Examples of solved problems in physics on the topic "the movement of a charge in a magnetic field in a spiral."

Some electrovacuum devices use the movement of electrons in a magnetic field.

Consider the case when an electron flies into a uniform magnetic field with an initial velocity v 0, directed perpendicular to the magnetic field lines. In this case, the moving electron is affected by the so-called Lorentz force F, which is perpendicular to the vector h0 and the intensity vector magnetic field N. The magnitude of the force F is defined by the expression: F= ev0H.

At v0 = 0, the force P is equal to zero, i.e., the magnetic field does not act on a stationary electron.

Strength F bends the electron trajectory into a circular arc. Since the force F acts at right angles to the speed h0, it does no work. The energy of an electron and its speed do not change in magnitude. There is only a change in the direction of speed. It is known that the motion of a body in a circle (rotation) at a constant speed is obtained due to the action of a centripetal force directed towards the center, which is precisely the force F.

The direction of rotation of an electron in a magnetic field in accordance with the left-hand rule is conveniently determined by the following rules. Looking in the direction of the magnetic lines of force, then the electron moves clockwise. In other words, the rotation of the electron coincides with rotational movement screw, which is screwed in the direction of the magnetic field lines.

Define the radius r the circle described by the electron. To do this, we use the expression for the centripetal force, known from mechanics: F = mv20/r. Equate it to the value of the force F=ev0H: mv20/r = ev0H. Now from this equation you can find the radius: r= mv0/(eH).

The greater the electron velocity v0, the stronger it tends to move rectilinearly by inertia, and the radius of curvature of the trajectory will be larger. On the other hand, with an increase H the force F increases, the curvature of the trajectory increases and the radius of the circle decreases.

The derived formula is valid for the motion of particles with any mass and charge in a magnetic field.

Consider the dependence r from m and e. A charged particle with a larger mass m more strongly tends to fly by inertia in a straight line and the curvature of the trajectory will decrease, i.e., will become larger. And the more charge e, topics more power F and the more the trajectory is curved, i.e., its radius becomes smaller.

Having gone beyond the magnetic field, the electron flies further by inertia in a straight line. If the radius of the trajectory is small, then the electron can describe closed circles in a magnetic field.

Thus, the magnetic field only changes the direction of the electron velocity, but not its magnitude, i.e., there is no energy interaction between the electron and the magnetic field. Compared to an electric field, the effect of a magnetic field on electrons is more limited. That is why the magnetic field is used to influence electrons much less frequently than electric field.

In all electronic and ionic devices, electron flows in a vacuum or gas under one pressure or another are subjected to an electric field. The interaction of moving electrons with an electric field is the main process in electronic and ion devices. Consider the motion of an electron in an electric field.

Fig.1 - Electron motion in accelerating (a), decelerating (b) and transverse (c) electric fields

Figure 1a shows the electric field in vacuum between two flat electrodes. They can be the cathode and anode of a diode, or any two adjacent electrodes of a multi-electrode device. Let us imagine that an electron is emitted from an electrode having a lower potential, for example, from the electrode, with a certain initial velocity Vo. The field acts on an electron with a force F and accelerates its movement to an electrode with a higher positive potential, for example, to the anode. In other words, the electron is attracted to the electrode with a higher positive potential. Therefore, the field in this case is called accelerating. Moving at an accelerated rate, the electron acquires top speed at the end of its path, i.e., when it hits the electrode to which it flies. At the moment of impact, the kinetic energy of the electron will also be the largest. Thus, when an electron moves in an accelerating field, the kinetic energy of the electron increases due to the fact that the field does work to move the electron. The electron always takes away energy from the accelerating field.

The speed acquired by an electron when moving in an accelerating field depends solely on the potential difference U passed through and is determined by the formula

It is convenient to express the electron velocities conditionally in volts. For example, the speed of an electron is 10 volts, which means the speed that the electron acquires as a result of moving in an accelerating field with a potential difference of 10 volts. From the above formula, it is easy to find that at U - 100 V the speed is V ~ 6,000 km/sec. At such high speeds, the time of flight of an electron in the space between the electrodes turns out to be very small, on the order of 10 V minus 8 - 10 V minus 10 sec.

