Maxwell's Equations for Nanotechnology

Milan Perkovac I. Technical School TESLA, Klaiceva 7, Zagreb milan@drives.hr

Abstract - Classical physics, which includes Maxwell's equations, a hundred years ago couldnt explain the stability of atoms, the periodic table of elements, the chemical bond, the discrete excitation energies of atoms and their energetic state, the ionization of atoms, the spectra, including its fine structure and transition rules, experimental evidence about x-ray spectra and the behavior of atoms in electric and magnetic fields, the properties of the matter in solid state. That was the reason why classical physics fails when applied to the atomic area, i.e. to the area of nanometers or below. Now the situation has changed. It's because within the framework of classical physics, with the help of Maxwell's equations, we can derive Schrdinger's equation, which is the foundation of quantum physics. With this new knowledge all of the above can now be explained within the framework of classical physics. This article describes the procedure for obtaining the equation, which replaces the Schrdinger's equation using classical physics.

I. INTRODUCTION
The whole twentieth century in physics is mostly passed in the search for answers to fundamental questions of matter, primarily on the relationship between waves and particles. Discussions about this are still going on. All the studies are based on unique natural laws that apply universally in the universe and the atom. But this general approach to the macroscopic and microscopic scale seemed questionable in the case of Maxwell's equations. In fact, Maxwell's equations have reached excellent results in the macroscopic scale. Their application in the microscopic scale at first glance seemed disappointed. Namely, Maxwell's equations previously couldnt explain the stability of atoms, the periodic table of elements, the chemical bond, the discrete excitation energies of atoms and their energetic state, the ionization of atoms, the spectra, including its fine structure and transition rules, experimental evidence about x-ray spectra and the behavior of atoms in electric and magnetic fields, the properties of the matter in solid state. That was the reason why classical physics fails when applied to the atomic area, i.e. to the area of nanometers or below. This is why the entire physics divided into two branches: (traditional) classical and (new) quantum physics.

Despite the uncertain physical meaning of the wave function at the famous Schrdinger's equation, 2+82m(E-U)/h2=0, the equation has played an important role in the development of quantum physics. The wave function (x,y,z)=(r) is the solution of the Schrdinger equation, i.e. it is a mathematical expression involving the coordinates (x, y, z) of a particle in space , where are unit ( r = x i , j and k i + yj + zk vectors; all vectors below are bold letters). If we solved the Schrdinger equation for a particle in a given system then, depending on the boundary condition, the solution is a set of allowed wave functions of the particle, each corresponding to an allowed energy level. The physical meaning of the wave function is that the square of its absolute value, ||2, at a given point is proportional to the probability of finding the particle in a small element of volume, dxdydz, at that point. For an electron in an atom, this gives the idea of atomic and molecular orbital. All of this was obtained without the use of Maxwell's equations. After the explanation of radiation [1, 2], Maxwell's equations can give a lot more contributions to quantum physics, than has been the case so far. It's because from Maxwell's equations we can derive Schrdinger's equation, which is the foundation of quantum physics. The meaning of physical quantities in Maxwell's equations is completely clear. Therefore, the meanings of the wave functions of these equations, which are derived from Maxwell's equations, are completely clear. This article describes how Schrdinger's equation is obtained using classical physics, with special emphasis on Maxwell's equations. With this equation all the previous disadvantages of Maxwell's equations can now be explained in the framework of classical physics.

II. DERIVATION OF WAVE EQUATION

Maxwells equations are the four differential equations describing the space and time, i.e. (r, t), dependence of the electromagnetic field [3, 4, 5, 6]:
Gausss law for electric flux (electric flux begins and ends on charge or at infinity);

 D (r , t ) = (r, t ) ,

(1a)

a) Lets take curl of Faradays law, i.e. of (2c):

( E ) = ( B ) . t

Gausss law for magnetism (magnetic field lines dont begin or end);

(3)

B (r, t ) = 0 ,

(1b)

Faradays law of electromagnetic induction; a changing B produce E;

E (r, t ) =

On the other hand, for each vector E is true:

( E ) = ( E ) 2E ,

(4)

B (r, t ) , t

(1c)

and the substitution in (3) Ampere's law (2d) gives:

( E ) 2E =

Ampres circuital law (with Maxwells correction); H is produced by current J and by changing D;
H(r, t ) = J (r, t ) + D (r, t ) , t

