Circuit Theory Based on New Concepts and Its ...

Journal of Signal Processing, Vol.21, No.1, pp.37-45, January 2017

LECTURE

Circuit Theory Based on New Concepts and Its Application to Quantum Theory 8. Application of Circuit Theory to Schrödinger Equation Nobuo Nagai１(Hokkaido University) and Takashi Yahagi2 (Signal Processing Technology Laboratory) E-mail: [email protected], [email protected] Abstract The Schrödinger equation is derived from the concept that particles of matter such as electrons have a wave nature, and it is used to express phenomena specific to quantum mechanics, such as the advancing wave and the quantum tunneling effect. In this session, we apply circuit theory to the Schrödinger equation and define voltage and current using wave functions that satisfy the Schrödinger equation. By this method, we show that a lossless transmission line or circuit can be obtained. Thus, a circuit that satisfies the Schrödinger equation is lossless and is not directly related to heat, but the reactive power is related to the regulation of energy. Therefore, such a circuit can be discussed in a theoretical system that does not rely on probability theory. Keywords: Schrödinger equation , wave function, advancing wave, quantum tunneling effect, complex equiva-

lent circuit, lossless transmission circuit, reactive power, imaginary resistor, backward wave, stop-band

1. Introduction

2. Handling Wave Equations in Physics

According to Ref. 1: “Many science historians consider that the year 1926 was a turning point in physics and involved some innovative events. In this year, the equivalence of Schrödinger’s wave mechanics and Heisenberg’s matrix mechanics was demonstrated, and as a result of their combination, quantum mechanics was considered to be complete. Currently, however, few physicists think that the solution of the wave equation, ψ (wave function), physically exists. Although ψ certainly expresses something, what it expresses cannot be determined. As a result, the wave function deviates from orthodox research on quantum mechanics.” In this session, we regard the Schrödinger equation as an extension of Maxwell’s equations in terms of circuit theory. Then, we determine the equivalent circuit of the Schrödinger equation and attempt to express its wave equation using two functions, one of voltage and one of current. Namely, we show what the wave function expresses by applying circuit theory and that this application may be related to quantum theory and elementary particle theory.

In physics, the Schrödinger equation is handled as a wave equation. From the viewpoint of circuit theory, incorrect buttonholes may be included in the wave equations used in physics. This is explained below. In physics, Maxwell’s equations are expressed using two functions, one of electric field (voltage) and one of magnetic field (current), but are often analyzed using the wave equations alone, as described in Ref. 2. Unless the two functions of voltage and current are used to analyze the wave equations, as conventionally used, the cascade matrix essential for circuit theory cannot be obtained. In addition, a Laplace transform is also required to obtain the cascade matrix. Without the Laplace transform, the transient response cannot be determined. As a result, the steady-state response cannot be distinguished from the transient response. In other words, one who does not use the Laplace transform can discuss only the steady-state response. Therefore, it is unclear how the system undergoes a transient phenomenon to achieve resonance in the steady state. If a wave formed through resonance is regarded as a wave packet, it cannot be understood why the decomposition using the Fourier transform of the wave packet is inappropriate. Moreover, the conservation law is examined only for instantaneous