Consider now the motion of an electron, for which starting speed Vo is directed against the force F acting on the electron from the field (Fig. 1b). In this case, the electron flies out with a certain initial speed from the electrode with a higher positive potential. Since the force F is directed towards the velocity Vo, then the deceleration of the electron is obtained and the field is called retarding. Consequently, the same field for some electrons is accelerating, and for others it is decelerating, depending on the direction of the initial electron velocity.

The kinetic energy of electrons moving in a decelerating field decreases, since the work is done not by the field forces, but by the electron itself, which overcomes the resistance of the field forces. The energy lost by the electron goes to the field. Thus, in a decelerating field, an electron always gives energy to the field.

If the initial speed of the electron is expressed in volts (Uo), then the decrease in speed is equal to the potential difference U that the electron passes in the decelerating field. When the initial speed of the electron is greater than the potential difference between the electrodes (Uo> U), then the electron will pass the entire distance between the electrodes and hit the electrode with a lower potential. If Uo< U, то, пройдя разность потенциалов, равную Uq, электрон полностью потеряет свою энергию, скорость его станет равна нулю, он на-момент остановится и начнет ускоренно двигаться обратно (рис.1 б).

If an electron flies in with a certain initial speed Vo at a right angle to the direction of the field lines of force (Fig. 1c), then the field acts on the electron with a force F directed towards a higher positive potential. Therefore, the electron simultaneously performs two mutually perpendicular motions: a uniform motion by inertia with a speed vQ and a uniformly accelerated motion in the direction of the force F. As is known from mechanics, the resulting motion of the electron should occur along a parabola, and the electron deviates towards the more positive electrode. When the electron leaves the field (Fig. 1 c), then it will move further, by inertia, rectilinearly evenly.

From the considered laws of electron motion, it can be seen that the electric field always acts on kinetic energy and the speed of the electron, changing them in one direction or another. Thus, there is always an energy interaction between an electron and an electric field, i.e., an energy exchange. In addition, if the initial velocity of the electron is not directed along the lines of force, but at some angle to them, then the electric field bends the trajectory of the electron, turning it from a straight line into a parabola.
Consider now the motion of an electron in a magnetic field.

A moving electron is an elementary electric current and experiences the same action from the magnetic field as a current-carrying conductor. It is known from electrical engineering that straight conductor with a current in a magnetic field, a mechanical force acts at right angles to the magnetic lines of force and to the conductor. Its direction is reversed if you change the direction of the current or the direction of the magnetic field. This force is proportional to the field strength, the magnitude of the current and the length of the conductor, and also depends on the angle between the conductor and the direction of the field.

It will be greatest if the conductor is perpendicular to the lines of force; if the conductor is located along the lines of the field, then the force is zero.

Fig.2 - Movement of an electron in a transverse magnetic field.

If an electron in a magnetic field is stationary or moves along the lines of force, then the magnetic field does not act on it at all. Figure 2 shows what happens to an electron that flies into a uniform magnetic field created between the poles of a magnet with an initial velocity Vo perpendicular to the direction of the field. In the absence of a field, the electron would move by inertia in a straight line and uniformly (dashed line); in the presence of a field, a force F will act on it, directed at right angles to the magnetic field and to the velocity v0. Under the action of this force, the electron bends its path and moves along an arc of a circle. His line speed Vo and energy remain unchanged, since the force F always acts perpendicular to the speed Vo. Thus, the magnetic field, unlike the electric field, does not change the energy of the electron, but only twists it.

Some electrovacuum devices use the movement of electrons in a magnetic field.

Let us consider the case when an electron flies into a uniform magnetic field with an initial velocity v0 directed perpendicular to the magnetic field lines. In this case, the moving electron is affected by the so-called Lorentz force F, which is perpendicular to the vector h0 and the vector of the magnetic field H. The magnitude of the force F is determined by the expression: F = ev0H.

