Lecture notes: Dirac particle simulations in 1+1 dimensions · Lecture notes: Dirac particle simulations in 1+1 dimensions Je rey Yepez ... mc2 ˝ ~ 2 ˙ xei‘p z˙ z mc ... u ic

Phys 711 Topics in Particles & Fields — Spring 2013 — Lecture 5 — v0.3.1

Lecture notes: Dirac particle simulations in 1+1 dimensions

Jeffrey Yepez

Department of Physics and AstronomyUniversity of Hawai‘i at ManoaWatanabe Hall, 2505 Correa RoadHonolulu, Hawai‘i 96822E-mail: [email protected]/∼yepez

(Dated: February 13, 2013)

Contents

I. Quantum algorithm for the Dirac equation 1

II. Continuity relations for scalar and spinor fields 2

III. Dirac particle in a square well potential in 1+1dimensions 3

IV. Dirac particle in a harmonic potential in 1+1dimensions 6

References 8

I. QUANTUM ALGORITHM FOR THE DIRACEQUATION

At the end of Lecture 1, we showed that the unitaryevolution operator generated by the Dirac Hamiltoniancan be accurately written as a composition of two rota-tions UzsUC ' e−i`hD/(~c), where the stream and collideoperators have the form

UzS = e−iβ22 σz (1a)

and

UC = e−iβ12 (cos β22 σx−sin

β22 σy), (1b)

respectively. We found that the angle β1 is fixed by

sinβ1

2=mc2τ

~, (2a)

and we also know that

β2

2= −`kz (2b)

because the stream operator UzS = ei`kzσz = eσz`∂z isjust the shift operator that displaces the spin-up andspin-down components of the Dirac field by ±`, respec-tively. Finally, to map the composite rotations UzsUConto e−i`hD/(~c), we found that the grid sizes must sat-isfy the transcendental equation

Eτ

~= sin

(E`

~c

). (3)

With the above results, we may rewrite UC in an analyt-ical form that is useful for the simulation of the quantumdynamics of a Dirac field. We begin by writing the colli-sion operator (1b) as

UC = e−iβ12 [σx(cos β22 −iσz sin

β22 )] (4a)

= e−iβ12 σxe

−iσzβ22 (4b)

= cosβ1

2− σxe−iσz

β22 sin

β1

2(4c)

(2b)

(2a)=

√1−

(mc2τ

~

)2

− σxei`pzσzmc2τ

~. (4d)

Since

σxei`kzσz =

(0 11 0

)(ei`kz 0

0 e−i`kz

)=(

0 e−i`kz

ei`kz 0

),

(5)the collision operator in matrix form is

UC =

√1−(mc2τ

~)2 −ie−i`kz mc2τ~

−iei`kz mc2τ

~

√1−

(mc2τ

~)2 . (6)

To be useful for quantum simulation, we wish to writeUC = UC(m, `, γ), where γ ≡ E/(mc2). Since pz =√E2 − (mc2)2, we have pz = ~kz = mc

√γ2 − 1, and in

turn the collision operator may be written as

UC =

√1−

(mc2τ

~)2 −i e−imc`~

√γ2−1mc2τ

~

−i eimc`~

√γ2−1mc2τ

~

√1−

(mc2τ

~)2

.

(7)Now we can rewrite (3) as

mc2τ

~=

1γ

sin(γmc`

~

), (8)

which allows us to eliminate the explicit τ -dependence inthe collision operator

II CONTINUITY RELATIONS FOR SCALAR AND SPINOR FIELDS

UC =1γ

√γ2 − sin2

(γmc`

~

)−i e−imc`~

√γ2−1 sin

(γmc`

~

)−i eimc`~

√γ2−1 sin

(γmc`

~

) √γ2 − sin2

(γmc`

~

) . (9)

So in natural lattice units (~ = 1 and c = 1), the quantumalgorithm for the Dirac equation is represented by thefollowing stream and collision operators (Yepez, 2010):

UzS = eσz`∂z (10a)

UC =1γ

( pγ2 − sin2(γm`) −i e−im`

√γ2−1 sin(γm`)

−i eim`√γ2−1 sin(γm`)

pγ2 − sin2(γm`)

).

(10b)

We will demonstrate the numerical performance of thisquantum algorithm below.