Journal of Signal Processing, Vol. 21, No. 1, January 2017

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energy because the transient phenomenon is not taken into consideration. In circuit theory, however, power is defined as the product of voltage and current, and therefore both active and reactive powers are obtained in the steady state. With only the wave function, the reactive power cannot be obtained and hence the regulation using the reactive power cannot be examined. Because the energy conservation law holds in the steady state, causal laws that hold in the steady state cannot be taken into consideration. If the reactive power cannot be obtained, a paradox of energy, shown in Ref. 3, may arise. As described in Ref. 4, scattering theory is treated in physics, while refraction is excluded. Because refraction is related to reflection and transmission, these phenomena are not discussed in the scattering theory treated in physics.[4] In Ref. 5, reflection and transmission are considered in relation to the tunneling effect expressed by the Schrödinger equation. However, only reflection and transmission in the steady state are considered but they are not considered in the transient solution of the Schrödinger equation in physics. Because a lossless circuit is obtained using cascade matrices, we cannot judge whether the circuit is lossless when the cascade matrices cannot be obtained. Moreover, impedance is considered for the cascade matrices, and reflection and transmission coefficients of the circuit are determined on the basis of the difference in impedance of two media. Hence, reflection and transmission are phenomena essentially different from the scattering defined in Ref. 4. Thus, two functions of voltage and current are defined in circuit theory. Although circuit theory has only treated onedimensional waves to date, it is a powerful tool for clarifying the behavior of waves. Now let us examine how to obtain the two functions of voltage and current from Schrödinger’s wave equation. The electric field (voltage) and magnetic field (current) that satisfy Maxwell’s equations also satisfy the wave equations derived from Maxwell’s equations. Moreover, the voltage and current also satisfy the telegrapher’s equations. In physics, it is important to treat phenomena in a unified manner. Therefore, the analysis of wave equations is emphasized because, if wave equations can be analyzed using circuit theory, both the voltage and current can be analyzed simultaneously. However, the voltage and current in circuit theory have their own specific properties, as described in Session 7; for example, the voltage and current wave digital filters for a circuit have different values for digital components corresponding to reflection. Therefore, cascade matrices that simultaneously use both voltage and current are used to effectively utilize their properties. If the Schrödinger equation is assumed to be a wave equation and an extension of Maxwell’s equations, we may obtain two functions of voltage and current that satisfy the Schrödinger equation on the basis of the above-mentioned concept.

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The wave function that satisfies the Schrödinger equation was obtained in Ref. 6. When this wave function is defined as voltage and the function obtained by differentiating the wave function with respect to x, which also satisfies the Schrödinger equation, is defined as current, the voltage and current that satisfy the Schrödinger equation can be obtained. In the following, by referring to Ref. 6, we show an equivalent circuit that satisfies the Schrödinger equation as well as its voltage and current. The Schrödinger equation is a partial differential equation and may be obtained as an equation that expresses a uniform transmission line. Moreover, the Schrödinger equation may express a lossless line. If the equation expresses a uniform lossless transmission line and both forward and backward waves are obtained, the transient response may be obtained.

3. Schrödinger Equation for Circuit Theory The Schrödinger equation is a basic equation in quantum mechanics and can represent the energy level of hydrogen atoms. By using the Schrödinger equation, the tunneling effect can be explained, and resonance can be obtained using resonant tunneling diodes based on the tunneling effect. Through this lecture series, we have shown that many in-phase waves are required to achieve resonance. Therefore, a lossless reciprocal transmission line or circuit is required to achieve resonance. To this end, here we examine the Schrödinger equation in the form to which circuit theory is applied, rather than the form used in conventional physics. 3.1 Derivation of Schrödinger equation Einstein considered that light has the nature of both electromagnetic waves and particles and assumed that photons, which are the constituents of light, have an energy and momentum, respectively, given by (8.1a) E  h  

p

h 

(8.1b)

where ν and λ are the frequency and wavelength of the light wave, ħ is Planck's constant, and ω is the angular frequency. de Broglie considered that matter particles, such as electrons, have a wave nature. The momentum p of a mass m with an energy of E is given by

p  2mE  U 

(8.2)

where U is the potential of the force. On the basis of Eq. (8.2) and the matter wave proposed by de Broglie, Schrödinger proposed an equation of a matter wave with a constant energy of E in the steady state as



2 2    U  H  E 2m

(8.3)

Here, the left-hand side includes the Hamiltonian operator of a harmonic oscillator and is rewritten as H.


Equation (8.3) is independent of time. We attempt to In Ref. 7, extend it to a time-dependent equation. Nakayama commented that Schrödinger’s discussions were somewhat circuitous. We apply the Hamiltonian operator to Eq. (8.3) twice, denoting the time-dependent function as Ψ, and obtain 2

 H        t  2

2

(8.4)

The extension method is examined using circuit theory. Because one-dimensional equations are treated in circuit theory, the time-independent equation in the steady state, as given by Eq. (8.3), may be rewritten as 2 d 2   x   U x   H x    x   E x  (8.5) 2m dx 2 Here, the energy E is expressed using Eq. (8.1a). Circuit theory uses a Laplace transform, and the angular frequency ω always appears with the imaginary number j (i.e., in the form of jω) in the steady state. Considering this, we assume that j has been omitted from Eq. (8.5). Then, the equation is raised to the second power to obtain an equation with real coefficients (without j). To extend the equation to a time-dependent equation, we denote the time-dependent function as Ψ(x,t) and obtain [7] 2

