2 b 11 l 2b21(r) , (34)

where a04 4 pe0ˇ2O me2, n 41, 2, 3, R, and l40, 1, 2, R, (n21). From eq. (3) and the definition of l , the bound-state energy eigenvalues are obtained as

(35) Enl4 mc2

C

`

D

1 1 (Za) 2

g

n 2l21₂ 1

o

g

l 11₂

h

22 (Za)2

h

2

E

`

F

21 O 2 4 4 mc2

C

`

D

1 2 (Za) 2 2 n2 2 (Za)4 2 n4

u

n l 11₂ 2 3 4

v

1 R

E

`

F

.

These bound-state energy eigenvalues are the same as those predicted by the Klein-Gordon theory [2]. Furthermore, _Ncnlm(r , u , f) N2d3r 4NRnl(r) N2NYlm(u , f) N2d3r

represents the probability of finding the pion in a volume element d3r . On the contrary,

the probability density as defined in the Klein-Gordon theory is not

positive-definite [1, 2]; thus, it cannot represent the pionic probability distribution. The negative probability density problem casts a serious doubt on the adequacy of the Klein-Gordon equation as the wave equation for spin-0 particles, within the scope of the single-particle interpretation [17].

The two examples above show that the alternative formalism provides the familiar technique of nonrelativistic quantum mechanics for solving relativistic quantum problems.

* * *

The author gratefully thanks the valuable comments of Dr. C. M. L. LEONARDin the

preparation of this paper.

R E F E R E N C E S

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