II. CONTINUITY RELATIONS FOR SCALAR ANDSPINOR FIELDS

If we multiply the free Schroedinger wave equation bythe complex conjugate of the scalar field, then we have

ϕ∗(i~∂tϕ+

~2

2m∇2ϕ

)= 0, (11a)

and the complex conjugate of this equation is

ϕ

(−i~∂tϕ∗ +

~2

2m∇2ϕ∗

)= 0. (11b)

Taking the difference of these two equations, and dividingthe result through by ~, gives a continuity relation for thescalar particle

∂t(ϕ∗ϕ) +∇ ·[−i ~

2m(ϕ∗∇ϕ− ϕ∇ϕ∗)

]= 0, (12)

where one identifies the probability density and the 3-vector probability current density as respectively

ρ ≡ ϕ∗ϕ (13a)

j ≡ −i ~2m

(ϕ∗∇ϕ− ϕ∇ϕ∗

). (13b)

Thus, the conservation of probability (or more preciselythe conservation of particle number) is manifestly covari-ant

∂µjµ = 0, (14)

where the conserved 4-current is jµ ≡ (ρc, j).Now let us consider the effective wave equation for a

quantum gas in a confining lattice(i ∂t + σx ⊗ 1

`2

τ∇2

)ψ + · · · = 0, (15)

which we derived earlier.1 Changing our frame of refer-ence with the transformation R = (σx⊗ 1 + σz ⊗ 1)/

√2,

the wave equation for the spinor field η = Rψ becomes(iσz ⊗ 1 ∂t +

`2

τ∇2

)η + · · · = 0. (16)

If we multiply this effective wave equation for a quantumgas in a confining lattice by the adjoint spinor field, thenwe have

η†(iσz ⊗ 1 ∂t +

`2

τ∇2

)η = 0, (17a)

and the adjoint of this equation is

η†(−i←−∂t σz ⊗ 1 +

`2

τ

←−∇2

)η = 0. (17b)

Taking the difference of these two equations gives a con-tinuity relation for the spinor particle

∂t(η†γ0η) +∇ ·[−i `

2

τη†(∇−

←−∇)η

]= 0, (18)

where γ0 ≡ σz ⊗ 1. One thus identifies the temporaland spatial parts of the probability flux density as re-spectively

u0 ≡ c η†γ0η = c η η (19a)

u ≡ −ic`(η†∇η − η†

←−∇η), (19b)

with c = `/τ and η ≡ η†γ0. So the particle dynamicsobeys a covariant flux continuity relation

∂µuµ = 0, (20)

where the conserved 4-flux density is uµ ≡ (u0,u). Sinceu0 is not a positive-definite scalar, this quantity cannotrightly be interpreted as a probability. Yet, its physicalinterpretation as the time component of a 4-flux densityis justified by (20). The covariant probability 4-currentfor the nonrelativistic spinor particle is

j0 ≡ c ηγ0η = c η†η (21a)

j ≡ −ic`(η∇η − η

←−∇η), (21b)

1 This is Eq. (35) in Lecture 4, where the unitary collision operatorU = 1 + J is chosen such that

`σzJ−1σz + 1/2

´= σx.

2

III DIRAC PARTICLE IN A SQUARE WELL POTENTIAL IN 1+1 DIMENSIONS

satisfying a continuity relation akin to (14).Finally, if we multiply the relativistic wave equation

for a Dirac particle by the adjoint 4-spinor, then we have

ψ†(i~c γµ∂µ −mc2

)ψ = 0, (22a)

and the adjoint of this equation is

ψ†(−i~c

←−∂µγ

µ −mc2)ψ = 0. (22b)

Taking the difference of these two equations, and dividingthe result through by ~, gives a continuity relation for the4-spinor particle

∂µ(ic ψ†γµψ) = 0, (23)

where one identifies the probability flux density as

uµ ≡ c ψ†γµψ, (24)

a 4-vector governed by a flux continuity relation of form(20). The reason for dropping the overall phase factorof i is to have the temporal component u0 = c ψψ bethe same as (19a), matching the relativistic and nonrel-ativistic spinor expressions. Furthermore, the 4-currentdensity may be defined as

jµ ≡ ic ψγµψ, (25)

which satisfies a continuity relation like (14).2

III. DIRAC PARTICLE IN A SQUARE WELL POTENTIALIN 1+1 DIMENSIONS

Here we consider a quantum particle in a confining one-dimensional lattice with grid points z = n`, for integern (0 ≤ n ≤ L), with stream and collide operators givenby (10) and with the similarity transformation given byR = (σx+σz)/

