Dynamics of Quantum Particles in Perturbed Parabolic 2d Potential

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( 12.02.2016; online 29.11.2016)

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DOI: 10.21272/jnep.8(4(1)).04014 PACS numbers: 03.65. – w, 02.30.Jr

* _{knigorua@mail.ru}

1. ВВ

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2. А В А А А

2.1

- 

1 2  21 2  2

( , , ) ( ( )) ( ( ))

2 x x 2 y y

V x y t m x f t m y f t , (1)

fx(t), fy(t) – - ;

m, x, y – .

(

-)   ( , , ;x y t x0,y0,0)

 

_{ }  _  _{  }

 

 _  _

2 2 2

2 2 ( , , )

2 V x y t

i t m x y , (2)

ħ– .

, 

-:

t00,

0 0

(x ,y ), t ,

-( , )x y .

2.2

-, x y

-  -x x( , ;x t x0,0)

  y y( , ;y t y0,0):









   _{ }   _{  }    _{ }   _{  }   2 2 2 2 2 2 2 2 2 2 1 ( ) , 2 2 1 ( ) . 2 2 x x

x x x

y y

y y y

m x f t

i t m x

m y f t

i t m y

(3)

,

-( , , ;x y t x0,y0,0) x( , ;x t x0,0)y( , ;y t y0,0). (4)

(4) (2).





 _  _{  }  _{  }     y x

x y y x

i t i t i t i t . (5)

(3) (5),









          _{    }   _{         }             2

2 2 2

2 2 2 2 2 2 1 1 ( ) ( )

2 2 2 2

y y

x x

y x m x x f tx x y x m y y fyt y

i t i t m x m y . (6)

(6) ,

.

, (4) .

( ),          _{ }             _{  }   _       _{ }       _   _  _{  }       1/2 ₂

2 2 2

0 ₂ 0

0 2 2 0 2 0 0

( , ; ,0) exp ( ) ( )

( 1) 2 2

exp ( ) ( ) ( ) ,

2 ( 1)

s

s

s s s

s

i t t

s s

s _{i t} s

t t

s i t i t i

s s _{i t} s

me m m

s t s s s i f d

e

m

i F d sF t s e s e F e d

m e m

(7)





 ,

s x y ,









  2  _    _ 

0

( ) s ( ) exp .

s s s

m

F i f i d (8) (4)

             _{   }       _{ }         _      _            1/2

0 0 _{2 2}

2 2

2 2 2 2 2 2

0 0 0 0 2 0 2 0

( , , ; , ,0)

( 1) ( 1)

exp ( ) ( ) ( ) ( )

2 2 2 2

exp ( ) ( )

2 ( 1)

y x y x x x x i t i t i t i t t t y xy x x x y t

x i t i t

x x _{i t}

me me

x y t x y

e e

m m

m m

x x y y i f d i f d

m

i F d xF t x e x e F

m e m

                 _{ }   _{ }    _{ }     _{ }     _   _   _{ }         2 0 2 2 0 2 0 0 ( )

exp ( ) ( ) ( ) .

2 ( 1)

x

y y y

y

t

i x

t t

y i t i t i

y y _{i t} y

e d

m

i F d yF t y e y e F e d

m e m

(9)

(x,y,t) (2) c

(x0,y0,0) t=0

0 0 0 0 0 0

( , , )x y t (x ,y t, ) ( , , ;x y t x ,y ,0)dx dy

         . , -2 ( , , ) ( , , )

p x y t  x y t .





1/4 _{2 2}

0 0

( , ,0) exp

2 2

y x

x y

m

m m

x y  x  y

  



 

 _  _

 .

,

x = 0, y = 0.

    













   

















 _{   } 

 

  _   _  _   _

     

   

2 2

0 0

( , , ) exp ( ) sin ( ) sin

t t

x y x y

x x x y y y

m m m

p x y t x f t d y f t d . (10)

, m/ћ 1. ,

fx(t)  fy(t)  0. , -x  0, y  0 ( . 1).

. 1– (10)

t  0.

