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Analytic solution for one-dimensional quantum oscillator with a variable frequency
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1999 Acta Phys. Sin. (Overseas Edn) 8 641
(http://iopscience.iop.org/1004-423X/8/9/001)
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Volume 8, Number 9 ACTA PHYSICA SINICA September, 1999 
1004-42~X/1999/08(09)/0641-06 (Overseas Edition) @ 1999 Chin. Phys. Soc. 
ANALYTIC SOLUTION FOR ONE-DIMENSIONAL QUANTUM 
OSCILLATOR WITH A VARIABLE FREQUENCY* 
Xu XIU-WEI(%%%), REN T I N G - Q I ( E ~ @ ) , 
and LIU SHENG-DIAN(#~@&) 
Department of Physics, Yantai Teachers University, Yantai 264025, China 
(Received 26 March 1999) 
According to generalized linear quantum transformation theory, we give the exact expression 
of the evolution operator, wave function and expectation value of observable quantity for one- 
dimensional quantum oscillator with a variable frequency: w ( t ) = - ( W O , P > 0). WO 1 + Pt 
PACC: 0413; 0411 
I. INTRODUCTION 
Time-dependent quantum oscillator (TDQO) was widely introduced in various areas of 
physics, such as the quantum motion of a particle in a Paul trap[132], a quantized electromag- 
netic field in a Fabry-PBrot cavity[3], etc. In these cases, the system can be transformed to 
a corresponding TDQO. Therefore, the problem to analytically solve TDQ0[4-9] becomes 
an important part in quantum physics. 
Using the generalized linear quantum transformation (GLQT) theory presented by 
Zhang et a1.[10-12], we exactly solve an one-dimensional quantum oscillator with a vari- 
able frequency w ( t ) = - (wo,P > 0). And we obtained exact formulae of evolution, 
wave function and expectation value of observable quantity. 
In fact, the system of a charged particle in a variable-intensity (B(t)) magnetic field 
is equivalent to a TDQO with frequency w 0: IBI. Thus, the results of this paper can be 
applied to this case, which is easy to implement in practical experiments. 
II. EXACT EXPRESSION OF EVOLUTION OPERATOR 
1 + Pt 
W 
Let us consider a quantum oscillator with time-dependent frequency 1 (here wo, 
1 + Pt 
a > 0): 
where we choose m = A = 1. 
operator 6(t) as follows: 
According to the GLQT theory, let us choose the normal ordering form of time evolution 
*Project supported by the Natural Science Foundation of Shandong Province, China (Grant No. Y95A0202). 
642 Xu Xiu-wei et al. Vol. 8 
where a, b, c , d are real parameters and the notation : - . . : represents the normal ordering 
(such as: s2$ + $0 : = $5’ + $0). U(”) ( t ) is the normal ordering form of U(t), hence 
c(t) = 8 ” ) ( t ) . Using GLQT theory, we have 
U .. (:,A) - x U - - A l - (f,?) - (l id), and a c + b d = l 
C 
From Schrodinger equation, we get the equation of o ( t ) , 
(3) 
Substituting Eq.(2) into Eq.(4) and simplified by Eq.(3), we obtain the equations for real 
parameters a , b, c, d as follows: 
di+ ( A ) z d = 0 , d ( 0 ) = 0, d(0) = -1; 
1 + Pt 
c = -d. 
Here a = da/dt and so on. 
The above differential equations have the following solution: 
[cos(v+i 
(5) 
No. 9 Analytic Solution for One-Dimensional Quantum ... 643 
III. FORMULA OF WAVE FUNCTION AND EXPECTATION VALUE 
A. Wave function 
Using the normal ordering form of U ( t ) and the completeness of momentum represen- 
tation, we obtain the wave function of TDQO 
where $(z, 0) is the initial wave function. If the initial state of TDQO is the nth eigenstate 
of a time-independent oscillator: 
where a = 6, then the wave function will read 
(P 2wo), 
1 
2 
v: - v+ cos(v+z) sin(v+z) + - sin2(v 
-112 
(1 - z + ;2) (p = 2w0), 
< 
\ (P > 2wo), 
22 (P = 2wo), 
zsin(v+z)/[sin(v+z) - 21 (P < 2w0), 
(P > 2wo), 
(1 + @)--U- - (1 + pty - 
- + v- (1 +@)-U- - (; - v-) (1 + Pt)”- (1 ) 
v;csc2(v+z) - v+cot(v+z) + ‘1 (P < 2w0), 2 I ‘ (l/z2 - l / z + 1/2) (P = 2w,), 
644 Xu Xiu-wei et al. Vol. 8 
B. Expectation value 
For an arbitrary observable f($, i), we can easily prove 
U+f^($ , i ) U = f ( c $ + bi, ai - dfi) i($, 2 ) . (12) 
Then the expectation value of f at t moment 
J ( t ) = (+(t)lfl+(t)> = (.ICl(o)P+.NI+(o)) = (+(O)lGl+(O)> = B(0) (13) 
is changed into the expectation value of i in the initial state. Such as, the initial wave 
function is nth eigenstate (refer to Eq.(9)), we have 
[Ap(t)12 = (n+ i) WO [C’ (b)l] 
WO 
[Ax(t)I2 = ( n + 1) L [ a ’ + ( ~ ~ d ) ~ ] , (14) 
2 WO 
[Ap(t)]2[Az(t)]2 = ( n + iy [l + 
- :y] , 
where [Ap(t)12 = (+(t)lk -p(t)121+(t)) and the like. 0bviously)the uncertainty o f t moment 
is not less than t = 0 moment. 
From Eq.(6), we get 
X 
No. 9 Analvtic Solution for One-Dimensional Quantum ... 645 
I -4(1+ ,8t)-2u-]} (p > 2w0) . 
n o m the above equalities we find out that, if t + 00, then [Ap(t)12 + 0. This shows 
that the system would have some sqeeae property. 
IV. CONCLUTION 
We have presented the solution for a special one-dimensional TDQO. Applying the re- 
sults of this paper, we can study the motion of a charged particle in a variable-intensity 
magnetic field. The method in this paper may be extended to other time-dependent prob- 
lems. 
ACKNOWLEDGEMENT 
The authors would like to thank Prof. Yong-de Zhang and Dr. Guang Hou for valuable 
discussions. 
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