Добавил:

ivanov666 Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

Башкирский Государственный Аграрный Университет

Предмет:

[НЕСОРТИРОВАННОЕ]

Файл:

Квантовая физика и нанотехнологии

.pdf

Скачиваний:

Добавлен:

12.08.2026

Размер:

806 Кб

Скачать

☆

<<< < Предыдущая 1 2 3 4 5 6 7 8 9 10 11 1213 / 1313

Appendix 2. Quantum particles motion in stationary external fields

	( P)	2	2		2	2 2
( )							.
	2m					4m x 2
			8m 2	x

We re-arrange Equation (Ap. 2.1) in the form:

m	P	P	0.

P t		x

We calculate this equation to within P % 0. Then

	m
		dx t
	P	dx t
P J	P		,
P J		T	,
		T

where Т is the specific problem time. Let us designate

m	dx t

P		.

	T

We substitute solution (П2.4) into (П2.3) and obtain:

( P)	2			m 2 .		1	dJ	2	2 d2J		1
		;			0						3.
2m		;			0				J d		3.
2m			8T 2P 2 0 J 2				d		J d	2 3
			/								2

The satisfying solution has the form:

		m
			dx t
		P	dx t
J J 0	cos2	P		.
J J 0	cos2		T	.
			T

Substituting (Ap. 2.6) into (Ap. 2.5) we get:

T m const.

P P

Taking into account that

( P)2 ( )2 . 2m 8m

We obtain from (Ap. 2.7):

P(x) (x) 2m const.

(Ap. 2.3)

(Ap. 2.4)

(Ap. 2.5)

(Ap. 2.6)

(Ap. 2.7)

Appendix 3. Solving quantum hydrodynamic equations for free particle

In the approximation considered the probability flow density in the coordinate space remains constant.

Thus, finally we find:

J 0

dx t

cos2

;

(Ap. 2.8)

P(x)

P(x)2

U (x)

m 2

(Ap. 2.9)

2T 2P 2

The solutions obtained are similar to the quasi-classical approximation, which is used in traditional quantum mechanics, in the method of derivation. This approximation is true under the following condition:


JT		dP
JT		dx	!! 1.	(Ap. 2.10)
		dx

m	dJ
m	d
	d

APPENDIX 3

SOLVING QUANTUM HYDRODYNAMIC EQUATIONS

FOR FREE PARTICLE

Let us consider motion of a free particle with mass m and given momentum Р. The system of Equations (4.2) and (4.3) takes on form:


	m		P 0,				(Ap. 3.1)
	m		P 0,				(Ap. 3.1)
		t
( P)2	2			( )2	2	const.	(Ap. 3.2)
2m	8m 2			( )2	4m	const.	(Ap. 3.2)
2m	8m 2				4m

We will look for a general solution to equation (Ap. 3.1) in the following form:

(t, r) ( ), где P(r Pt / m) / .

(П3.3)

Appendix 3. Solving quantum hydrodynamic equations for free particle

Then
		P		P		,

t	t			m
			P		.

Substituting and into Expression (Ap. 3.1) we make sure that

(Ap. 3.3) is a solution to this equation. Then we calculate

d2 P)2 . d 2

Substituting and into expression (П3.2) we find

	1			2	2	2
4	1		d		2		d	.	(Ap. 3.4)
	2	d					d

To solve this equation we reduce the order of derivatives. We designate:

1 d	u.	(Ap. 3.5)

d

Then

du	2			1				2
		1	d			d	.

d d				2	d

Substituting these expressions into (П3.4) we obtain:

2 du u2 4 0. d

We calculate this equation using the method of separation of variables and get:

arctg u . 2

Substituting the solution into (Ap. 3.5) and integrating again we finally obtain:

(t, r) 0cos2 0cos2( P(r Pt / m) / ).

The solution obtained matches Formula (4.6).

Appendix 4. Charged particle motion in electromagnetic field

APPENDIX 4

CHARGED PARTICLE MOTION

IN ELECTROMAGNETIC FIELD

Let us consider the motion of a particle with charge e and mass m in an arbitrary electromagnetic field in quasi-hydrodynamic representation [1]. We suppose that the electric field intensity is and the magnetic field intensity is H. These variables can be expressed in terms of magnetic vector and scalar potentials: А and 6:

	1		A	6.	(Ap. 4.1)
	c
			t
H		rot A.			(Ap. 4.2)

In this case the Hamiltonian operator has the following form [2]:


	1		e		2
:		P		A	e6 U .	(Ap. 4.3)
:		P	c	A	e6 U .	(Ap. 4.3)
	2m		c

It has been taken into account here that beside electromagnetic forces there are some other forces described by the force function U. Using the Coulomb gauge divA = 0, the Hamiltonian operator can be rewritten in a different form:

	1		e		e2
:		P 2		( AP)		A2	e6 U .	(Ap. 4.4)
			mc
	2m				2mc2

Let us write necessary initial Scroedinger equations for further calculations with operator (Ap. 4.4):

:, i

:*.

