In this approximation, the wavelength of the electromagnetic radiation is taken to be long, compared with the spatial dimension of the system, hence only the.
Commun. Theor. Phys. (Beijing, China) 41 (2004) pp. 45–47 c International Academic Publishers
Vol. 41, No. 1, January 15, 2004
Variable Frequency Harmonic Oscillator in an Electromagnetic Field HUANG Bo-Wen,1 WANG Jing-Shan,2 GU Zhi-Yu,1 and QIAN Shang-Wu3 1
Physics Department, Capital Normal University, Beijing 100037, China
2
Physics Department, New Jersey Institute of Technology, Newark, NJ 07201, USA
3
Physics Department, Peking University, Beijing 100871, China
(Received March 7, 2003)
Abstract The invariant, propagator, and wavefunction for a variable frequency harmonic oscillator in an electromagnetic field are obtained by making a specific coordinate transformation and by using the method of phase space path integral method. The probability amplitudes for a dissipative harmonic oscillator in the time varying electric field E(t) = E0 sin(Ωt) are obtained. PACS numbers: 03.65.Bz, 03.65.Db
Key words: invariant, propagator, canonical transformation, phase space path integral, variable frequency harmonic oscillator
1 Introduction
chosen to be zero.
There has been considerable interest in variable frequency harmonic oscillator (VFHO), and the behavior of VFHO in an electromagnetic field is also a very important problem in practice. In Sec. 2 we shall use a specific canonical transformation to obtain the invariant of VFHO, which is very useful when we use the Lewis– Riesenfeld method[1] to obtain the wave functions of this case. In Sec. 3 we shall use an alternative method — phase space path integral method, to obtain wave functions. The Hamiltonian for a one-dimensional variable frequency harmonic oscillator with unit mass is 1 1 (1) H = p2 + ω 2 (t)x2 . 2 2 As to the consideration of influences of electromagnetic field on this oscillator, we shall use the Coulomb gauge in electric dipole approximation (EDA). EDA is widely used in quantum optics to treat the interaction of electromagnetic radiation with matter in the long-wavelength ~ limit. In EDA[2] only the effect of the electric field E ~ is on the system is considered, and the magnetic field B neglected. In this approximation, the wavelength of the electromagnetic radiation is taken to be long, compared with the spatial dimension of the system, hence only the electric field at the origin needs to be considered, thence the spatial variation of the field can be neglected. Thus in EDA, for the one-dimensional case, we have ~ ~ t) = E(t), ~ E(x, t) ≈ E(0,
~ B(x, t) ≈ 0 .
(2)
In the Coulomb gauge in the electric dipole approximation,[3] the Hamiltonian is i2 1 1h q H = p − A(t) + ω 2 (t)x2 , (3) 2 c 2 where A(t) is the vector potential at the origin, which is chosen to be a transverse field. The scalar potential A0 is
2 Invariant We define a canonical transformation qA , Q = x + a(t), P = p + a˙ − c where a(t) satisfies
(4)
q A˙ . (5) c The generating function of this canonical transformation is qA F2 (x, P, t) = (x + a)P − a˙ − x, (6) c a ¨ + ω2 a =
and the new Hamiltonian is[4] ˜ = H + ∂F2 = 1 P 2 + 1 ω 2 Q2 + dG , H ∂t 2 2 dt
(7)
where q A˙ ∂F2 = aP ˙ − a ¨− x, ∂t c
(8)
Z 1 2 1 2 2 a˙ − ω a dt . 2 2
(9)
and G=
It is obvious that the invariant of this system is[5,6] I=
1 1 Q2 (ρP − ρQ) ˙ 2+ , 2 2 ρ2
(10)
where ρ(t) satisfies ρ¨ + ω 2 ρ =
1 . ρ2
Returning to the original coordinate, we have i2 1h qA I = ρ p + a˙ − − ρ(x ˙ + a) 2 c 2 1 (x + a) + . 2 ρ2