At v0 = 0, the force P is equal to zero, i.e., the magnetic field does not act on a stationary electron.

The force F bends the electron trajectory into a circular arc. Since the force F acts at right angles to the speed h0, it does no work. The energy of an electron and its speed do not change in magnitude. There is only a change in the direction of speed. It is known that the motion of a body in a circle (rotation) at a constant speed is obtained due to the action of a centripetal force directed towards the center, which is precisely the force F.

The direction of rotation of an electron in a magnetic field in accordance with the left-hand rule is conveniently determined by the following rules. Looking in the direction of the magnetic field lines, the electron moves clockwise. In other words, the rotation of the electron coincides with the rotational movement of the screw, which is screwed in in the direction of the magnetic field lines.

Let us determine the radius r of the circle described by the electron. To do this, we use the expression for the centripetal force known from mechanics: F = mv20/r. Let us equate it to the value of the force F = ev0H: mv20/r = ev0H. Now from this equation you can find the radius: r= mv0/(eH).

The greater the electron velocity v0, the stronger it tends to move rectilinearly by inertia, and the radius of curvature of the trajectory will be larger. On the other hand, with increasing H, the force F increases, the curvature of the trajectory increases, and the radius of the circle decreases.

The derived formula is valid for the motion of particles with any mass and charge in a magnetic field.

Consider the dependence of r on m and e. A charged particle with a larger mass m tends to fly rectilinearly by inertia and the curvature of the trajectory will decrease, i.e., will become larger. And the greater the charge e, the greater the force F and the more the trajectory is curved, i.e., its radius becomes smaller.

Having gone beyond the magnetic field, the electron flies further by inertia in a straight line. If the radius of the trajectory is small, then the electron can describe closed circles in a magnetic field.

Thus, the magnetic field only changes the direction of the electron velocity, but not its magnitude, i.e., there is no energy interaction between the electron and the magnetic field. Compared to an electric field, the effect of a magnetic field on electrons is more limited. That is why a magnetic field is used to influence electrons much less frequently than an electric field.

Objective.

Devices and accessories: e

Introduction

e, speed of light With, Planck's constant h cl∙kg -1 .

A magnetic field. AT B B q, moving at a speed V

F l = q∙[ V∙B] or F l = |q|VB∙sin a(1)

where α – V AT .

». B

q> I

Fig.1

q>q< 0) current directions I and speed V V B r determined from the condition

, (2)

where α is the angle between the vectors V and B .

When α = 90 0 , sinα

Δ BUT = F l. Δ r

or Δ BUT = F l. Δ r cosβ, (4)

where β F Δ r .

F l Δ r , β = 90 0 and cosβ

r

V directed at an angle α to power lines AT V // = V∙cosα and uniform

V ┴ = V∙sinα.

V //

h = VTcos, (7)

Substituting this expression for T in (7), we get

. (8)

B .

Electric field. For a point charge q, E , the force acts

F= q E , (9)

Force direction F E E .

According to Newton's second law F= m a

q E = (10)

X with speed V

Charge movement along the axis X x= x 0 + Vt(x 0 – start coordinate, t– time), V= const,x 0 = 0. ℓ equals .

Movement along the axis Y , E y = V y = V 0y + at. At , where FROM− t= 0) V 0 y = 0 we get C = 0. .

Y according to the formula .

u,

AT E , then the resulting force F

F Em = q E + q[V∙B ]. (11)

UVV << скорости света c ) having the form

where e m

From (12) the electron speed

. (13)

u, B r

Experimental setup

3 - power supply IP1 Helmholtz coils; 4 - Helmholtz coils; 5 - power supply IP2 cathode ray tube.

Functional parts of the experimental setup and their connection diagrams

Helmholtz coils(Helmholtz rings) are two coaxial ring conductors of the same radius with n the number of turns, located in parallel planes coaxially, so that the distance between them is equal to the radius of the rings (Fig. 8).

On fig. 9 shows a diagram of connecting Helmholtz coils to a power source IP1 .