 H   x, t         x, t   t  2

2

(8.6)

This equation can be factorized as     (8.7)  H  j  H  j x, t   0 t  t   Therefore, the time-dependent Schrödinger equation can be expressed in two ways:

 2 2      U x, t    j x, t  2 2 m  t  x    2 2      U x, t   j x, t  2 t  2m x 

(8.8a) (8.8b)

In physics, the time-dependent Schrödinger equation is given by Eq. (8.8b). When circuit theory is used, however, the Schrödinger equation is given by Eq. (8.8a).[6] When the variables of the Schrödinger equation are separated, its time-dependent term is assumed to be given by (8.9) exp jt 

where   0 . That is, we assume that the time-dependent Schrödinger equation is given by (8.10) x, t    x  exp jt  By substituting Eq. (8.10) into Eq. (8.8b), as done in physics, the propagation constant obtained from the secondorder derivative becomes a real number, and no wave function is determined. However, when Eq. (8.10) is substituted into Eq. (8.8a), the propagation constant


obtained from the second-order derivative becomes purely imaginary, showing that a wave function can be obtained. 3.2 Wave function satisfying the Schrödinger equation Here, we attempt to solve the Schrödinger equation, as shown in Ref. 5, instead of obtaining an equivalent circuit. To this end, a method of deriving the sole wave function from Maxwell’s equations is used. Equation (8.5) is a second-order linear homogeneous differential equation and can be rewritten as (8.11)  ( x)  K exp(x) To understand how a particle with an energy of E behaves within a material at a constant potential U, we determine the behavior of the particle within the material under the following potential conditions: (a) U(x) = Uw, E > Uw (b) U(x) = Ub, E < Ub (a) Solution in the case of U(x) = Uw, E > Uw Under this condition, Eq. (8.5) may be rewritten as



2 d 2  ( x)  (U w  E ) ( x)  0 2m dx 2

(8.12)

By substituting Eq. (8.11) into Eq. (8.12), we obtain

2 2   (U w  E )  0 (8.13a) 2m 2 m( E  U w )   j 1 (8.13b) ∴  j  Thus, the wave function ψ(x) has a propagation constant consisting of the phase constant β1 alone and is given by (8.14)  ( x)  Aw exp(  j1 x)  Bw exp( j1 x) 

Here, Aw and Bw depend on the boundary conditions. The material just discussed is called a quantum well and serves as a glass for photons[8]. Note that the second term on the right-hand side of Eq. (8.14) represents a wave traveling from right to left, which is defined as the advancing wave specific to quantum mechanics and conveys future information to the present phase. If the voltage and current waves are defined on the basis of circuit theory, it can be shown that the wave is a backward wave that goes back from the present to the past phases, as discussed in Session 1, rather than the advancing wave specific to quantum mechanics. (b) Solution in the case of U(x) = Ub, E < Ub Under this condition, Eq. (8.5) may be rewritten as



2 d 2  ( x)  (U b  E ) ( x)  0 2m dx 2

(8.15)

By substituting Eq. (8.11) into Eq. (8.15), we obtain

2 2   (U b  E )  0 2m 2m(U b  E )   ∴    

(8.16a) (8.16b)

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In this case, the wave function ψ(x) has a propagation constant consisting of the attenuation constant α alone and is given by (8.17)  ( x)  Ab exp( x)  Bb exp(x) Here, Ab and Bb depend on the boundary conditions. In this material, electrons are transmitted via the tunneling effect, which is a quantum effect, rather than as waves. The potential domain in this case is called the quantum barrier. If voltage and current are defined on the basis of circuit theory, the phenomenon is not specific to quantum mechanics but serves as the stop band, which is based on iterative parameter theory. When the Schrödinger equation is assumed to be specific to quantum mechanics, as in conventional physics, the advancing wave and tunneling effect may be considered as phenomena specific to quantum mechanics. However, if the voltage and current waves that satisfy the Schrödinger equation can be obtained, such phenomena are expected to be explained by applying circuit theory.