E J + , t t

(5)

where = (1d)

is operator called 2 2 2 x y z the Laplacian. Now in (5) add Gauss's law for electric flux (2a), and J = gE:
2E g E 2E 2 = 0 . t t

where D is the electric displacement, E is the electric field strength, B is the magnetic flux density, H is the magnetic field strength, is the volume charge density, and J is the electric current density. The differential operator = i / x + j / y + k / z is called "del" operator. There are a total of 16 variables in (1a)-(1d) (the 15 components of the five vectors E, D, B, H, J, and the scalar ). If the source densities and J are given (four known variables) still remain 12 unknown variables. There are, however, eight equations; one from (1a), one from (1b), and three for components in (1c) and (1d) each. Obviously, to make system determinate we need additional relations. For a completely linear medium there are constitutive relations describing the properties of the media in which the fields exist: D = E, B = H, J = gE, where is dielectric constant, is the magnetic constant of proportionality, and g is the conductivity of the media. Hence [without writing (r,t) which is implied] the Maxwell equations become:

(6)

b) Lets take curl of Ampres law, i.e. of (2d):

( B ) = ( J + E ). t

(7)

On the other hand, for each vector B is true:

( B ) = ( B ) 2B ,

(8)

Using J = gE gives:
( B ) 2 B = g ( E ) + ( E ) . (9) t

Now in (9) add Faraday's law (2c):

( B ) 2 B = g B 2B 2 . t t

(10)

Using (2b), i.e. B = 0 , and B = H, and after sharing with , we get from (10):
2 H g H 2H =0. t t 2

Gausss law for electric flux; E =

(2a) (2b) (2c)

(11)

Gausss law for magnetism; B = 0 , Faradays law;

E = B = J + B , t E . t

Equations (6) and (11) are required wave equations. These two vector equations constitute six component equations with seven unknowns (Ex, Ey,

Ampres law;

(2d)

Ez, Hx, Hy, Hz, ). The system becomes determined if viewed without charge, i.e. = 0 . So we get:

L'dx / 2

Fig. 2a

C'dx L'dx / 2

C
Fig. 2b

Figure 1. Lecher line (section); twin-lead transmission line consisting of pair of wires of diameter 2, separated by .

Figure 2. Lecher line is presented with an infinitely small capacitors C ' dx and inductors L ' dx (Fig. 2a), which are concentrated at its end; C at the open end of the line, and L on its short-circuited end (Fig. 2b).

2E g

E 2E 2 = 0 , t t H 2H =0. t t 2

(12) (13)

2 H g

only a mutual adjustment of parameters. Then the voltage u of transmission lines represents the wave function E, while the current i of transmission lines represents the wave function H.

In empty space, i.e. in vacuum, what is also the inside of atoms, we have g = 0 , so the equations (12) and (13) becomes:
E 2 H
2

III. ADJUSTMENT THE PARAMETERS OF WAVE AND LECHERS LINE

Lecher line can be regarded as a limiting case of an LC network with infinitely small capacitors and inductors, Fig.2a, [10]. If all the capacitors of the network are put on the open end, and inductances on the short-circuited end, Fig 2b, the natural angular frequency of such oscillatory circuit will be, [11]:

2E t 2 t 2

= 0, =0.

(14) (15)

2H

Wave equations of the electromagnetic wave on Lecher's lines, Fig. 1 (using Maxwell's equations and Kirchhoff's laws, [7]), have the same form as equations (14) and (15) of the electromagnetic wave in vacuum [8],
2u 2i x
2

= 1 / LC .
The product of L' and C' is:
L' C ' = ln( / )+1/4 ln[( / )/2+ ( / )2 /41]

(20)

2u =0, L' C' x 2 t 2 L' C' 2i t 2 =0,

(16) (17)

(21)

The coefficient LC in (16) and (17) determines the phase velocity of the wave on the line (Fig. 3), in
Ag Mo Rb

where L is inductance of Lechers line per unit length, and C is its capacitance per unit length [9]:
L' = C' =
ln( / )+1/4

.8
, (18) , (19)

uem ()1/2
Pb U

La 0.74

.6 .4 .2 0 1

p0 - structure constant p0 = 8.277 56

2 ln / (2 ) + [ / (2 )] 1

UNSTABLE Z - atomic number

Uuo 2.398

where is the radius of each conductor and is the distance between the conductors. Transmission line can be a model for the observation of electromagnetic waves, given that they have the same differential equation. It takes

= ep /Z 5 6 /

2 0

Figure 3. Phase velocity of electromagnetic wave on Lechers line as a function of ratio /. Areas of stability of atoms is determined in accordance with [12] (Z=82, lead, /=2.398).