√2. The grid-level equation of motion for

this finite quantum system is

η′ = RUzsUCR†η. (26a)

We will use (26a) for the purpose of modeling the dynam-ical behavior of a Dirac particle in a square well poten-tial. The low-energy effective field theory (in the rotatingframe) of (26a) is the Dirac equation

i~∂tη(z, t) = pc σzη(z, t) +mc2σxη(z, t), (26b)

where the momentum operator is identified as p = −i~∂z.

2 The conventional definition is jµ ≡ c ψγµψ, which ensures thatj0 = ψ†ψ is a real-valued and positive-definite scalar.

Exercise:

Verify that the grid-level dynamical equationη′ = RUzs (`)UC(m, `, γ)R†η, when Taylor ex-panded in `, yields the Dirac equation at low-est order i~∂tη+ · · · = pc σzη+mc2σxη+ · · · .

We will work in the nonrotating frame ψ ≡ Rη wherethe effective equation of motion is

i~∂tψ(z, t) = pc σxψ(z, t) +mc2σzψ(z, t). (26c)

Using separation of variables, the 2-spinor field is

ψ(z, t) =

(φ(z)ξ(z)

)e−i(pz−Et)/~, (27)

and in turn the eigenequation is

Eψ =

(mc2 pc

pc −mc2

)ψ. (28)

Explicitly writing out the coupled component equations

Eφ = mc2φ+ pc ξ (29a)Eξ = pc φ−mc2ξ, (29b)

one immediately identifies two solutions types

φ =pc

E −mc2ξ (30a)

ξ =pc

E +mc2φ. (30b)

That is, the eigensolutions for a free Dirac particle havethe form

ψ(z, 0) =

φ(z)

(1pc

E+mc2

)e−ipz/~, for E > 0

ξ(z)

(pc

E−mc2

1

)e−ipz/~, for E < 0.

(31)

Let us consider the positive-energy solution for a planewave with momentum eigenvalue p = ~k. Normalizingsuch that 〈ψk|ψk′〉 ≡

∫dz ψ†k(z, 0)ψk′(z, 0) = δ(k − k′),

we find that 〈ψk|ψk〉 = 2E/(E +mc2), so in turn we maywrite the plane-wave solution as

ψk(z) =

√E +mc2

2E

(1√

E−mc2E+mc2

)e−ikz, (32a)

where we made use of the relativistic energy relationE2 = (pc)2 + (mc2)2. We can rewrite this solution bysplitting it into its right-going (spin-up) and left-going

3


(spin-down) components3

ψk =

√1

2E

√E +mc2

(10

)︸︷︷︸right-goer (spin up)

+√E −mc2

(01

)︸︷︷︸left-goer (spin down)

e−ikz.(32b)

As a way to avoid the Klein paradox, we can modelthe external confining square well barrier as regions inspace where the mass of the Dirac particle is large. Thisis depicted as

V (z) ∼= m(z)OO

z//

M

m

E_______

IIIIII

0 L(33)

In (33) the mass of the Dirac particle is m for 0 ≤ z ≤ L(region II) and its mass is M > m for z > L (region I)and for z < 0 (region III). So the plane-wave solutions in(33) are

ψIk(z) = Aeik

′z

(1

~k′cE+Mc2

)(34a)

ψIIk (z) = B eikz

(1

~kcE+mc2

)+ C e−ikz

(1−~kcE+mc2

)(34b)

ψIIIk (z) = De−ik

′z

(1

−~k′cE+Mc2

). (34c)

For convenience, let us write the the 2-spinor field inregion II as

ψIIk (z) =

(B eikz + C e−ikz(B eikz − C e−ikz

)P

), (35)

where

P ≡ ~kc(E +mc2)

. (36)

Choosing boundary conditions to ensure that the Diracparticle is appropriately confined within the square wellin region II is a rather subtle matter. We will choose

3 Right and left-going chirality and spin-up and spin-down proper-ties of a Dirac particle are the physically the same properties inone spatial dimension. This is not the case in higher dimensions.

boundary conditions such that the probability flux den-sity vanishes at z = 0 and z = L (Alberto et al., 1996).It is a remarkable property of the spinor structure of aDirac particle that we do not have to choose ψ(0) = 0and ψ(L) = 0 even when M =∞, nor do the componentsof the 2-spinor field individually have to be continuousfunctions of position across the boundaries.4 Bound-ary conditions with continuous probability flux densityψψ ≡ ψ†γ0ψ vanishing at the container walls were origi-nally introduced in the MIT bag model of bound hadrons(Chodos et al., 1974a,b).