(11)
(12)
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HUANG Bo-Wen, WANG Jing-Shan, GU Zhi-Yu, and QIAN Shang-Wu
3 Phase Space Path Integral, Propagator It is well known that the two sets of canonical coordinates (x, p) and (Q, P ) satisfy[4] ˜ + dF , px˙ − H = P Q˙ − H dt
where we introduce another new Hamiltonian J, 1 1 (16) J = P 2 + ω 2 Q2 . 2 2 In this case we can use the method of phase space path integral[7,8] to find the propagator, i.e. Z hi Z T i K = Dx Dp exp (px˙ − H)dt . (17) ¯h 0 The transformation of measure is[8] ∂x ∂x 1/2 ∂Q ∂P DQ DP = Dx Dp ∂p ∂p ∂Q ∂P 1 0 1/2 Dx Dp = Dx Dp , = 0 1 hence equation (17) becomes T i hi K = exp (F − G) K0 , h ¯ 0 where Z i hi Z T K0 = DQDP exp (P Q˙ − J)dt , h 0 ¯
(18)
(19)
(20)
1/2 ω . (22) 2πi¯ h sin ωT Returning to the original coordinate, we have T i h hi iω K = exp (F − G) F1 (T ) exp {[(x + a)2 ¯h 2¯ h sin ωT 0 i + (x0 + a0 )2 ] cos ωT − 2(x + a)(x0 + a0 )} , (23) F1 (T ) =
4 Wave Function The wave function can be obtained by[9] Z ∞ dx0 K(x, t; x0 , 0)Ψn (x0 , 0) . Ψn (x, t) =
(24)
−∞
Let Ψn (x0 , 0) =
pω/¯h 1/2 r ω √ Hn x0 ¯h 2n n ! π ω × exp − x20 , 2¯h
(25)
we obtain pω/¯h 1/2 ω Ψn (x, t) = n √ exp − (x + a − a0 cos ωT )2 2¯ h 2 n! π r ω × Hn (x + a − a0 cos ωT ) ¯h T i 1 × exp −i n + ωT exp (F − G) 2 ¯h 0 iω sin ωT [a20 cos ωT − 2(x + a)a0 ] .(26) × exp 2¯h
5 Probability Amplitude
which is just the propagator of ordinary harmonic oscillator, i.e. n iω K0 = F1 (T ) exp [(Q2 + Q20 ) 2¯h sin ωT o × cos ωT − 2QQ0 ] , (21) where
where x0 and a0 are the initial values of x and a, respectively.
(13)
˜ is the Hamiltonian corresponding to (Q, P ), F is where H any function of the phase space coordinates with continuous second derivatives. Let q F (x, p, t) = −P Q + F2 = A − a˙ x , (14) c then we have d(F − G) px˙ − H = P Q˙ − J + , (15) dt
Vol. 41
From Eq. (26) we readily know the probability amplitude for the n-th excited state, p ω/¯h 2 Pn (x, t) = |Ψn (x, t)| = n √ 2 n! π n ω o2 × exp − (x + a − a0 cos ωT )2 2¯ h n r ω o2 . (27) × Hn (x + a − a0 cos ωT ) ¯h Now we shall illustrate the process of calculation for a particular case, in which t ω 2 = ω02 exp − , T E(t) = E0 sin(Ωt) , (28) i.e., we discuss a dissipative harmonic oscillator in a timevarying electric field E(t) = E0 sin(Ωt) . Since A˙ = −E(t), equation (5) becomes h t i q a ¨ + ω02 exp − a = − E0 sin(Ωt) . (29) T c Solving the second order p differential equation (29) for a, then calculating z = ω/¯h(x + a − a0 cos ωT ), we can easily obtain Pn (x, t) from Eq. (27). For example, when we let ω0 T = 2, Ω = ω0 , a0 = 1/ω0 , qE/c = 1, ¯h = 1, we obtain three-dimensional figures for P0 (x, t), P1 (x, t), and P2 (x, t) in Figs. 1, 2, and 3.
No. 1
Variable Frequency Harmonic Oscillator in an Electromagnetic Field
Fig. 1
Fig. 2
47
Probability amplitudes P0 (x, t).
Probability amplitudes P1 (x, t).
References [1] S.W. Qian, G.Q. Xie, and Z.Y. Gu, Ann. Phys. 266 (1998) 497. [2] S.W. Qian, J. Phys. A: Math. Gen. 20 (1987) 607. [3] S.W. Qian, Y.T. Hu, and J.S. Wang, J. Phys. A: Math. Gen. 20 (1987) 2833. [4] H. Goldstein, Classical Mechanics, Addison-Wesley, Singapore (1980) pp. 378–385.
Fig. 3
Probability amplitudes P2 (x, t).
[5] J.R. Ray and J.L. Reid, Phys. Lett. A71 (1979) 317. [6] D.C. Khandekar and S.V. Lawande, Phys. Rep. 137 (1986) 214. [7] Namik K. Pak and I. Sokmen, Phys. Rev. A30 (1984) 1629. [8] L. Chetouani and L. Guechi, Phys. Rev. A40 (1989) 1157. [9] H.G. Oh, H.R. Lee, and Thomas F. George, Phys. Rev. A39 (1989) 5515.