70 60 50 40 30 20

(/) 82 (2.398) (2.398) = = 82 82 0.835 0.836 6 9~ ~ 68.518 82 A0=(0/0)1/2e2Z (/)

A0=3772.5610-3882(2.4) =6.6210-34 Js 2He

Z
1 2 3

0 0 n Decay modes: p+e- + e

10

103 10 2 101 10 0

Atomic number

2 2

3 4

10 0 2.4

15 Z = 82 (2.397 973)=0.835 585 5.04410^29

10^6 10^12 10^18 10^24

Be 4 65

Li

LECHER LINE

H Elementary particles He Li All elements of the periodic table 82 Pb 118 Uuo 10 0 101000 10 2000 103000 104000

10^30

10 5000

10 6000

10

-1

Figure 4. Structural coefficient of twin-lead transmission line,

Figure 5. Structural coefficient of twin-lead transmission line,

( / ) =

[ ln( / ) + 1 / 4] ln[( / ) / 2 + ( / ) / 4 1 ]

( / ) =

[ ln( / ) + 1 / 4] ln[( / ) / 2 + ( / ) / 4 1 ]

(Lecher line) in function of / (log-lin scale), Ref. 12.

(Lecher line) in function of / (log-log scale), Ref. 12, (Elements and Particles).

accordance with the equation, [13]:

uem = 1 / L' C' ,

T = 1 / = 2 LC .

(27)

(22)

i.e.,
1 ln[( / ) / 2 + ( / ) 2 / 4 1] ln( / ) + 1 / 4

Electromagnetic energy of an LC circuit, Eem , theoretically can be presented as the electrical energy stored in a capacitor that has a capacitance (or equal to C and a maximum charge equal to Q C magnetic energy stored in a coil that has inductance equal to L and maximum current equal ), so: to I L
Eem = 2 1 1 2 1 Q C = LI L = 2 2 C 2 2 Q C C C L L

uem =

. (23)

Using (18) and (19), the characteristic (wave) impedance of LC circuit, as well as the characteristic impedance of the Lechers line read:
Z LC =
= =

L/ C =

L'/ C'

[ ln( / )+1/ 4]ln[( / )/ 2 + ( / )2 / 41 ]

( / ) / ,

/ (24)

1 L 2 2 = A , = = Z LC Q QC C C 2 LC

(28)

where
2 A = Z LC Q C

(29)

where from a mathematical point of view is a function of / ,

is an action of electromagnetic oscillators, defined as the product of energy, Eem , and period, T, of the LC oscillator:
A = EemT = Eem / .

[ ln( / ) + 1 / 4] ln[( / ) / 2 +

( / ) / 4 1 ]

(25) From the physical point of view is a coefficient of the structure of transmission lines, so we will call it structural coefficient dependent on the parameters and , more specifically on / , (see Fig. 4 and Fig. 5). Natural frequency of electromagnetic LC oscillator is determined by (20), i.e.,

(30)

All above is generally true at the macroscopic and the microscopic level.

IV. THEORY ON MICROSCOPIC SCALE

Now we will adjust the previous equations for the microscopic scale. From Newtons second law, F = ma , we substitute a = v 2 / ra and Coulombs law for F, and also from relativistic mass we obtain:

= 1 / 2 LC .

(26)

This means that the oscillatory period of a LC oscillator is equal to

2 ra/(0.1R0) K Eem 2 1 /(mc ) 1E U

ra.=.R0

1- ____
2

K Eem _.. v .=. c. E

0 -1

When the electromagnetic wave leaves the atom it carries out the energy Eem , and vice versa, when it enters the atom, it brings in the same energy. This means that due to the reduction of energy Eem , the atom should be compensated with equally large energy eV :
Eem = E = eV .

