On an Electron in a Magnetostatic Field

Eugen Grycko

On an Electron in a Magnetostatic Field

Lehrgebiet Stochastik Forschungsbericht

Fakultät für

Mathematik und

Informatik

ON AN ELECTRON IN A MAGNETOSTATIC FIELD

Eugen Grycko

Department of Mathematics and Computer Science Division of Applied Mathematics

University of Hagen Universit¨atsstr. 1 58084 Hagen / Germany

Absract: It is well known that a voltage is induced in a wire exposed to an alternating magnetic field. The present contribution is guided by the questi- on whether a similar phenomenon can be predicted and observed for a wire in a magnetostatic field. A classical and a semi-classical models describing a valence electron in a metal imply that the electron would not acquire ki- netic energy from the field. A computer-supported evaluation of a quantum model suggests, however, that the the second moment of the velocity ope- rator for the electron increases if a stronger magnetostatic field is applied, which motivates an empirical experiment whose outcome seems to confirm the computational quantum phenomenon.

Keywords: Gibbs state, tensor power, velocity operator

1. Introduction

In [4] and [5] the phenomenon of amplification of the noise level in metals that are exposed to an external electrostatic field, is reported and explained in accordance with the Quantum Theory. The required strength of the field entails the necessity of applying very high voltages.

Recently alloys were synthetized for producing strong permanent magnets and we pose the question whether the quasi-free valence electrons within a metal (wire) can be transferred by a strong magnetostatic field into a state

which entails a stronger noise voltage between the ends of the wire.

In Section 2 we show classically that the electron gas cannot take up kine- tic energy in a magnetostatic field, which motivates quantum considerations.

Similarly to [3] we introduce in Section 3 a Schr¨odinger operator in a 1- dimensional discrete position space (lattice). The operator admits (Section 4) the derivation of a quantum Gibbs-state of an electron which serves as a mathematical model for the thermodynamic equilibrium. As an application of the tensor calculus we introduce in Section 5 the Gibbs state of an electron in a 3-dimensional lattice and define a semiclassical Lorentz-Force-Operator which describes the action of the magnetic field on a quasi-free electron. It turns out that the semiclassical Lorentz-force does not change the kinetic energy of the electron. Similarly to [1] we introduce in Section 6 a quantum Hamiltonian for an electron in a magnetostatic field and define the corre- sponding Gibbs state accordingly to the quantum formalism. A numerical example with realistic parameter values provides evidence for the influence of the strength of the magnetostatic field on the second moment of the velo- city operator for the electron which motivates empirical investigations that we report in Section 7.

2. A Classical Electron in a Magnetostatic Field

We consider a homogeneous magnetostatic field in R³. This field is described by a vector B = (B₁, B₂, B₃). If an electron of charge −e moves with the momentary velocity v ∈R³, then the Lorentz force

(2.1) F =−e·(v ×B) =e·(B×v)

acts on it where × denotes the vector product in R³ (cf. [6], p. 260). The momentary magnetic power P providing the electron by the magnetic field is given by:

(2.2) P =hF, vi=e· hB ×v, vi= 0, where h·,·i denotes the scalar product.

(2.2) implies that the classical Lorentz force does not change the kinetic

energy of the electron in the time interval [t₁, t₂]:

∆E_kin =

t2

Z

t1

P dt= 0.

We remind the reader that the vector potential A:R³ →R³ of the homoge- neous magnetostatic field B is given by:

(2.3) A(x, y, z) = 1

2 ·B×(x, y, z) cf. [7], p. 162. In the special case B = (0,0, B3) we have

(2.4) A(x, y, z) = B₃

2 ·(−y, x,0).

3. A Schr¨odinger Operator in a Discrete Position Space We consider a finite lattice

L_a :={na|n= 1, . . . , N}

consisting of N points which models a discrete position space; parameter a > 0 is called lattice constant. La serves as the position space of a valence electron in a 1-dimensional electric conductor.

A quantum state of an electron within the conductor is described by a func- tion ϕ:L_a→C where

N

X

n=1

|ϕ(na)|² = 1.

holds. In this context we interprete|ϕ(na)|² as the probability of the spatial proximity of the electron to the lattice point na ∈ L_a. By a standard iden- tification the set of all quantum states of the electron is given by the unit sphere in C^N.

The momentum operator pb:C^N →C^N for an electron in La is defined by (3.1) (pϕ)(na) =b −i~· ϕ((n+ 1)a)−ϕ((n−1)a)

2a (n = 1, . . . , N),

where~= 1.05·10⁻³⁴Js denotes the Planck constant; in (3.1) the convention ϕ(na) = 0 f¨ur n <1 und f¨ur n > N

is applied and can be interpreted as a Dirichlet boundary condition (cf. [2], p. 28ff).pbis self-adjoint and can be interpreted as a discrete central difference approximation of the 1-dimensional momentum operator

−i~· d dx

for the position space R¹. Accordingly, the kinetic energy can be expressed by p/2mb where m denotes the mass of the electron.