Let us write the Dirac equation (26c) as

i~∂tψ(z, t) =(−i~cαz∂z +m(z)c2β

)ψ(z, t), (37a)

where αz = σx and β = σz and where m(z) = m for0 ≤ z ≤ L and m(z) = M otherwise. In the chiral rep-resentation, this equation may be written in manifestlycovariant form

i~c(γ0∂0 + γz∂z)ψ −m(z)c2ψ = 0, (37b)

where ∂0 ≡ ∂ct, γ0 = β, and γz = βαz = −iσy. Inthis representation, the probability flux density is thedifference of the particle’s probability occupancy of itsspin-up and spin-down states

ψψ =

(ψ†R ψ†L

) (1 00 −1

)(ψRψL

)= ψ†RψR − ψ

†LψL.

(38)Equating the 4-current density (25) along the −z-

direction to the probability flux density (24) at the leftwall, the boundary condition at z = 0 is

jz(0) = −ic ψk(0)γzψk(0) = c ψk(0)ψk(0) = u(0) (39a)

or (0 −11 0

)(ψIIR(0)

ψIIL (0)

)= i

(ψIIR(0)

ψIIL (0)

). (39b)

One chooses the boundary condition (39) to force bothjz(0) = 0 and u(0) = 0, which represents both vanishingprobability current escaping the square well and vanish-ing probability flux at the left wall. We will now verifythat this is indeed the case.

From (39b) we see that the components are constrainedby the relation ψII

L (0) = −i ψIIR(0). Using the plane-wave

solution (35), the left wall boundary condition implies

i(B − C)P = B + C (40a)

or

C = BiP − 1iP + 1

. (40b)

4 Albeit the probability density and probability flux density areboth continuous across the boundaries.

4


Inserting (35) into (38), we verify that the probabilityflux density vanishes at the left wall

ψ(0)ψ(0) = |B + C|2 − |B − C|2P 2 (40a)= 0. X (41)

Now we can apply similar boundary conditions at theright wall. The boundary condition at z = L is

jz(L) = ic ψk(L)γzψk(L) = c ψk(L)ψk(L) = u(L)(42a)

or (0 −11 0

)(ψIIR(L)

ψIIL (L)

)= −i

(ψIIR(L)

ψIIL (L)

). (42b)

The boundary condition (42) forces both jz(L) = 0 andu(L) = 0, which represents both vanishing probabilitycurrent escaping the square well and vanishing probabil-ity flux at the right wall. So the components are con-strained by the relation ψII

L (L) = i ψIIR(L). Using the

plane-wave solution (35), the right wall boundary condi-tion implies

− i(B eikL − C e−ikL

)P = B eikL + C e−ikL. (43a)

Defining eiθ ≡ C/B and multiplying through by e−iθ/2,(43a) becomes

−i(ei(kL−θ/2) − e−i(kL−θ/2)

)P = ei(kL−θ/2)+e−i(kL−θ/2)

(43b)or

cot(kL− θ

2

)= P. (43c)

From (40b) we have

eiθ =iP − 1iP + 1

=P 2 − 1P 2 + 1

+ i2P

P 2 + 1, (44)

so the phase angle is determined by tan θ = 2P/(P 2−1).Additionally, we can write (43b) as

− (iP + 1)eikL = (−iP + 1)eiθe−ikL (45)

or

e2ikL =iP − 1iP + 1

eiθ(44)= e2iθ. (46)

This implies that θ = kL, so we can write (43c) as

cot(kL

2

)(36)=

~kc(E +mc2)

. (47)

This is a transcendental equation whose solution for wavenumber k ensures both vanishing probability current es-caping the square well and vanishing probability flux atthe right wall. The solution of (47) is shown graphicallyin Fig. 1 for a relativistic case where ~k > mc2. With

0 1 2 3 4 5 6

-5

0

5

10

k

Cot

HkL

�2L

FIG. 1 Solution to transcendental equation (47) for a squarewell of size L = 2 for a Dirac particle of mass m = 1/2 inlattice units ~ = c = 1. The first crossing occurs at k =0.860334.

this k value, we can determine the value of P and in turndetermine the value of the coefficient C in terms of B byusing (40b).