4. Preparation for Application of Circuit Theory We attempt to apply circuit theory, in particular, the theory of the transmission line, to the Schrödinger equation. To this end, we start by simply outlining the method of solving the telegrapher’s equations given in Ref. 6. 4.1 Telegrapher’s equations and transmission line It is assumed that the resistance and inductance of a round-trip unit length of a uniform transmission line are R and L and the leakage conductance and capacitance of a round-trip unit length between the two conductors are G and C, respectively. The voltage v(x, t) and current i(x, t) along the round-trip unit length between the two conductors satisfy    ( x, t )  L i ( x, t )  Ri( x, t ) x t    i ( x, t )  C  ( x, t )  G ( x, t ) x t



(8.18a) (8.18b)

By substituting Eq. (8.18b) and the equation obtained by differentiating Eq. (8.18b) with respect to t into the equation obtained by differentiating Eq. (8.18a) with respect to x, we obtain 2 2  ( x , t ) LC  ( x, t )  t 2 x 2  (8.19)  ( RC  LG )  ( x, t )  RG ( x, t ) t

Equation (8.19) indicates that the voltage wave satisfies the telegrapher’s equations. Similarly, it can be shown that the current wave i(x,t) also satisfies the telegrapher’s equations. To apply circuit theory to the Schrödinger equation, we should examine the correspondence between Eqs. (8.8a) and (8.19) and derive two functions, similar to those expressed by Eqs. (8.18a) and (8.18b), from the correspondence. In

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this case, a Laplace transform is used to emphasize that the Schrödinger equation includes purely imaginary coefficients. That is, Eq. (8.19) is rewritten by applying a Laplace transform and expressing the partial derivative with respect to t using the Laplace variable s as follows. d2 V  x   s 2 LCV  x   sRC  LG V x   RGV x  dx 2

(8.20)

4.2 Circuit theory of complex coefficients Focusing on the similarity between the Schrödinger equation given by Eq. (8.8a) and the telegrapher’s equations given by Eqs. (8.19) and (8.20), we attempt to apply circuit theory to the Schrödinger equation. To this end, a Laplace transform is applied to the Schrödinger equation by Eq. (8.8a), similarly to the case of Eq. (8.20), which is obtained by applying a Laplace transform to the telegrapher’s equations, and the partial derivative with respect to t is expressed using s. Then we obtain

2m 2mU d2  x   j s  x   2   x    dx 2

(8.21)

Since the similarities between quantum phenomena and circuit theory were proved by Kron [9] and others, many attempts to apply circuit theory to clarify quantum phenomena have been made. In these attempts, however, the property that the time-dependent Schrödinger equation given by Eq. (8.8a) has imaginary coefficients was ignored. When the time-dependent Schrödinger equation including imaginary coefficients is rewritten as the steady-state solution, i.e., a time-independent equation, we obtain a differential equation with real coefficients, as given by Eq. (8.5). In this session, we attempt to apply circuit theory to the Schrödinger equation. Let us examine in detail the fact that the time-dependent equation has purely imaginary coefficients. The standard function frequently used in circuit theory is a positive real function and is a real function of s after a Laplace transform. In contrast, the Schrödinger equation is a complex function of s, as expressed by Eq. (8.21), where the function of s obtained by applying a Laplace transform has purely imaginary coefficients. Therefore, it is necessary to extend the positive real function. Note that in system theory including control engineering, even the function of s obtained after a Laplace transform has complex coefficients and that the function group called the class-C group may be treated.[10] Even in circuit theory, Belevitch [11] proposed a circuit element with impedance including a purely imaginary constant regardless of the frequency and called it an imaginary resistor. He [11] also added the imaginary resistor to the conventional circuit elements, demonstrating the possibility of extending circuit theory. Thus, an imaginary resistor is required to realize circuit equations including complex coefficients. An imaginary resistor is required for circuits expressed by an equation that has complex coefficients before a


Laplace transform. According to Belevitch [11], the imaginary resistor is represented by a small circle, as shown in Fig. 8.1(a), and satisfies the following relation. (8.22a) v(t )  jX  i(t ) Here, v(t) and i(t) are the voltage and current expressed as functions of t, respectively. Even when a Laplace transform is applied to Eq. (8.22a), the steady-state response is similarly obtained as (8.22b) V  jX  I Here, V and I are the voltage and current in the steady state, respectively.

jX

v(t)=jXi(t)

jB

i(t)=jBv(t)