3.1 Remark:

All entries in the matrix pbare purely imaginary; therefore all entries in the matrix representing the operator pb²/2m are real.

In our simplified discrete model the electrostatic potential being generated by the ions positioned at the lattice points is neglected.

The Schr¨odinger operator H : C^N → C^N describing the electron in the position space L_a is given by

(3.2) H = pb²

2m. 3.2 Remark:

The set of quantum states of an electron inL_a can be embedded into the set of positive operators with trace 1 by associating every unit vector ϕ ∈ C^N with the orthogonal projection onto the 1-dimensional subspace spanned by ϕ. Therefore we call every positive operator Z : C^N → C^N with trace 1 a generalized quantum state of the electron. Analogously to the classical case the velocity operator v :C^N →C^N of the electron is given by:

v := 1 m ·bp.

4. The Gibbs State of an Electron

We consider the Schr¨odinger operator from Section 3. Let T > 0 be the (absolute) temperature of an electric conductor. Operator GT : C^N → C^N which models the Gibbs state of an electron in L_a is defined by

(4.1) GT := 1

Z(T) ·exp

− 1 k_B·T ·H

, where

(4.2) Z(T) := trace

exp

− 1 k_B·T ·H

is the partition function and k_B the Boltzmann constant. Operator G_T is motivated by the entropy principle (cf. [5], p. 384) and describes the state of the electron in L_a at a thermal equilibrium with the interpretation of a diagonal entry G_T(n, n) as the probability of the spatial association of the electron with a lattice point na.

4.1 Remark:

All entries in the matrix GT are real numbers; cf. Remark 3.2

According to the quantum formalism the expectation of an observable (i.e. of a self-adjoint operator) S :C^N →C^N w.r.t. the state G_T is given by:

Eq(S) := trace(G_TS).

We know from Linear Algebra that the expectation is always a real number.

Since all entries of matrix v are purely imaginary, it follows that:

(4.3) Eq(v) = trace(G_Tv) = 0.

5. The Semiclassical 3-Dimensional Case

To make the model more realistic we consider the 3-dimensional lattice L³_a. We introduce the third tensor power

V₃ := (C^N)^⊗3 :==C^N ⊗C^N ⊗C^N

of the vector spaceC^N. From the Multilinear Algebra it is known that (C^N)^⊗3 is isomorphic to C^N

3 where the quantum states of an electron in L³_a are located. It is natural to introduce the Gibbs state GT :V3 →V3 according to

G_T :=G_T ⊗G_T ⊗G_T. G_T is positive and self-adjoint and the equalities

trace(GT) = trace(GT)3

= 1

hold. We interpret G_T as the thermal equilibrium state of an electron in the lattice L³_a.

The velocity operator v of the electron has now three components:

(5.1) v_x :=v ⊗I_N ⊗I_N, v_y :=I_N ⊗v⊗I_N, v_z :=I_N ⊗I_N ⊗v, where I_N denotes theN ×N-identity matrix. We have (cf. (4.3)):

(5.2) Eq(v_x) = trace(G_Tv_x) = trace(G_Tv)· trace(G_TI_N)2

= 0, and analogously

(5.3) Eq(v_y) =Eq(v_z) = 0.

We have, moreover:

v_xv_y =v⊗v⊗I_N =v_yv_x,

which means that the operatorsv_x, v_y andv_zcommute. It follows that (cf. (4.3)):

(5.4) Eq(v_xv_y) = trace(G_Tv)²

·trace(G_TI_N) = 0 and

(5.5) Eq(v_xv_z) =Eq(v_yv_z) = 0.

We assume now that the lattice L³_a is exposed to an external homogeneous magnetostatic fieldB = (B₁, B₂, B₃). Analogously to the classical case (cf. (2.1)) we introduce the Lorentz force operator F = (F1, F2, F3) according to

(5.6) F :=e·(B×_q(v_x, v_y, v_z)) :=e·(B₂v_z−B₃v_y, B₃v_x−B₁v_z, B₁v_y−B₂v_x).

Fj : V3 → V3 is obviously a self-adjoint operator (cf. (5.1) and (5.2)) such that

Eq(F_j) = 0

forj = 1,2,3. The tupleF = (F₁, F₂, F₃) is analogous to the classical Lorentz force acting on the electron L³_a. Similarly to the classical formula (2.2) the power operator P :V₃ →V₃,

P :=hF,(v_x, v_y, v_z)i_q :=F₁v_x+F₂v_y +F₃v_z

can be introduced. P describes the power which is transferred from the ma- gnetostatic field to the electron. (5.4),(5.5) und (5.6) imply that

P = 0 holds which is reminiscent to the classical case.