Plots of the 2-spinor components <{ψR} and ={ψL}and plots of the probability density ρ ≡ ψ†ψ = ψγ0ψ andparticle flux density’s time component ψ†γ0ψ = ψψ areshown in Fig. 2 for a relativistic case. The physical inter-pretation of the particle dynamics is that our plane-wavesolution represents a perfectly matched situation wherethe Dirac particle is trapped because of total internal re-flection occurring with ρ 6= 0 and |ψR| = |ψL| at theboundaries. At z = 0 the spin-down (left-going) statescatters off the wall into the spin-up (right-going) state,which is the bounce-back collision ψL(0)→ ψR(0). Like-wise, at z = L the spin-up (right-going) state scattersoff the wall into the spin-down (left-going) state, whichis the bounce-back collision ψR(L) → ψL(L). This is anexample where one can interpret the spin state dynamicsin terms of kinetic particle motion in position space atthe grid level.

In the nonrelativistic regime where ~kc ≪ mc2, thenthe transcendental equation (47) simplifies to

cot(kL

2

)≈ 0, (48)

which is analytically solvable: the wave number solutionis k ' π/L. In this limit, P ≈ 0 and B = eiπC = −C, sothe spinor field in region II reduces to

ψIIk (z) ≈ 2iB

(sin kz

0

). (49)

The lower (fast) component vanishes while the upper(slow) component is just the usual ground state solutionof the Schroendinger wave equation for a square well po-tential, and these components are shown in Fig. 3.

A quantum simulation of a Dirac particle confined to asquare well is shown in Fig. 4. The quantum simulation

5

IV DIRAC PARTICLE IN A HARMONIC POTENTIAL IN 1+1 DIMENSIONS

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

k

ΨR,

-ä

ΨL,

Ψ†

Ψ,

ΨΨ

FIG. 2 Plane-wave solution for a relativistic case with k =0.860334, square well size L = 2, and Dirac particle mass m =1/2 in lattice units ~ = c = 1. The spin components ψR (blue)and ψL (red) are matched but in the relativistic regime (k >m) do not vanish at the z = 0 and z = 2 walls, nor does theprobability density ψ†ψ (gold) vanish at the walls. However,the 4-flux 0-component ψψ (green) does indeed vanish at theboundary walls.

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

k

ΨR,

-ä

ΨL,

Ψ†

Ψ,

ΨΨ

FIG. 3 Plane-wave solution for a nonrelativistic case withk = 1.5704, square well size L = 2, and Dirac particle massm = 2000 in lattice units ~ = c = 1. The spin componentsψR (blue) and ψL ≈ 0 (red) are matched near zero at thez = 0 and z = 2 walls. That is, in the norrelativistic regime(k � m) these components nearly vanish at the z = 0 andz = 2 walls. The probability density ψ†ψ (gold) also vanishesat the walls. The 4-flux 0-component ψψ (green) still vanishesat the boundary walls and ψψ ≈ ψ†ψ because ψL ≈ 0, so thegreen curve overlaps the gold curve.

is in excellent agreement with theory. Another way to seethe behavior of a Dirac particle confined to a square wellpotential is shown in Fig. 5 as a parametric plot. The2-spinor field is not a perfect approximation of a non-relativistic scalar field because the 2-spinor field twistsinside of the potential barrier. Despite this twisting (arelativistic effect), the shape of the wave function withinthe square well (region II) remains sinusoidal.