When current is passed through the coils, a magnetic field arises in the space between them, which is characterized by a high degree of uniformity. It is the result of a superposition of magnetic fields induced by each current-carrying turn of a ring conductor and, in general, a system of two ring conductors (Fig. 8).

The magnetic field induction at the center of a ring current-carrying conductor containing one turn is expressed by the formula

where R is the radius of curvature of the conductor, I- current strength in it, µ- magnetic permeability, µ 0 - magnetic constant (µ 0 = 4π·10 -7 Gn/m).

The magnitude of the magnetic field induction on the axis of the coils is proportional to the current I flowing in the winding of each of the ring conductors and the number of turns in them n. Theoretical calculation of the magnetic induction of the field of Helmholtz coils using the Biot-Savart-Laplace law and the principle of superposition on the axis X in the center of the system leads to an adapted formula for calculating AT used in this work

. (15)

where R- radius of the ring conductor, µ 0 = 4π·10 -7 Gn/m (magnetic constant).

Figure 10 shows the distribution of the magnetic field induction in space between the Helmholtz coils along the axis x, coinciding with the axis of symmetry of the coils. The dotted line shows the distribution of magnetic fields created by each of the ring conductors.

The inhomogeneity of the generated field with the appropriate adjustment of the coils may not exceed 5%.

Cathode-ray tube (CRT ) used in the experimental setup is shown in Fig.11. The photo (top view) also illustrates its location in the space between the Helmholtz coils in the region of a uniform magnetic field. CRT is a beam tetrode in a spherical glass bulb with a vacuum. In the flask there is an electron gun - an indirectly heated cathode, fixed on a metal traverse with jumpers. To visualize the electron beam, a glass flask is filled with hydrogen at low pressure.

Fig.11. Cathode-ray tube with Helmholtz coils (top view):

1 - electron gun; 2 – traverse with jumpers used as a scale for estimating the radius of the electron trajectory;

3 - Helmholtz coils.

The electrons emitted by the cathode as a result of thermionic emission are focused by the electrodes of the electron beam gun in the form of a beam and move along a rectilinear trajectory vertically upwards. When applying voltage to the Helmholtz coils from a power source IP1 in the field of accommodation CRT, a uniform magnetic field is created. The trajectory of the electron beam changes from a straight line to an annular annular.

The effect is observed visually by a weak bluish glow inside the glass bulb, corresponding to the trajectory of the electron beam. The diameter of the visualized electron trajectory is estimated using a crossbar located in the flask with several jumpers coated with a phosphor (Fig. 12).

Figure 13 shows the connection to the power supply IP2

cathode ray tube indicating the ranges of source parameters.

Rice. 14. Power supply for Helmholtz coils ( IP1 ) (front panel photo).

Rice. 15. Power supply of the cathode ray tube ( IP2 ) (front panel photo).

Work order

NOTE 1.

All devices and functional elements of the installation are connected by connecting cords.

DO NOT TOUCH!

ATTENTION.

When performing work, it is necessary to strictly observe the safety regulations established at the workplace and in the laboratory.

ATTENTION.

ATTENTION.

Measurements must be carried out in a darkened room in order to observe the trajectory of the electron beam.

NOTE 4

On the experimental setup, it is also possible to measure the radius of the electron beam trajectory using the third jumper from the left of the scale located in a glass flask for registration. CRT. It corresponds to the radius of the electron beam r3= 0.03 m (Fig. 12).

14. These measurements are carried out at the request of the teacher. Repeat steps 11 and 12 several times, observing the intersection of the electron beam with the third jumper.

15. Measurement data of the corresponding pairs of characteristics: accelerating voltage U and current in the coils I and for each experiment r3= 0.03 m enter in the table. 2.

16. Switch off the measuring system.

Shutdown order:

a) use the adjustment knobs to reduce the current in the Helmholtz coils to zero (turn to the leftmost position). On the IP1 set the left and right knobs to 0.

b) use the adjustment knobs to reduce the accelerating voltage of the cathode ray tube to zero (turn to the extreme left position by IP2 2nd and 3rd handles).

c) turn off the power sources IP1 and IP2 (toggle switches on the rear panel).