-2

R0=|q.Q|/(40mc2) E.=.K+U.=.-Eem
0.2 0.4 0.6 0.8

(36)

U
1

From the expressions (35) and (36) we obtain

K,

Figure 6. Radius of the atom

ra , kinetic energy

1 2 = 1 eV / mc 2 ,

(37)

electromagnetic energy Eem , total mechanical energy E , and potential energy U of the electron in an atom as a function of = v / c , ( 0 1 ).

i.e.,

2 = 2eV / mc 2 1 eV / 2mc 2 ,
or

)(

(38)

mv

= 2eV / mc 2 1 eV / 2mc 2 .
= | qQ |
2 4 ra

(39)

ra 1 2

(31)

Using (32), (37), and (38), we obtain

ra = | qQ | 1 eV / mc 2 , 8 eV 1 eV / 2mc 2

or
ra = | qQ | 4 mc
2

(40)

(32)

while from (40) we obtain eV using (28) and (36):

eV = 2 | qQ | 1 eV / mc 2 1Q C . = 8 ra 1 eV / 2mc 2 2 C

where ra is the radius of the atom, q is the charge of the electron ( q = e , e = 1.602 177 33 1019 C ), Q is the core charge, m = 9.109 ...1031 kg is the electron rest mass, c = 299 792 458 m / s is the speed of light in vacuum, = v / c , where v is the electron velocity. The kinetic energy of electrons in the atom is mc 2 K= mc 2 , (33) 2 1 and, using (32), noting that an electron is of opposite charge of the nucleus, the potential energy of electrons is
U= qQ mc 2 = 2 . 2 4 ra 1

(41)

From a single equation, (41), we need to determine two unknown sizes, i.e., parameter C and . Therefore, we seek solutions with the variable Q C help of Diophantine equations, [14]:
C = 4 ra ,
2 2 = | qQ | 1 eV / mc , Q C 1 eV / 2mc 2

(42) (43)

Using (24), (26), (41) and (42) we get the following, which is similar to the expression (28):
eV = | qQ | 1 eV / mc
2 2

(34)

8 ra 1 eV / 2 mc

=
2 2

| qQ | 1 eV / mc

2 2

2C 1 eV / 2 mc

(44)

Thus, the electromagnetic energy, Eem , (see Fig. 6) is equal to the total mechanical energy E with negative sign,
Eem = E = ( K + U ) = mc 2 1 1 2 .

= Z LC | qQ |

1 eV / mc

1 eV / 2 mc

= A .

From (29) and (43) follows

A = Z LC | qQ | 1 eV / mc
2 2

(35)

1 eV / 2 mc

(45)

70

60

= p 2 /Z =68.518 0

H1+ ;

A0 0 /0 e2Z

Eem | 0 = A0 .

(51)

50 40 30 20 10

2 Z = p0

2 /2 = 34.259 He2+ ; = p0

A p0 = 1 0 =8.277 56 e 0c

This, in the case of A0 = h , proves the validity of empirically derived Planck-Einstein's relation Eem = h , but also shows that Planck-Einstein's relation is limited only to relatively low frequencies . If this frequency is high, then
Eem | = mc .
2

p0

Li3+;22.839 Be4+;17.129 Ca20+; Zr40+;

2 = d 2 =(11.706 237) 2 = 137.036 -1 = 2p0 0

1.142
50 60

Nd60+;

00

d0 10
p0

3.426
20

1.713
40

0.856 0.692 Atomic number Z

70
2

Hg80+; Es99+

(52)

30

80

90

99

Figure 7. Structural coefficient = p0 / Z as a function of atomic number Z (equilateral hyperbole of atoms).

From (28) and (50) follows the action of electromagnetic oscillators:

mc 2 mc 2 + .
2

A = A0 +

If eV = 0 then A = A0 . This means that (45) can be written as

A = A0 1 eV / mc
2 2

2 A0

(53)

1 eV / 2mc

(46) (47)

where
A0 = Z LC | qQ | .

We see that the action of electromagnetic oscillators, A, is not constant for all frequencies . Momentum of electromagnetic wave, pem , in accordance with [16], is equal to the ratio of its energy ( Eem ) and phase velocity
uem = ,

A0 does not depend on the energies eV, which

(54) (55)

enter or leave the atoms. This ultimately means that A0 is the universal constant of any atom (see Fig. 4 and Fig. 7), [15]:
A0 = 0 / 0 e p0 = 6.626 075 10
2 2

i.e.,
pem = Eem E = em , uem

34

J s (48)
7

where

p0 = 8.277 56 ,
12

0 = 4 10 H / m ,
F/ m.