(a) (b) Fig. 8.1 Imaginary resistors and its representations

Thus, the imaginary resistor has a certain imaginary impedance for waves of any frequency and has no resistance component. Hence, it is more understandable to consider that the imaginary resistor has a certain reactance. Anyway, the imaginary resistor has a certain imaginary impedance regardless of frequency and has no resistance component. Equation (8.22a) can be rewritten using an admittance component as 1 (8.23a) i(t )  v(t )  jB  v(t ) jX When the imaginary resistor is represented using the admittance component, it is illustrated as a double circle, as shown in Fig. 8.1(b). Similarly to Eq. (8.22b), the steadystate response is expressed by (8.23b) I  jB  V In circuit theory, it is important to realize the desired frequency characteristics by effectively using lossless circuit elements, which are related to the reactive power. Coils and capacitances are lossless elements but are expressed by a first-order equation of the angular frequency ω, and therefore, they have frequency characteristics. In contrast, imaginary resistors are independent of frequency and lossless elements with a certain reactance, and their effectiveness can be demonstrated by applying them to the Schrödinger equation.


5. Circuit Theory for the Schrödinger Equation Conventionally, devices based on the quantum effect do not satisfy causality and are not to transmit signals or energy. However, quantum-effect-based devices called resonant tunneling diodes use resonance. This indicates that the maximum active power supplied by a power source can be transmitted to a load. That is, signals can be transmitted with energy and the resonant tunneling diodes can satisfy causality. Considering these points, quantum-effect-based devices may be regarded as a type of circuit. In the following, we show a complex equivalent circuit obtained by assuming the Schrödinger equation to be an extension of the telegrapher’s equations, with the aim of applying the theory of distributed constant circuits to quantum mechanics. 5.1 Complex equivalent circuit The time-dependent one-dimensional Schrödinger equation treated in circuit theory is given by Eq. (8.8a) and corresponds to the telegrapher’s equation given by Eq. (8.19). In Ref. 6, an equivalent circuit is obtained by relating Eq. (8.8a) to Eq. (8.19) on the basis of this concept. In this session, we attempt to obtain an equivalent circuit by referring to Ref. 6. Concretely, a real resistor (the real part of the impedance) of a circuit is replaced with an imaginary resistor (the imaginary part of the impedance), i.e., a lossless circuit. By leaving the real coefficients unchanged, the capacitance (C) and coil (L) remain as lossless elements, and thus a lossless circuit can be obtained. That is, the wave function of the one-dimensional Schrödinger equation ψ(x) [Eq. (8.21)] is considered to be related to the voltage V(x) of the telegrapher’s equation [Eq. (8.20)] as (8.24)  ( x)  V ( x) Equation (8.20) includes a second-order term of s, whereas Eq. (8.21) does not. Therefore, either L or C should be zero. Here, we assume L = 0. Because of the correspondence of the coefficients of the first-order terms of s, RC should be given by the imaginary number j2m/ħ. Therefore, R or C should be imaginary. As mentioned previously, a circuit expressed by an equation having complex coefficients can be realized using an imaginary resistor. Hence, R should be imaginary. Here, RG should be the real number 2mU/ħ2. Therefore, G should also be selected for an imaginary resistor. Thus, the coefficients are assumed to be L = 0, C = 2 (8.25) R  jm /  , G   j 2U /  By substituting Eq. (8.25) into Eqs. (8.18a) and (8.18b), we obtain respectively m d (8.26a)  V ( x)  j I ( x) dx  d 2U (8.26b)  I ( x)  2sV x   j V ( x) dx 

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Therefore, a complex equivalent circuit including an imaginary resistor expressed by a one-dimensional Schrödinger equation is given by using a serial element with an impedance of jm/ħ and a parallel element (an imaginary resistor) with an admittance of –j2U/ħ, as shown in Fig. 8.2.

Z 0 sinh l  V l  V 0  cosh l       1   I 0   Z 0 sinh l cosh l   I l  

2Δx

Z0 

Fig. 8.2 Complex equivalent circuit for Schrödinger equation obtained by using imaginary resistors

Thus, the voltage V(x) has been expressed with respect to the wave function ψ(x) of the Schrödinger equation, as given by Eq. (8.24). The current I(x) can also be expressed using ψ(x) similarly to Eq. (8.26a).