Table 1

table 2

Bibliography

1. Yavorsky B.M., Detlaf A.A. Physics course. - M .: Publishing house "Academy", 2005 onwards. – 720 p.

2. Trofimova T.I. Physics course. - M .: Higher School, 2004 onwards. – 544 p.

3. Saveliev I.V. Course of general physics in 3 vols. – M.: Astrel AST, 2007 and beyond.

Zakharova T.V. (General ed.) Physics. Collection of tasks in test form, part 2. – M.: MIIT, 2010 – 192 p.

MOTION OF ELECTRONS IN A MAGNETIC FIELD

Objective. Determination of the specific charge of an electron from a known trajectory of an electron beam in electric and alternating magnetic fields.

Devices and accessories: e experimental plant of the PHYWE brand from HYWE Systems GmbH & Co. (Germany) consisting of: cathode-ray tube; Helmholtz coils (1 pair); universal power supply (2 pcs.); digital multimeter (2 pcs.); multi-colored connecting cords.

Introduction

The specific charge of an elementary particle is the ratio of the charge of a particle to its mass. This characteristic is widely used to identify particles, as it allows one to distinguish different particles having the same charges from each other (for example, electrons from negatively charged muons, pions, etc.).

The specific charge of an electron refers to the fundamental physical constants, such as the electron charge e, speed of light With, Planck's constant h and others. Its theoretical value is = (1.75896 ± 0.00002)∙10 11 cl∙kg -1 .

Numerous experimental methods for determining the specific charge of particles are based on studies of the features of their motion in a magnetic field. Additional possibilities are the use of the configuration of the magnetic and electric fields and the variation of their parameters. In this work, the specific charge of an electron is determined on a German-made PHYWE experimental setup. In it, to study the trajectories of electrons in a magnetic field, a method based on a combination of the possibilities of varying the parameters of homogeneous magnetic and electric fields in their mutually perpendicular configuration is implemented. This manual was developed using the documentation supplied with the unit.

A magnetic field. Experiments show that a magnetic field acts on charged particles moving in it. The force characteristic that determines such an action is magnetic induction - a vector quantity AT .The magnetic field is depicted using magnetic induction lines of force, the tangents to which at each point coincide with the direction of the vector B . For a uniform magnetic field, the vector B constant in magnitude and direction at any point in the field. Force acting on a charge q, moving at a speed V in a magnetic field, was determined by the German physicist G. Lorentz (Lorentz force). It is expressed by the formula

F l = q∙[ V∙B] or F l = |q|VB∙sin a(1)

where α – angle formed by the velocity vector V moving particle and magnetic field induction vector AT .

A stationary electric charge is not affected by a magnetic field. This is its essential difference from the electric field.

The direction of the Lorentz force is determined using the "left hand" rule ». If the palm of the left hand is positioned so that it includes the vector B , and point four outstretched fingers along

direction of movement of positive charges ( q>0), coinciding with the direction of the current I(), then bent thumb

Fig.1

will show the direction of the force acting on a positive charge ( q>0) (Fig. 1). In the case of negative charges ( q< 0) current directions I and speed V movements are opposite. The direction of the Lorentz force is determined by the direction of the current. Thus, the Lorentz force is perpendicular to the velocity vector, so the modulus of the velocity will not change under the influence of this force. But at a constant speed, as follows from formula (1), the value of the Lorentz force also remains constant. It is known from mechanics that a constant force perpendicular to the velocity causes movement in a circle, that is, it is centripetal. In the absence of other forces, according to Newton's second law, it informs the charge of a centripetal or normal acceleration. The trajectory of the charge in a uniform magnetic field at V B is a circle (Fig. 2), the radius of which r determined from the condition

, (2)

where α is the angle between the vectors V and B .

When α = 90 0 , sinα= 1 from formula (2) the radius of the circular trajectory of the charge is determined by the formula

The work done on a moving charge in a magnetic field by the constant Lorentz force is

Δ BUT = F l. Δ r

or Δ BUT = F l. Δ r cosβ, (4)

where β is the angle between the direction of the force vectors F l. and the direction of the displacement vector Δ r .