From (44) follows eV = A , and using (46) we get:

eV 1 eV / 2mc 2 . = A0 1 eV / mc 2
(49)

0 = 8.854 187 817 10

where is the wavelength of the electromagnetic wave. On the other hand, in accordance with the law of conservation of momentum, the momentum of electrons is equal to the momentum of the electromagnetic wave, Eem /( ) = m v / 1 = mc / 1 .
2 2

(56)

From the expression (56), and using (28), (37), (39), (41), and (46), we obtain

A0

(1 eV / mc )

2 2 2 3

V. RELATIONSHIP OF MAXWELLS AND SCHRDINGERS EQUATIONS

From (36) and (49) we get the general relativistic expression for the energy of electromagnetic waves emitted or absorbed by the atom,
Eem = A0 + mc - ( A0 ) + ( mc ) .
2 2 2 2

2meV

(1 eV / 2 mc )
2

(57)

If the voltage is low, then eV m v , and

| eV / m c 2

A0 mv
2

(58)

(50)

If the voltage approaches ( mc

e) 511 kV, then

If the frequency is low, then

1 0.8

0.6 10 1/2 (1) 0.4 0.2 0 0

(2)

A0 mc

(3)

(5) (4)

eV =v / c mc 2 mc2 A r 0
10
(6)

0 0 0 0

2m eV 2m eV 1

1 eV / 2 mc 1 eV / mc
2

= r 0 ,

(65)

uem /v uem /c

(7)

=
eV mc2
0.6 0.8 1

1 eV / 2mc 1 eV / mc 2m eV
2

= r 0 ,

(66)

i.e.,

0.2

0.4

r = r =

1 eV / 2 mc 1 eV / mc
2

Figure 8. Phase velocity u em , (1) and (2), wavelength , (3), frequency , (4), of standing-electromagnetic wave; speed of the particles = v / c , (5); r = r , (6); frequency of matter waves (quantum mechanics), (7); all as a function of eV / mc .
2

0 0

(67)

Current values of linearly standing wave, [17], in the atom read:

polarized

E x ( z,t ) = E 0 sin

z cos (t ) ,

(68)

| eV / m c 2 1 0 m .

(59)

H y ( z,t ) =

E0 2 cos /

z sin (t ) ,

(69)

Phase velocity, according to (49), (54) and (57), (see Fig. 8) is

uem = = eV 2m 1 eV / mc
2 2

where E 0 is the maximum value, i.e., the amplitude of electric field strength, E x ( z,t ) , the x-component of the electric field dependent on the z-axis of the rectangular coordinate system and the time t, and H y ( z,t ) is the y-component of the magnetic field dependent on the z-axis of the rectangular coordinate system and the time t. If we use the second derivation of the equations (68) and (69) with respect to z, we get equations:
E x ( z,t ) 2 + E x ( z,t ) = 0 , 2 z H y ( z,t ) z
2 2 2 2 2

1 eV / 2 mc

. (60)

If the voltage is low, then

uem | eV / m c 2 0 = eV / 2 m = m v / (2 m) = v / 2 .
2

(61)

(70)

On the other hand, Maxwell's equations also require that the following relationship be satisfied for the phase velocity:
uem = 1 /

(62)

2 + H y ( z,t ) = 0 . (71)

while the wave impedance / , because of energy reasons, should remain unchanged, i.e.,

After substitution of wavelengths, (57), in the expressions (70) and (71), we get:
E x ( z ,t )
2

/ = 0 / 0 .
From (60) and (62) we get
1/

(63)

8 meV (1 eV A0
2 (1

/ /

2 mc mc

2 3 )

eV

2 4 )

E x ( z ,t ) = 0 ,

(72)
2 2

eV 2m

1 eV / mc

H y ( z ,t )

1 eV / 2 mc

(64)

8 meV (1 eV A0
2 (1

/ /

2 mc mc

2 3 )

eV

2 4 )

H y ( z ,t ) = 0.

(73) Since eV = E , we can write,

System of equations (63) and (64) have solutions and as follows:

 E x ( z ,t ) z
2

8 mE (1 + E / 2 mc )

2 3

A0

(1 + E / mc )

2 4

E x ( z ,t ) = 0,

(74)
H y ( z ,t ) z
2 2

8 mE (1 + E / 2 mc

2 3 )

A0

(1 +

E / mc

2 4 )

H y ( z ,t ) ) = 0,

the y-component of the magnetic field dependent on the z-axis of the rectangular coordinate system and the time t. Meaning of all the other physical quantities in this article is also completely clear. The solutions of differential equations (76) and (77) are the same as the known solutions of the Schrdingers equation only the meaning of the wave function is different.