According to the considered semiclassical model, the magnetostatic field does not change the kinetic energy of the electron.

6. The 3-Dimensional Quantum Case

We consider the momentum operator pbfor an electron inL_a, cf. (3.1). It is natural to introduce the tuple p = (p_x, p_y, p_z) of the momentum operators p_x, p_y, p_z :V₃ →V₃ according to

(6.1) p_x :=pb⊗I_N ⊗I_N, p_y :=I_N ⊗pb⊗I_N, p_z :=I_N ⊗I_N ⊗p.b By (6.1) one can recognize that the components of p commute:

p_xp_y =p_yp_x, p_xp_z =p_zp_x, p_yp_z =p_zp_y.

We again specify the homogeneous magnetostatic field B := (0,0, B₃). To specify the tuple s = (s_x, s_y, s_z) of position operators s_x, s_y, s_z :V₃ →V₃ for the electron in L³_a we define the diagonal matrix

s :=













1 0 · · · 0 0 2 0 · · 0

· 0 · · · ·

· · · 0 0 · · · 0 N











 .

and the tuple s= (s_x, s_y, s_z) of position operators s_x, s_y, s_z :V₃ →V₃ by (6.2) s_x:=a·s⊗I_N⊗I_N, s_y :=a·I_N⊗s⊗I_N, s_z :=a·I_N⊗I_N⊗s.

Tuple s describes the position of an electron inL³_a. Its componentss_x, s_y, s_z are self-adjoint operators in V₃. The natural quantization Ab = (A_x, A_y, A_z) of the vector potential A from (2.4) is given by:

(6.3) A_x :=−B₃

2 ·s_y A_y := B₃

2 ·s_x A_z := 0.

Operators Ax, Ay und Az are obviously self-adjoint.

Analogously to [1], p. 210, the Hamiltonian H : V₃ → V₃ of an electron in the lattice L³_a which is exposed to the magnetostatic field B, is given by:

H:= 1

2m · (p_x+e·A_x)²+ (p_y +e·A_y)²+p²_z .

Operator H is again self-adjoint and positive. According to the quantum formalism the Gibbs state G_T :V₃ →V₃ of the electron can be defined by:

(6.4) G_T := 1

Z(T)·exp

− 1 k_BT ·H

where the partition function Z(T) := trace

exp

− 1 k_BT ·H

,

corresponds to the temperature T > 0.

The second momentµ₂ of the velocity operatorv_x for the stateG_T is defined by

µ₂ := trace G_Tv²_x

≥0.

6.1 Example:

Put N := 15, a = 10⁻¹⁰ m and T := 300 K. In the diagram of Fig. 1 the horizontal axis corresponds to the strength −100 T ≤ B₃ ≤ 100 T of the magnetostatic field; the physical unit is T (Tesla). The vertical axis corre- sponds to the square root √

µ2 of µ2 with the physical unit m/s. The Graph of the numerically determined function confirms that the second moment of

the velocity operator depends on the strength B₃ of the external magneto- static field. Although the influence of B₃ on µ₂ is minimal, Fig. 1 motivates an empirical experiment.

Fig. 1: The second moment of the velocity operator

7. Empirical Measurements

We take a ring-shaped permanent magnet and wind it with an isolated wire of diameter 0.3 mm. The ends of the wire are connected with the imput of a Delon rectifier (cf. Fig. 2). The rectifier consists essentially of two diodes

with the limiting frequency of 3 GHz.

Fig. 2: Delon rectifier

The output of the rectifier is connected with a voltmeter VC 270 produced by Voltcraft. The impedance of the voltmeter for the range [-400 mV, 400 mV] is given by R=4 GΩ.

Fig. 3: The measurement arrangement

We start the observation of the display while the low impedance button is being pushed; at this stage the measured voltage 0.4 mV is shown. After releasing the button the voltage on the display increases and exceeds the voltmeter range of 400 mV after about 15 s. Although the observed values are exemplary and difficult to keep a record of, we conjecture that they are causally connected with the quantum phenomenon reported in Example 6.1 in particular because the course of the observations depends on the position of the observer during the experiment. Note that there are also alternative explanations for the observed phenomenon in discussion, for example, electro- magnetic induction by electro-smog. For making a final decision about the dominant cause of the measurements further experimental research, prefera- bly in a professional laboratory, is necessary.

Acknowledgments

The author would like to thank Silke Hartlieb, Matthias Miehl, Verena Sam- met, Joachim Warzecha and Volker Winkler for technical support and Martin Roos for valuable comments on the first draft of the article. The encourage- ment and advice of Wilfried Arends, Stefan Helfert, Joachim Kerner, G¨unter Kl¨utzke, Helmut Meister, Tobias M¨uhlenbruch, Thomas M¨uller, Frank Re- cker, Andreas Wiegner, and of Avanti Publishers are greatly appreciated.

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