Quantum Mechanical Expansion of a Charged Particle in the Presence of a Non-Uniform Magnetic Field

Poh Kam CHAN

Candidate for the Degree of Master in Engineering

Supervisor: Associate Professor OIKAWA Shun-ichi

Division of Division of Quantum Science and Engineering

1. Introduction It is well known that a charged particle in the presence of a non-uniform magnetic field B tends to move to regions of weak magnetic field via collision/interaction with other particles. In quantum mechanics the probability density function (PDF) for a charged particle in the presence of a uniform magnetic field 0 zBB e in the x y plane is given by

2

0 0, exp ,qB qB

t t

r r r

where tr stands for the expectation value of the position r that is the same as the time-dependent position of the corresponding classical particle. The tendency towards the weak B B field region stated above makes the PDF of the particle broader than that for a uniform field case. Uncertainty in the position of a single particle in the presence of a non-uniform magnetic field is studied. The numerical expansion rate of variance in position and the expansion rate of variance in total momentum are studied.

In the case of a non-uniform field, we have developed a code to solve the time-dependent Schrödinger equation in the presence of a non-uniform magnetic field. In the paper [1], we have shown that the quantum mechanical variance in position may reach the square of the interparticle separation in a time interval of the order of 10

-4 sec for

typical magnetically confined fusion plasmas with a number density of n ~10

20 m

-3 and a temperature of

T~10keV. After this time the wavefunctions of neighbouring particles would overlap, as a result the conventional classical analysis may lose its validity.

2. Solving Schrödinger Equation Using Finite Difference Method (FDM) and Finite Element Method (FEM) We have solved the one-dimensional and two- dimensional Schrödinger equation numerically using FDM and FEM.

In the FEM calculation, we integrate both sides of the time dependent Schrödinger equation and after some derivation we arrive at the following expression: For a one-dimensional free particle case [1], we get

2

2i

2t

t m x

S D , (1)

where is the wavefunction, m is the mass and t is the time under consideration, i 1 the imaginary unit, and

/ 2h the reduced Planck constant. The relationship of S and D are described as

1

10S I + D . (2)

In the two-dimensional case, we adopt the simplest form for FEM which is based on a triangle model, thus we get,

2

2i

2t

t m h

S D , (3)

where the relationship S and D are described as

2 1

3 12S I + D . (4)

The inverse of the matrix S is difficult to obtain. Thus, we use an approximation that is known as Cayley’s form. One-dimensional Cayley’s form with FEM is written as,

eff 2

*

eff 2

i

4i

4

tm

xt t tt

m -x

D

D

, (5)

where the spatial operator is approximated with meff, the effective mass. In one-dimensional case, the effective mass is written as,

2

eff

1 1 2i

5

x

m m t

, (6)

and in two-dimensional case, the effective mass, meff is, 2

eff

1 3 1 1i

2 2

h

m m t

. (7)

By comparing the FDM result with the FEM result as shown in Table 1, it is found that the FEM wave function has higher velocity than the FDM wave function. In the FEM case, the final position at timestep t = 10,000 is closer to the exact expectation 0x .

However, from the viewpoint of the particle conservation, the energy conservation and the momentum conservation, we conclude that the FDM is more suitable for solving the one-dimensional and two-dimensional Schrödinger equation numerically.

We will continue to use the FDM until we can make further improvement of the FEM scheme for solving the two-dimensional Schrödinger equation.

Table 1 Comparison between the FDM and the FEM at

timestep t = 10,000 in periodic boundary case.

Method Finite Element

Method (FEM)

Finite Difference

Method (FDM)

Position x

(normalized)

2.821 -21.893

Momentum

conservation error

2.595×10-12

1.024×10-13

Particle

conservation error

5.408×10-12

2.148×10-13

Energy

conservation error

7.033×10-13

2.627×10-14

3. Expansion rate of variance in position and

total momentum

3.1 Schrödinger Equation We have solved the two-dimensional time-dependent Schrödinger equation for a single particle in the presence of a non-uniform magnetic field for a wavefunction at position r and time t, i.e.,

21

i i ,2

qt m

A (8)

where A is the vector potential, m and q are the mass and the electric charge of the particle. We assume a non-uniform magnetic field of 0,0, B yB and

B y 01 / By L B , where BL is the gradient scale length.

The initial condition for wavefunction at 0r r with

0r being the initial center of , is given by

2

0

02

00

1,0 exp i ,

2

r rr k r (9)

where 0 is the initial standard deviation, and

0 0 /mk v is the initial wavenumber vector. Where m is the mass of the particle under consideration,

0v is the initial velocity of the corresponding classical particle.