(75) while in
2

the

case

of

low-energy

VI. CONCLUSION
The article shows how Maxwell's equations adapted to the use in the nanometer range. The resulting equations are similar at the end of Schrdingers equation, which was expected. Given the great success of the application of Schrdingers equation, it is also a confirmation of the success of Maxwell's equations. The advantage of Maxwell's equations is that they are part of a comprehensive theory, while Schrdingers equation is only one isolated part adapted to the atomic scale. Hence the resulting difficulty in interpreting the meaning of the wave function in the Schrdingers equation. Such difficulties in Maxwells theory, of course, do not exist. Therefore it seems appropriate to use Maxwell's equations in the nanometer range, in which up to now exclusively used Schrdingers equations.

(eV= m v = K = E U ),
E x ( z ,t ) z
2 2 2

8 m

A0

( E U ) E x ( z ,t ) = 0,

(76)

H y ( z ,t ) z
2

8 m

A0

( E U ) H y ( z ,t ) = 0,

(77)

which in this form resembles the Schrdinger's equation. In the same way as we got (70) and (71), using the second derivation with respect to time t, with = 2 , we get from (68) and (69):
E x ( z ,t ) t
2 2 2

+ E x ( z ,t ) = 0 ,

(78)
.[1].J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, . . . 157 (1945), p.162. .[2].M. Perkovac, Phys. Essays 15, 41 (2002); 16, 162 (2003); . . http://arxiv.org/abs/1010.6083. .[3].D. R. Frankl, Electromagnetic theory (Prentice-Hall, Inc., . . Englewood Cliffs 07632, 1986). .[4].J. D. Jackson, Classical Electrodynamics, 3rd edition (John . . Wiley & Sons, Inc., New York, 1998). .[5].T. Bosanac, Teoretska elektrotehnika 1, Tehnika knjiga, . . Zagreb, 1973. .[6].Z. Haznadar and . tih, Elektromagnetizam, kolska . . . knjiga, Zagreb, 1997. .[7].R. Rdenberg, Elektrische Schaltvorgnge, Verlag von . . . Julius Springer, Berlin, 1923, p. 329. .[8].H. Czichos, HTTE, Die Grundlagen der . . . . . . . . Ingenieuewissenschaften, 29. Auflage, Springer-Verlag, . . . Berlin, 1989, p. B 199. .[9].T. Bosanac, Teoretska elektrotehnika, (Ref. 5), p. 371. [10].R. H. Good, Classical Electromagnetism, Saunders Golden . . Sunburst Series, Fort Worth, 1999, p. 420. [11].R. Rdenberg, Elektrische Schaltvorgnge, (Ref. 7), p. . . . 352. [12].M. Perkovac, http://arxiv.org/abs/1010.6083, (Ref. 2). [13].H. Czichos, HTTE (Ref. 7), p. B 200. [14].M. Perkovac, http://arxiv.org/abs/1010.6083, (Ref. 2). [15].Ibid. [16].E. W. Schpolski, Atomphysik, Teil I, VEB Deutscher . . . . Verlag der Wissenschaften, Berlin, 1979, p. 406. [17].Z. Haznadar and . tih, Elektromagnetizam, (Ref. 6), p. . . . 436. [18].B. Wong, Phys. Essays 22, 296 (2009).

H y ( z ,t ) t
2

+ H y ( z ,t ) = 0 ,

(79)

or, using (49) and eV = E ,

E x ( z ,t ) t
2 2

4 E A0
2

1 + E / 2 mc 2 2 1 + E / mc 1 + E / 2 mc 2 2 1 + E / mc

E x ( z ,t ) = 0 ,

(80)
H y ( z ,t ) t
2 2

4 E A0
2

H y ( z ,t ) = 0 .

(81) Unlike wave function in the Schrdinger's equation,

/ x + 8 m( E U ) / h = 0 ,
2 2 2 2

[18], physical meaning of wave functions, E x and

H y , in this article is entirely clear. As discussed

above, E x ( z,t ) is the x-component of the electric field dependent on the z-axis of the rectangular coordinate system and the time t, and H y ( z,t ) is