0 50 100 150 200 250-4

-2

0

2

4

z

t=0 Ù dz Ψ†HzLΨHzL=256.02

0 50 100 150 200 250-4

-2

0

2

4

z


0 50 100 150 200 250-4

-2

0

2

4

z


0 50 100 150 200 250-4

-2

0

2

4

z


FIG. 4 Numerical prediction of the dynamical behavior ofthe lowest energy plane-wave eigenstate for a nonrelativisticcase for a Dirac particle with wave number k = 1.5704 ≈ π/2confined to a square well. The grid size is L = 256. The Diracparticle’s mass is m = 1.94707 + 2000π and in lattice units~ = c = 1. The barrier (red) is modeled with M = 3.45218.The spin components ψR (blue) and ψL (purple) nearly vanishat the walls, and the probability density ψ†ψ (gold) vanishesat the walls too. The 4-flux 0-component ψψ (green) becomesnegative at t = 92. The green curve overlaps the gold curveat t = 0 and at t = 192.

A quantum simulation of a Weyl particle confined to asquare well is shown in Fig. 6. The quantum simulationis in excellent agreement with theory.

IV. DIRAC PARTICLE IN A HARMONIC POTENTIAL IN1+1 DIMENSIONS

Here we will use the quantum algorithm (10) to sim-ulate a nonrelativistic scalar particle in an externalparabolic potential. Let us begin with the effective non-relativistic particle dynamics in 1+1 dimensions as gov-erned by the Schroedinger wave equation

i~∂tφ(z, t) = − ~2

2m∂zzφ(z, t) +

12κz2φ(z, t). (50)

Using separation of variables

φ(z, t) = f(z) e−iEt/~ (51)

gives

− ~2

2mdzzf(z) +

(12κz2 − E

)f(z) = 0 (52a)

or

dzzf(z) +(

2mE~2− mκ

~2z2

)f(z) = 0. (52b)

6


-100

0

100

z

-2

-1

0

1

2

Ψ

-2-10

1

2

-äΨ¯

-100

0

100

z

-2

-1

0

1

2

Ψ

-2-10

1

2

-äΨ¯

-100

0

100

z

-2

-1

0

1

2

Ψ

-2-10

1

2

-äΨ¯

-100

0

100

z

-2

-1

0

1

2

Ψ

-2-10

1

2

-äΨ¯

FIG. 5 Parametric plot (blue curve) of <[ψR(z)] and=[ψL(z)] versus z for the same simulation as shown in Fig. 4for t = 0, 32, 96, 192. The numerical solution is seen to wraparound the initial state (red curve) as time progresses in re-gions I and III inside the potential barrier.

0 200 400 600 800 1000

-5

0

5

z


0 200 400 600 800 1000

-5

0

5

z


0 200 400 600 800 1000

-5

0

5

z


0 200 400 600 800 1000

-5

0

5

z


0 200 400 600 800 1000

-5

0

5

z


0 200 400 600 800 1000

-5

0

5

z


FIG. 6 Numerical prediction of the dynamical behavior ofthe lowest energy plane-wave eigenstate for the case of a Weylparticle (m=0) with wave number k = 1.5704 ≈ π/2 confinedto a square well. The grid size is L = 1024. The barrier (red)is modeled with M = 3.45218. The quantum particle initiallymoves to the right as the upper component of the 2-spinorfield is nonzero while the lower component is zero. The Weylparticle remains confined to the square well, reflecting off ofboundary walls.

Defining b ≡√mκ/(4~2), the eigenequation may be

written as

dzzf(z) +(

4bE√

m

κ~2− 4b2z2

)f(z) = 0. (52c)

Consider a solution of the form

f(z) = e−b z2h(ς z), (53a)

where ς ≡ (mκ/~2)14 . Then,

dzf = e−b z2dzh(ς z)− 2b z e−b z

2h(ς z) (53b)

dzzf = e−b z2dzzh(ς z)− 4b z e−b z

2dzh(ς z)

+(4b2z2 − 2b

)e−b z

2h(ς z). (53c)

7


Inserting (53) in the eigenequation (52c) leads to

dzzh(ς z)− 4b z dzh(ς z) +(

4bE√

m

κ~2− 2b

)h(ς z) = 0,

(54)or since ς2 = 2b this is

dςz,ςzh(ς z)−2ς z dςzh(ς z)+2(E

√m

κ~2− 1

2

)h(ς z) = 0,

(55)which is Hermite’s ordinary differential equation

dz′z′h(z′)− 2z′ dz′h(z′) + 2nh(z′) = 0, (56)

for z′ ≡ ς z and

n =(E

√m

κ~2− 1

2

). (57)