By using the finite difference method in space with Crank-Nicolson scheme for the time integration, Eqs. (8) and (9) above become as

1I H I H ,2i 2i

n nt t

(10)

where I is a unit matrix, H the numerical Hamiltonian matrix, and n stands for the discretized set of the two dimensional time-dependent wavefunction , ,x y t at a discrete time

nt n t to be solved numerically. We use successive over relaxation (SOR) scheme for

time integration in our numerical calculation. Calculation is done on a GPU (Nvidia GTX-580: 512cores/3GB @1.54GHz) [1-3].

3.1 Exact Wavefunction in a Uniform Magnetic

Field The exact solution , t r for the two-dimensional Schrödinger Eq. (2) with a uniform magnetic field with a Landau gauge [4], of , 0, 0,x y zA By A A is shown below,

2

2

2

0 0

2 2

1, exp

2

sin 2 sinexp ,

24

ikx

BB

B B

u tet y

y t yy t ti

r

(11) where /B qB is the magnetic length [4], the

/qB m is the cyclotron frequency,2

0 ,By k and

u t is classical velocity of the particle in x direction. By referring to Eq. (4) above, we can conclude that the standard deviation, variance, or uncertainty, in position remain constant throughout the time. In the case of uniform magnetic field, 2 2,B r BL t const.

3.2 Numerical Results Lengths are normalized by cyclotron radius of a proton

with a speed of 10 m/s in a magnetic field of 10 T. The cyclotron frequency in such a case is used for normalization of the time. Throughout this paper, we use normalization unit as shown in Ref. [2].

Throughout the calculation, we use normalized grid size of 0.02x y and normalized time step of

52 10 .t This normalized grid size is sufficiently small to use as noted in Ref. [1].

Figure 1 shows the time evolution of the variance in position and total momentum for 0 10v m/s in a uniform magnetic field. The variance of position oscillates with cyclotron period which half of the cyclotron period in total momentum. The time evolution of the variance in position for both uniform magnetic field and non-uniform magnetic field, which are defined as

2

2 * 2dr

r r r (12)

The time evolution of the variance in total momentum for both uniform magnetic field and non-uniform magnetic field, which are defined as

2

2 * 2i dP

P r , (13)

where P is the total momentum:

* 2i d

P r . (14)

Fig. 1 Normalized variance 2 r

in position, red plot, and 2 ,P in total momentum, blue line, for uniform magnetic

field .BL

3.3 Numerical error As for the numerical error in variance, our code is capable of accurately reproducing the time dependent variance in position 2

r t and total momentum 2

P t in uniform magnetic field which an exact solution is available [1]. For uniform magnetic field, ,BL the variance in position should remain constant: 2 2

r Bt as given by Eq. (4) [2]. Heisenberg uncertainty principle states that

/ 2r Pt t , therefore, the variance in total momentum also remains constant for uniform magnetic field. However, there is slight increment in variance as shown in Fig. 2. The difference between numerical calculation and theoretical value in this research is attributed to numerical errors [2]. Both variance in position and variance in total momentum are assumed to consist of numerical error, therefore, we perform numerical subtraction as in Ref. [2]. For non-uniform magnetic field, the peak variance grows with time as results of diffusion. The variance grows faster in higher magnetic flux density region as shown in Fig. 3.

In our numerical calculation, both non-uniform

magnetic field variance and uniform magnetic field variances behave non-linearly, as shown in Fig. 4. This is due to numerical error accumulated throughout the calculation. In this case, both non-uniform magnetic field and uniform magnetic fields’ increments in variance are consisting of the same numerical errors. Since this numerical error is undesirable in our calculation, we subtract the increment in variance for the non-uniform magnetic field from that for the non-uniform magnetic field as shown in Fig. 5.

Fig. 2 Comparison of 2 ,P between uniform magnetic field

,BL in red plot, and non-uniform magnetic field 45.219 10 m,BL in blue line. With magnetic flux

density B = 10T.

Fig. 3 Comparison of 2 ,P between uniform magnetic field

,BL in red plot, and non-uniform magnetic field 55.219 10 m,BL in blue line. With magnetic flux

density B = 50T.

Fig. 4 Comparison of 2 ,P normalized peak variance in total

momentum of uniform magnetic field ,BL

between numerical calculation, in red and theoretical 2 ,P in blue line.

Fig. 5 Comparison of normalized increment of peak variance in

total momentum, 2 ,P between uniform magnetic field

,BL in blue circle, compared with 45.219 10 m,BL in red square.

After subtraction of non-uniform magnetic field’s peak

variance with uniform magnetic field’s peak variance, we obtain a linear relationship for the increment in variance of total momentum with time as shown in Fig. 6. By considering particle gyration base time, we equalize each result to the same cyclotron frequency. From Fig. 6, we obtain a single data for our final expression, which after we perform multiple sets of calculation, we reach to the final results shown in Fig. 7 and Fig. 8.

Fig. 6 Linear normalized increment of peak variance in total

momentum 2 ,P for non-uniform magnetic field 45.219 10 mBL after subtraction of its numerical

3.4 Results The parameters used in this research are as follow: for initial speed 0v is 8 100 m/s, mass of the particle of 1 10 ,pm where pm is the mass of a proton. The magnetic field at the origin 0B of 5 10 T, charge of 1 4 ,e where e is the elementary charge, and BL of

52.610 10 – 55.219 10 m. Totally 38 sets of data were examined.