-: x = y  1

, t  0

,

:









 

 

 

 _  _

 

1

2

( ) 2 sin( ) 1 / 2 ,

( ) sin( ) 1 / 2, x

y

f t t

f t t (11)

θ(·)– ( . 2).

. 2– .

-: x y 4, 1  2 1

, (2)

fx(t) fy(t). fx(t) fy(t) (11)

.

. 3.

. 3–

(8) t _∈ [0, 25]

.

-: x y 4, 1 2  1

(8) ,

, . . 3

(8). ,

-, а п и д ая -–

( )

( )

( )

( )

( )

. 4–

(10)

-t = 5 ( ), t = 10 ( ), t = 15 ( ), t = 20 ( ),

t = 30 ( ). : x  y  4, 1 2 1

(10)

. 4. , (10)

t,

(10). . 4

-,

-.

(10),

.

fx(t), fy(t).

,

,

-fx(t), fy(t).

-. . 4

. 3 ,

-, , . А -





          

2 2 ₂

2 2

1

( )

2 2m x f t

i t m x ( .1)

2

0 0 0 1 2

( , ;x t x t, ) expQ t( ) Q t x( ) Q t x( ) 

  _   _. ( .2)

-,

-, Q0(t),

Q1(t), Q2(t) (Q2(t)<0) ,

-.









2

0 1 2

1 2

2

2

1 2 2

2

( ) ( ) ( ) ,

( ) 2 ( ) ,

( ) 2 ( ) 2 ( ) ,

d d d

Q t x Q t x Q t

t dt dt dt

Q t xQ t

x

Q t xQ t Q t

x  _{ } _{ } _       _{   }   _{    }      ,













         _   _ 2 2 2

0 1 2

2

2

1 2 2

1

( ) ( ) ( ) ( )

2

( ) 2 ( ) 2 ( ) . 2

Q t xQ t x Q t m x f t

i

Q t xQ t Q t

m

-,

Q0(t), Q1(t), Q2(t).

       _  _      2

2 2 2

0 1 2

2 2

1 1 2

2 2 2 2 2 1 2 , 2 2 2 , 1 2 . 2

Q m f Q Q

i m

Q m f Q Q

i m

Q m Q

i m

( .3)

, -

-0 0 0 0

( , ,x t x t, ) (x x )

   (

( .1)) ( .2) t→0:

0

( ,0)_{x e} ikx_{dx e} ikx

       ,





2

0 1 2

2 1 0 2 2 exp exp . 4 ikx

Q xQ x Q e dx

Q ik Q Q Q           _         





2

1

0 0

0 _{2 2}

1 lim ln 2 4 t Q ik Q ikx Q Q    _{  }   _ _     _{ }    . ( .3) –

Q2(t),

-Q1(t) , ,

Q0(t). ,

(A.1).

Dynamics of Quantum Particles in Perturbed Parabolic 2d Potential

A.S. Mazmanishvili

1

_{, I.A.}

Knyaz’

2

1 _{NSC "Kharkov Institute of Physics and Technology", 1, Academicheskaya st., 61108 Kharkov, Ukraine} 2 _{Sumy State University, 2, Rimsky-Korsakov st., 40007 Sumy, Ukraine}

2d quantum-mechanical problem of the time evolution of a particle in a quadratic potential is studied. We suppose that the center of the potential is displaced in arbitrary way in time. An analytical expression for the wave function in arbitrary instant time was built. It is shown the dynamic shift of the center of the

potential doesn’t change the variance. Moreover, the system can exhibit the resonance: when the frequency of the potential perturbation approaches to the natural frequency the amplitude of the wave packet of par-ticle is increased.

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1 « а _- ч », . А а ч а, 1, 61108 а , а а

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(Cambridge University Press: 2001).

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C 19, 1607 (2008).

10.F. Iacob, Centr. Eur. J. Phys.12, 628 (2014). 11. F. Iacob, M. Lute, J. Math. Phys.56, 121501 (2015) . 12. Wei-Ping Zhong, Milivoj Belić, and Yiqi Zhang, Optic.