I ( x)  j

 d  ( x) m dx

(8.27)

Therefore, the voltage and current can be determined from the wave function ψ(x). Thus, it has been demonstrated that the Schrödinger equation can be dealt with circuit theory. 5.2 Cascade matrix for the Schrödinger equation By applying circuit theory to the Schrödinger equation, the equations of electromagnetic waves consisting of voltage and current waves expressed by Eqs. (8.26a) and (8.26b), respectively, were obtained using the wave function ψ(x). Similarly to the solution of the telegrapher’s equations, we determine the propagation constant γ and characteristic impedance Z0, as well as a cascade matrix with a length of l. From Eqs. (8.26a) and (8.26b), γ is given by d2 (8.28a) V ( x )   2V ( x ) dx2 (8.28b)  2  2m(  U ) /  2 Therefore, V(x) and I(x) [Eq. (8.27)] are respectively expressed using the constants of integration, Ka and Kb, as (8.29a) V x   K a exp  x   K b exp x 

I x  

Ka K exp  x   b exp x  Z0 Z0

Z0 

m 2(  U )

Here,

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(8.30)

Here, γ and Z0 depend on the relation between U and the wave energy E as follows. (i) m(ħω − U) > 0 2 m (   U )   j  j (8.31a) 

( jm /  )x

( j 2U / )x

Assuming that the potential U and the effective mass m are constant in the range from x = 0 to x = l, the cascade matrix is obtained using Eq. (8.29) as

(8.29b)

(8.29c)

m  R0 2(  U )

(8.31b)

(ii) m(ħω − U) < 0 2m(U   )     Z0  j

(8.32a)

m  jX 0 2(U   )

(8.32b)

5.3 Equivalent circuit of quantum well The quantum phenomenon in the case of (i) corresponds to the quantum well. The propagation constant γ for this quantum well consists of jβ, as given by Eq. (8.31a), and the phase constant β is not su−1 given by Eq. (2.4) in Session 2 and cannot be directly subjected to a z-transform. However, by substituting Eq. (8.31) into Eqs. (8.29a) and (8.29b), the quantum well can be expressed using voltage and current waves as (8.33a) V x   K a exp jx   K b exp jx 

Ka K (8.33b) exp jx   b exp jx  R0 R0 The quantum well is not the transmission line explained in Session 2 but a transmission line that has different properties depending on frequency; for example, a ztransform cannot be applied to the quantum well, and the characteristic resistance R0 is a function of ω. I x  

E

l Fig.8.3

l

(a) (b) (a) Quantum well of width l ，and valent representation

(b) its equi-

The quantum well has been shown to have four types of wave, i.e., voltage forward, voltage backward, current forward, and current backward waves, and the properties of the quantum well should be determined with respect to


frequency (energy in quantum mechanics). The cascade matrix of a quantum well with a width of l is expressed using Eq. (8.33) as jR 0 sin  l   cos  l   j (8.34)   sin  cos  l l    R0



This is the cascade matrix for a reciprocal lossless circuit explained in Session 1 (Ref.[12]). The equivalent circuit of the quantum well with a width of l shown in Fig. 8.3(a) is expressed as a lossless line, as shown in Fig. 8.3(b). 5.4 Equivalent circuit of quantum barrier The quantum phenomenon in case (ii) corresponds to the quantum barrier. The propagation constant γ for this quantum barrier consists of the attenuation constant α, a real number, as given by Eq. (8.32a). That is, a wave traveling along the quantum barrier attenuates. Therefore, the cascade matrix is considered to represent a circuit having a lossy element. To confirm this idea, we determine the cascade matrix of a quantum barrier with a width of L obtained on the basis of Eqs. (8.32a) and (8.32b) as  coshL  j   X sinhL  0

jX 0 sinhL   coshL  

band can pass along the lossless transmission lines that satisfy the lossless telegrapher’s equations. However, a stop band is generated when the telegrapher’s equations are generalized. It is reasonable that there is a stop band, through which waves ooze out owing to the tunneling effect, because the Schrödinger equation written as a differential equation is considered to be an extension of Maxwell’s equations.