Since the condition is always satisfied F l Δ r , β = 90 0 and cosβ= 0, then the work done by the Lorentz force, as follows from (4), is always zero. Consequently, the absolute value of the charge velocity and its kinetic energy when moving in a magnetic field remain constant.

The period of rotation (the time of one complete revolution), is equal to

Substituting in (5) instead of the radius r its expression from (3), we obtain that the circular motion of charged particles in a magnetic field has important feature: the period of revolution does not depend on the energy of the particle, it depends only on the induction of the magnetic field and the reciprocal of the specific charge:

If the magnetic field is uniform, but the initial velocity of the charged particle V directed at an angle α to power lines AT , then the motion can be represented as a superposition of two motions: a uniform rectilinear motion in a direction parallel to the magnetic field with a speed V // = V∙cosα and uniform

rotation under the action of the Lorentz force in a plane perpendicular to the magnetic field with a speed V ┴ = V∙sinα.

As a result, the trajectory of the particle will be a helix (Fig. 3).

The pitch of the helix is ​​equal to the distance traveled by the charge along the field with the velocity V // for a time equal to the period of rotation

h = VTcos, (7)

Substituting this expression for T in (7), we get

. (8)

The spiral axis is parallel to the magnetic field lines B .

Electric field. For a point charge q, placed in an electric field characterized by a strength vector E , the force acts

F= q E , (9)

Force direction F coincides with the direction of the vector E if the charge is positive and opposite E in case of negative charge . In a uniform electric field, the intensity vector at any point in the field is constant in magnitude and direction. If the movement occurs only along the lines of force of a uniform electric field, it is uniformly accelerated rectilinear.

According to Newton's second law F= m a the equation of motion of a charge in an electric field is expressed by the formula

q E = (10)

Let's assume that the dot negative charge, moving initially along the axis X with speed V , falls into a uniform electric field between the plates of a flat capacitor, as shown in Fig. four.

Charge movement along the axis X is uniform, its kinematic equation x= x 0 + Vt(x 0 – start coordinate, t– time), V= const,x 0 = 0. Time of flight of the capacitor charge with the length of the plates ℓ equals .

Movement along the axis Y determined by the electric field inside the capacitor. If the gap between the plates is small compared to their length , edge effects can be neglected and the electric field in the space between the plates can be considered uniform ( E y = const). The movement of the charge will be uniformly accelerated V y = V 0y + at. At acceleration is determined with formula (10). After integrating (10), we obtain , where FROM− constant of integration. Under the initial condition ( t= 0) V 0 y = 0 we get C = 0. .

The trajectory and nature of the motion of a charged particle in a uniform electric field of a flat capacitor are similar to the similar characteristics of motion in the gravitational field of a horizontally thrown body. Deviation of a charged particle along the axis Y equals . Given the nature operating force it depends on according to the formula .

When moving a charge in an electric field between points having a potential difference u, work is done by the electric field, as a result of which the charge acquires kinetic energy. In accordance with the law of conservation of energy

If on a moving electric charge, in addition to a magnetic field with induction AT there is also an electric field with strength E , then the resulting force F , which determines its motion, is equal to the vector sum of the force acting from the electric field and the Lorentz force

F Em = q E + q[V∙B ]. (11)

This expression is called the Lorentz formula.

In this laboratory work the motion of electrons in the magnetic and electric fields. All relations considered above for an arbitrary charge are also valid for an electron.

We assume that the initial velocity of the electron is zero. Getting into an electric field, the charge is accelerated in it, and, having passed the potential difference U, acquires some speed V. It can be determined from the law of conservation of energy. In the case of nonrelativistic speeds ( V << скорости света c ) having the form

where e= –1.6∙10 -19 C – electron charge, m e \u003d 9.1 ∙ 10 -31 kg - its mass.