(Ap. 4.5)

Multiplying the first Schroedinger equation by

and the second by

and subtracting the second equation from the first one, we obtain:

*: : *

( * *)

mc( * A A *)

designate

* . Let us

introduce the

probability density

flow J / m:

( * * ).

(Ap. 4.6)

Appendix 4. Charged particle motion in electromagnetic field

Let us denote:

R	e	div A .	(Ap. 4.7)

	c

Then we obtain:

m div J R 0.

Let us introduce a new expression for the momentum flow density J

J * J e A. c

We will obtain the probability density conservation law in the following form:

m div J* 0.

	e
If J P, then J * P		A	P* , where P*
If J P, then J * P	c	A	P* , where P*
	c

(Ap. 4.8)

is a standard gen-

eralized momentum of a particle in the electromagnetic field. Further, the derivation will be calculated:

/ )

(ln * ln )

1 *

: * *:

Here Schroedinger equations written above are used. Substituting the explicit form of Hamiltonian operator, we obtain:

(J / )	e2			2e AJ	2	* *
		A 2	e6 U				. (Ap. 4.9)
t				mc
	2mc2				4m

The term with wave functions should be substituted by expressions sity. For this purpose, we calculate:

in the right part of this equation dependent on the probability den-

* * 2 *.
( )2 ( * * )2 ( * * )2 4 *	4J 2
		.

	2

Excluding the term with wave functions in (Ap. 4.9), we finally obtain:

(J / )

J 2

2( )2

A 2

e6 U

. (Ap. 4.10)

2m 2 2mc2

8m 2

Appendix 4. Charged particle motion in electromagnetic field

The new potential could be introduced

U *	e2		e6 U	2e	AJ	.	(Ap. 4.11)
		A 2
				mc
	2mc2

Then we get the equation:

		J 2		2( )2	2
	(J / )		U *			,	(Ap. 4.12)
t		2m 2		8m 2
					4m

which is in agreement with (Ap. 1.6).

If there is a macroscopic momentum J / P, then we have:

	P 2			2( )2	2
	P		U *			.	(Ap. 4.13)
t				8m 2
		2m			4m

which agrees with (Ap. 1.7), but the motion takes place in the effective force field with the potential U * from (Ap. 4.11). Let us introduce the expression for the particle velocity v = P/m, then Eq. (Ap. 4.13) can be rewritten as:

v		1		2( )2
	(v )v		U *		.	(Ap. 4.14)
t
		m		8m 2

This is a quantum equation for a laminar flow of compressible perfect fluid [1].

Reference

1.B.V. Alexeev, A.I. Abakumov. On Approach to Solving Schroedinger Equation. Doklady Akademii Nauk. V. 262, P. 1100. 1982

2.D.I. Blokhintsev. Fundamentals of Quantum Mechanics. M.: Nauka 1976. 664 p.

ABOUT THE AUTHOR

Vladimir Kirillovich Nevolin

Doctor of Physical and Mathematical Sciences, professor of Quantum Physics and Nanoelectronics chair at Moscow State Institute of Electronic Technology (Technical University) (MIET), head of Probe Microscopy and Nanotechnology research-and-educational center, Honoured Figure of Russian Higher Education, the Russian Federation Government Prize laureate in Science and Technology.

Expert in the field of probe microscopy, probe nanotechnology and nanoelectronics.

For contact

No. 162, Apart. 216, Zelenograd,

Moscow, Russia 124482 +7 (499) 734-49-88 (Home) +7(499) 732-72-41 (Office)

Mobile phone: 8-916-542-16-28 E-mail: vkn@miee.ru www.nanotube.ru

Производство книг на заказ Издательство «Техносфера»

тел.: (495) 234 01 10 e mail: knigi@technosphera.ru

Реклама в книгах:

•модульная

•статьи

Подробная информация о книгах на сайте http://www.technosphera.ru

Неволин Владимир Кириллович

Квантовая физика и нанотехнологии

Издание 2-е, исправленное и дополненное

Компьютерная верстка — В.Ю. Кознов Корректоры — А.Ф. Ануфриев, Н.И. Кононенко Дизайн книжных серий— С.Ю. Биричев Дизайн — А.В. Войткевич, М.А. Костарева Выпускающий редактор — С.Ю. Артемова Ответственный за выпуск — С.А. Орлов

Подписано в печать 09.08.2013. Формат 60Ї90/16. Печать офсетная. Гарнитура Ньютон.

Печ.л. 8. Тираж 200 экз. Зак. № Бумага офсет № 1, плотность 65 г/м2

Издательство «Техносфера» Москва, ул. Краснопролетарская, д. 16, стр. 2

Отпечатано в ООО «Чебоксарская типография № 1» 428000, Чувашская Республика, г. Чебоксары,

пр. И. Яковлева, 15. Тел. +7(8352) 28-77-98, 57-01-87. www.volga-print.ru

<<< < Предыдущая 1 2 3 4 5 6 7 8 9 10 11 1213 / 1313

Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]