We now demonstrate the quantum algorithm (10) byapplying it to the harmonic oscillator problem. Sinceς =√

2b = (mκ/~)14 , the quantum simulation is possible

to do by employing the analytical solution

φ(z) = Hn

[(mκ

~

)14

(z − L

2

)]e−√

mκ4~2 (z−L2 )2

, (58)

so long as we run the quantum algorithm in the nonrel-ativistic regime with ~k � mc2. What we need do is letthe Dirac particle’s mass be position dependent, using itto encode the confining potential

m(z) = m◦ +κ

2z2, (59)

and we also need to use (58) as the first component ofthe initial 2-spinor field that has a vanishing second com-ponent (ξ = 0)

ψ(z, 0) =

(φ(z)

0

). (60)

Adding together the coupled equations in (30), we have

φ+ ξ =−2iE(z)~c

E(z)2 − (m(z)c2)2∂z (φ+ ξ) . (61)

Let us work out the ground state solution to the harmonicoscillator problem, which has the form

ϕ(z) ≡ φ(z) + ξ(z) = ϕ◦e−bz2 , (62a)

and which upon inserting into (61) gives

ϕ =4iE(z)~c bz

E(z)2 − (m(z)c2)2ϕ =

4iE(z)c bz~ k(z)2

ϕ (62b)

or

k(z)2 = 4ic2 bz

√k(z)2 +

(m(z)c

~

)2

. (62c)

Therefore, in lattice units ~ = 1 and c = 1, the wavenumber must satisfy the fourth-order polynomial equa-tion

k(z)4 + 16b2z2(k(z)2 +m(z)2

)= 0. (62d)

A physical solution to this is

k(z) = 2b√−2z2 +

√4z4 −m(z)2z2/b2. (63)

A plot of |k(z)| is given in Fig. 7. The relativistic energy

0 100 200 300 400 5000.000

0.001

0.002

0.003

0.004

z

ÈkHz

LÈ

FIG. 7 Wave number variation for parameters L = 512 withκ = 0.01/L2 for a particle of mass m = 1/2.

is E(z) =√|k(z)|2 +m(z)2, so in turn the Lorentz factor

that we need for the quantum simulation is

γ(z) =

√|k(z)|2 +m(z)2

m(z). (64)

The particular quantum algorithm obtained by inserting(59) and (64) into (10) allows us to perform a numericalquantum simulation that unitarily evolves the 2-spinorinitial state (60). The observed numerical behavior isthat |ψ(z, t)| u |ψ(z, 0)| for all time, which is the ex-pected behavior of any energy eigenstate. The quantumsimulation agrees with theory, as shown in Fig. 8.

References

Alberto, P., C. Fiolhais, and V. Gil, 1996, Eur. J. Phys. 17,19.

Chodos, A., R. L. Jaffe, K. Johnson, and C. B. Thorn, 1974a,Phys. Rev. D 10, 2599, URL http://link.aps.org/doi/

10.1103/PhysRevD.10.2599.Chodos, A., R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F.

Weisskopf, 1974b, Phys. Rev. D 9, 3471, URL http://

link.aps.org/doi/10.1103/PhysRevD.9.3471.Yepez, J., 2010, arXiv:1106.0739 [gr-qc] .

8

http://link.aps.org/doi/10.1103/PhysRevD.10.2599





0 100 200 300 400 5000.0

0.2

0.4

0.6

0.8

1.0

z

Ψ†

HzLΨ

HzL


0 100 200 300 400 5000.0

0.2

0.4

0.6

0.8

1.0

z

Ψ†

HzLΨ

HzL


FIG. 8 Initial t = 0 probability density (blue curve) set equalto the ground state solution (red curve) of the harmonic os-cillator with external potential V (z) = κz2/2 for parametersL = 512 with κ = 0.01/L2 for a particle of mass m = 1/2.The numerical solution is shown at t = 10, 000, and onlyslightly departs from (or oscillates about) the exact groundsolution over time.

9

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Lecture notes: Dirac particle simulations in 1+1 dimensions · Lecture notes: Dirac particle simulations in 1+1 dimensions Je rey Yepez ... mc2 ˝ ~ 2 ˙ xei‘p z˙ z mc ... u ic