From (12) the electron speed

Substituting it into (3), we obtain a formula for finding the radius of the circle along which an electron moves in a magnetic field:

. (13)

Thus, knowing the potential difference u, accelerating electrons as they move in an electric field to nonrelativistic velocities, the induction of a uniform magnetic field B, in which these electrons move, describing a circular trajectory, and, experimentally determining the radius of the specified circular trajectory r, we can calculate the specific charge of an electron using the formula

Experimental setup

A photo of the measuring stand is shown in Fig.5.

On fig. Figure 6 shows a photo of the PHYWE experimental setup.

On fig. 7 shows the main components of the experimental setup with the designations of the functional parts.

Fig.7. Experimental setup:

1 - cathode ray tube; 2, 6 - digital multimeters;

3 - power supply IP1 Helmholtz coils; 4 - Helmholtz coils; 5 - source p

Consider the Pauli operator for the case of a constant magnetic field. For clarity, we will carry out calculations in rectangular Cartesian coordinates. If the magnetic field is weak enough, then the terms in the operator containing the square

vector potential, we can neglect, in the linear terms, we can replace the expressions

which give

where are the components of the orbital angular momentum of the electron's momentum (see (1) § 1).

Using (2), we obtain an approximate expression for

Adding to according to (19) § 5, terms depending on the spin, we will have

This expression includes the scalar product of the magnetic field and the vector of the magnetic moment of the electron

This vector consists of two parts: orbital and spin. The orbital part is proportional to the orbital angular momentum of the electron

and the spin part is proportional to the intrinsic (spin) moment

In this case, the proportionality factor between the magnetic and mechanical moment for the spin part is twice that for the orbital part. This fact is sometimes called the magnetic spin anomaly.

In a problem with spherical symmetry, the magnetic field-dependent correction part of the energy operator (4) commutes

with the main part (operator (7) § 5). Therefore, the correction to the energy level for the magnetic field consists simply in adding to it the eigenvalue of the correction term in (4). If the axis is directed along the magnetic field, then the addition will be equal to

where is the eigenvalue of the operator

However, the spin-based correction to that consists of replacing by does not introduce new levels, since there is an integer. Only corrections for the theory of relativity play an essential role here.

In the Pauli R energy operator [formula (4)], these corrections are not taken into account. Taking them into account leads to the fact that in a field with spherical symmetry, the equation for radial functions will contain not only the quantum number I of the Schrödinger theory, but also the quantum number entering the equation for spherical functions with spin

[formula (22) § 1] and related to the relation

[formula (20) § 1].

We know that for will have a single value, but for two values ​​are possible, namely, . As a result, the Schrödinger level corresponding to a given value of I (and a certain value of the principal quantum number) decays at into two close levels, which form a doublet. This doublet is usually called a relativistic doublet.

In the equation for radial functions, the order of magnitude of the relativistic correction term with respect to the principal (potential energy) term can be characterized by the value where

is a dimensionless constant, which is commonly called the fine structure constant. The influence of the magnetic field on the energy levels is characterized by the quantity (8).

The splitting of energy levels in a magnetic field is called the Zeeman phenomenon.

A complete theory of the Zeeman phenomenon for the hydrogen atom will be presented at the end of this book on the basis of Dirac's theory. Here we would only like to emphasize the fact that the behavior

electron in a magnetic field convincingly proves that it has a new degree of freedom associated with the spin.

The existence of this new degree of freedom of the electron plays a particularly important role in the quantum mechanical theory of a system of many electrons (for example, an atom or molecule), which cannot even be formulated without taking into account the symmetry properties of the wave function with respect to electron permutations. These properties consist in the requirement that the wave function of a system of electrons, expressed in terms of sets of variables related to each electron, change sign when two such sets related to two electrons are interchanged. This requirement is called the Pauli principle or the antisymmetry principle of the wave function. It is essential to note that the number of variables of each electron includes, in addition to its coordinates, also its spin variable a. This shows that the introduction of the spin degree of freedom of the electron is already necessary in the nonrelativistic theory.

The next part of this book will be devoted to the many-electron problem of quantum mechanics.