We perform calculation for uncertainty in position using numerical results for these parameter sets, we found a new relation between expansion rate of variance and the physical parameters. The final results are shown in Fig. 7 with logarithm of base 10 scale. Figure 7 is graph of physical parameter, 10 0 0log / BqB Lv against expansion rate of variance, 2

10log / .d dt r It was numerically found that the variance or the uncertainty in position can be expressed as [2],

2

0

0

4.3B

d

dt qB L

r

v[m

2/s]. (15)

Physical parameter, 10 0 0log / BqB Lv

Fig. 7 Expansion rate of variance in position with different

sets of parameters of 0 0, , ,m B q v and .BL

We compare this numerical result Eq. (15) with our

theoretical expansion rate of variance in position. In the presence of non-uniform magnetic field, we start the derivation of the expansion rate of variance of a single particle by using the Heisenberg’s equation of motion

22i ,H ,

d

dt

rr (16)

where H is the Hamiltonian, , and represent the commutator and the expectation value, respectively. Using this expression, we arrive at a preliminary result in the form

22

0 0 0

0 0

0

2 ,2

x

B B

qB yd

dt mL qB L

rr ev

v (17)

where the terms in the parenthesis are the sum of the initial velocity and the ∇B drift velocity, while the last term represents the expansion rate of variance. Expansion rate of variance in position is qualitatively the same as those in Eq. (15) and Eq. (17), except that numerical factor of 4.3 in Eq. (15) is replaced by unity in Eq. (17). The reason for this discrepancy is due to the fact that we have to derive the quantity relying only on the initial wave function for a non-uniform magnetic field, i.e., we only have the exact solution for the wave function in a uniform magnetic field.

Physical parameter, 10 0 0log / BqB Lv

Fig. 8 Expansion rate of variance in total momentum with

different sets of parameters of 0 0, , ,m B q v and .BL

Later on, using numerical analysis method, we have

developed a new expression for expansion rate in total momentum as a function of 0 0, , ,m B q v and BL [5],

2

0

00 . 5 7P

B

dqB

dt L

v[kg

2m

2/s

3]. (18)

Final results are shown in Fig.8. Figure 8 is graph of physical parameter, 10 0 0log / BqB Lv against expansion rate of variance, 2

10log / .Pd dt The numerical result Eq. (18) is compared with our theoretical expansion rate of variance in the total momentum. We derived the expansion rate of variance of a single particle by using Heisenberg’s equation of motion Eq. (16), so that we arrive at a preliminary result in the form

2

0 0

12 .

2

y y

y x

B

dm qB qB

dt L

pv v

v (19)

Expansion rate of variance in total momentum is qualitatively the same as Eq. (18) and Eq. (19), except that numerical factor of 0.57 in Eq. (18) is replaced by factor of 1/2 in Eq. (19). The reason for this discrepancy possibly comes from the fact that numerical error accumulated through the calculation.

In this expression, we found that mass, m , does not

affect both of our developed expressions in Eq. (15) [2] and Eq. (18) [5].

It is interesting to note that there is no mass, ,m dependence for the both expression above for variance in total momentum and position. In plasmas, however, the mass dependence, or the isotope effect, may appear through the replacement of

0 th 2 ,Bk T mv v where

thv is the thermal speed, Bk is the Boltzmann constant

and T is the temperature.

4. Summary We have solved the two-dimensional time-dependent Schrödinger equation for a single particle in the presence of a non-uniform magnetic field for difference set of parameters such as

0 0, , , Bm q v and .BL It is shown that the expansion rate for position increases linearly as

2

0 0/ 4.3 / Bd dt qB L r

v and the correspond expansion rate for total momentum increases linearly as

2

0 0/ 0.57 /P Bd dt qB L v . Both of these expressions are derived using numerical calculation. We are also interested in developing theoretical expansion rate of variance for both position and momentum. For these studies, we left it for future work.

References [1] S. Oikawa, T. Shimazaki and E. Okubo, Plasma Fusion Res.

6, 2401058 (2011).

[2] P.K. Chan, S. Oikawa, and E. Okubo, Plasma Fusion Res.

7, 2401034 (2012).

[3] S. Oikawa, E. Okubo and P.K. Chan, Plasma Fusion Res. 7,

2401106 (2012).

[4] L.D. Landau and E.M. Lifshitz, Quantum Mechanics:

Nonrelativistic Theory, 3rd ed., translated from the Russian

by J.B. Sykes and J.S. Bell (Pergamon Press, Oxford, 1977).

[5] S. Oikawa, P.K. Chan and E. Okubo, submitted to Plasma

Fusion Res.

[6] http://www.nvidia.com