L

L E

jX0

(b)

(a)

jX0tanhαL/2

jX0tanhαL/2

(8.35)

Here, we consider whether the cascade matrix is a Junitary matrix[13], which satisfies the conditions for lossless circuits, as discussed in Session 1. Contrary to our expectation, the cascade matrix given by Eq. (8.35) satisfies the conditions of the cascade matrix for reciprocal lossless circuits[12]. However, the circuit with this cascade matrix has no rotational vector, unlike lossless transmission lines expressed by Eq. (8.34), and is regarded as a lossless circuit. Thus, we found that the circuit can be lossless even when the propagation constant consists only of the attenuation constant, a real number. What are the differences between lossless circuits and lossless transmission lines? The differences are considered as follows: for lossless transmission lines, (1) the rotational vector exp(−jβx) exists and (2) the characteristic impedance is a real number (i.e., characteristic resistance). In contrast, for lossless circuits, (1') no rotational vector of the propagation constant exists [i.e., exp(−αL), a real number], and (2') the characteristic impedance is the purely imaginary number jX0. That is, the reactive power with the characteristic impedance given by the purely imaginary number jX0 attenuates but the active power (energy) does not. In other words, the active power does not exist in the quantum barrier. These differences between the lossless transmission lines and lossless circuits have already been discussed in iterative parameter theory, which is covered by classical circuit theory, as described in Session 6. Because the elements of lumped constant circuits have a mass, there are two types of frequency band, i.e., the pass band, along which waves can easily pass, and the stop band, through which waves cannot easily pass. In contrast, ideally, waves of any frequency


jX 0 sinh L

(c)

Fig.8.4 (a) Quantum barrier of width L， (b) Transmission-line type equivalent representation， and (c) Lumped circuit type equivalent representation From this discussion, it is inappropriate to represent the equivalent circuit of a quantum barrier as a conventional lossless transmission line. Therefore, we use a lumped constant circuit as a different transmission line. As a result, the equivalent circuit of the quantum barrier with a width of L shown in Fig. 8.4(a) is represented as a transmission line type with characteristic impedance jX0, as shown in Fig. 8.4(b), unlike Fig. 8.3(b). Moreover, the equivalent circuit can also be represented by a T-shaped circuit with lumped constants as shown in Fig. 8.4(c). Note that the values in the figures are those of impedance.

6. Research Challenges in the Future de Broglie considered that matter particles have a wave nature, and Schrödinger derived an equation that satisfies the wave nature, which is called the Schrödinger equation. This equation is a basic equation in quantum mechanics.

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The behavior of matter particles is discussed using probability theory because it is closely related to thermodynamics; for example, the behavior of matter particles is related to black-body radiation described in thermodynamics, and a random walk is related to thermal conduction equations. Thus, quantum mechanics is discussed using thermodynamics and probability theory. If matter particles exhibit random behavior, such as a random walk, their behavior can be expressed by the equations of electromagnetic waves including a lossy term given by the attenuation constant, as discussed in Session 4, thus requiring thermodynamics and probability theory. However, if matter particles are crystalline with a periodic structure, they behave as a lossless circuit, as discussed in Session 5. It may be necessary to take into consideration the reactive power, which is not directly related to heat, or resonance might be discussed. In this session, we successfully derived a lossless transmission line by applying circuit theory to the Schrödinger equation and obtaining an equivalent circuit of the Schrödinger equation. This equation is valuable for explaining the behavior of matter particles, such as elementary particles, using circuit theory, without using probability theory, on the basis of the results obtained. In Ref. 14, an image parameter circuit was derived by rewriting the Schrödinger equation as a difference equation and applying image parameter theory to the difference equation. In this case, a stop band may be generated in addition to the domain of the tunneling effect, resulting in properties different from theoretical values. Therefore, a difference in values from the properties of the transmission line shown in this session may be obtained. In this session, we used the Schrödinger equation in the form of Eq. (8.8a). If the form of Eq. (8.8b) is used, the wave expressed by exp(−jωt), rather than exp(jωt), is transmitted and is considered to represent antiparticles, as shown in Ref. 8. In other words, when circuit theory is applied to the Schrödinger equation, the behavior of antiparticles can be determined by Eq. (8.8b). In later session, the behavior of a group of particles, rather than individual particles, will be discussed on the basis of circuit theory. [Acknowledgments] We are deeply grateful to Professor Hiroshi Tanimoto at Kitami Institute of Technology for his continued cooperation in our discussion via e-mail. We also thank Dr. Jie Ren for her continued support in the development of the figures and calculations.

References [1] N. Yoshida: Field of Light, Sea of Electrons (Way to Quantum Field Theory), Shinchosha Publishing Co., Ltd., 2008. (in Japanese)

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[2] H. Matano and M. Jimbo: Heat, Waves, and Differential Equations (Introduction to Modern Mathematics), Iwanami Shoten Publishers, 2004. (in Japanese) [3] M. Kitano: Maxwell Equations (To Better Understand Electromagnetics) SGC Library 39, Saiensu-sha Co., Ltd., 2005. (in Japanese) [4] M. Ikawa: Scattering Theory, Iwanami Shoten, 2008. (in Japanese) [5] R. Saito: Quantum Physics, Electronic Engineering Basics Series 7, Baifukan Co., Ltd., 1995. (in Japanese) [6] N. Nagai: Circuit Theory Based on New Concept and Its Application to Quantum Mechanics, Seibunsha, 2013. (in Japanese) [7] M. Nakayama: Separate Volume: Mathematical Science, Development of Modern Physics, Creation of Wave Mechanics, E. Schrödinger, Saiensu-sha Co., Ltd., pp. 49–57, 1997. (in Japanese) [8] R. P. Feynman: QED －The Strange Theory of Light and Matter－, Princeton University Press, 1985. [9] G. Kron : Electric circuit model of the Schrödinger equation， Phys. Rev. 67, p.39, 1945. [10] H. Kimura : Application of classical interpolation theory，(in Linear Circuits，Systems and Signal Processing (Edited by N. Nagai))，Marcel Dekker, 1990. [11] V. Belevitch : Classical Network Theory，Holden-Day, 1968. [12] H. Tanimoto and N. Nagai: An investigation into lossless circuits and reciprocity via chain matrix, Paper of Tech. Meeting on Electronic Circuits, ECT-14-044, IEE Japan, 2014. (in Japanese) [13] A. V. Efimov and V. P. Potapov: J-expanding matrix functions and their role in the analytical theory of electrical circuits, Usp. Mat. Nauk, (in Russian) pp. 65–130, 1973. Russian Mathematical Surveys, Vol. 28, No. 1, pp. 69-140, 1973. [14] P. P. Civalleri, M. Gilli and M. Bonnin: A cascaded two-port model for quantum particles propagation in crystals, 20th ECCTD (European Conference on Circuit Theory and Design), 2011.

Nobuo Nagai received his B.S. and D.Eng. degrees from Hokkaido University in 1961 and 1971, respectively. In 1961, he joined Hokkaido University as an Assistant and in 1972 he became an Associate Professor, and from 1980 to 1992 he was a Professor in the Research Institute of Applied Electricity. From 1992 to 2001, he was a Professor in the Research Institute for Electronic Science, Hokkaido UniUniversity. In 2001, he retired and became an Emeritus Professor. His research interests are circuit theory and digital signal processing. He is interested in the application of above theory to quantum theory. Dr. Nagai is a Life Fellow of the Institute of Electronics, Information and Communication Engineers, Japan.


Takashi Yahagi received his B.E., M.S. and Ph.D. degrees all from the Tokyo Institute of Technology in 1966, 1968 and 1971, respectively. In 1971, he joined Chiba University as a Lecturer and in 1974 he became an Associate Professor, and from 1984 to 2008 he was a Professor at the same university. Since 2008 he has been with the Signal Processing Research Laboratory. In 1997, he founded the Research Institute of Signal Processing, Japan (RISP). Since 1997 he has been President of RISP. From 1997 to 2013 he was Editor-in-Chief of the Journal of Signal Processing (JSP). Since 2013 he has been Honorary Editor-in-Chief of JSP. He was the author of “Theory of Digital Signal Processing (Vols. 1-3)”, (1985, 1985, 1986), Corona Pub.Co., Ltd. (Tokyo, Japan). He was also the editor and author of “Library of Digital Signal Processing (Vols. 1-10)”, (1996, 2001, 1996, 2000, 2005, 2008, 1997, 1999, 1998, 1997), Corona Pub.Co., Ltd. (Tokyo, Japan). He was the editor of “ My Research History (Vols. 1 and 2)” (2003, 2003), RISP. Dr. Yahagi is a Life Fellow of the Institute of Electronics, Information and Communication Engineers, Japan.


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Circuit Theory Based on New Concepts and Its ...

Circuit Theory Based on New Concepts and Its ...

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