Quantum infinite square well with an oscillating wall

Chaos, Solitons and Fractals 41 (2009) 2067–2074

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Quantum infinite square well with an oscillating wall M.L. Glasser a, J. Mateo b, J. Negro b, L.M. Nieto b,* a b

Center for Quantum Device Technology, Clarkson University, Potsdam, NY 13699-5820, USA Departamento de Física Teórica, Atómica y Óptica, Universidad de Valladolid, 47071 Valladolid, Spain

a r t i c l e

i n f o

Article history: Accepted 16 July 2008

Communicated by Prof. G. Iovane

a b s t r a c t A linear matrix equation is considered for determining the time dependent wave function for a particle in a one-dimensional infinite square well having one moving wall. By a truncation approximation, whose validity is checked in the exactly solvable case of a linearly contracting wall, we examine the cases of a simple harmonically oscillating wall and a non-harmonically oscillating wall for which the defining parameters can be varied. For the latter case, we examine in closer detail the dependence on the frequency changes, and we find three regimes: an adiabatic behabiour for low frequencies, a periodic one for high frequencies, and a chaotic behaviour for an intermediate range of frequencies. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Solution of the Schrödinger equation subject to moving boundary conditions has been studied for many years and a small number of exactly solvable cases, in one spatial dimension, are known [1–11]. The simplest of these describes a particle in an infinite square well with one wall moving at constant speed. For the most part, the other solved cases are found by starting from the Schrödinger equation for a harmonic oscillator with a time-dependent ‘‘frequency”. By transforming the spatial coordinate to eliminate the wall motion, new time-dependent terms are introduced into the potential. By selecting the wall motion suitably, one can cancel inconvenient terms and end up with a solvable problem. A supersymmetry approach has recently been proposed to extend the list of solvable cases [12]. One situation which apparently cannot be studied in this way is the so-called quantum Fermi accelerator: an infinite square well with one oscillating wall. In 1949, Fermi [13] suggested that cosmic ray protons might have been accelerated to high energy by colliding with moving galactic magnetic fields. He proposed no specific model, but merely provided some estimates based on contemporary data. It appears to have been Ulam [14] who modeled this as a classical particle in a square well with a moving wall. His numerical studies displayed both regular and stochastic motion and the model has been popular in chaos studies. The quantum mechanical version soon emerged in the area of quantum chaos. In 1986, José and Cordery [15] formulated the solution to the Schrödinger equation for sawtooth motion of the moving wall and examined the statistical features of the energy spectrum. This was followed in 1990 by Seba’s work [16] on the absolute continuity of the spectrum. In these papers, dealing with a saw-tooth profile, the particle is subject to a periodic sequence of force discontinuities and subsequently this version of the model was shown to be equivalent to the periodically kicked rotator and many aspects of its dynamics have been investigated (several references are provided in [17]). Our aim in this article is to examine features of the wave function and the variation of the energy when the particle is initially in a given instantaneous eigenstate. We assume that the wall oscillates smoothly. The paper is organized as follows: in Section 2 the theoretical foundations are presented, in Section 3 some examples are developed, paying especial attention to the change in the frequency of the oscillations of the wall, and finally, Section 4 ends the paper with some conclusions. * Corresponding author. Fax: +34 983423013. E-mail addresses: [email protected] (M.L. Glasser), [email protected] (J. Mateo), [email protected] (J. Negro), [email protected] (L.M. Nieto). 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.07.055

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M.L. Glasser et al. / Chaos, Solitons and Fractals 41 (2009) 2067–2074

2. Theoretical background The system has the Hamiltonian

H0 ¼ o2x

ð1Þ

(here h ¼ 2m ¼ 1) over the interval ½0; LðtÞ with Dirichlet boundary conditions. We have the complete orthonormal set of instantaneous eigenstates

sffiffiffiffiffiffiffiffi 2 sin½kpx=LðtÞ; /k ðx; tÞ ¼ LðtÞ

2

k ðtÞ ¼

k p2 L2 ðtÞ

ð2Þ

:

The Schrödinger equation iot wðx; tÞ ¼ H0 wðx; tÞ has the general solution

wðx; tÞ ¼

1 X

An ðtÞ/n ðx; tÞ

ð3Þ

n¼1

whose substitution into the Schrödinger equation yields

A_ k ðtÞ þ ik ðtÞAk ðtÞ ¼

1 X

C kn ðtÞAn ðtÞ;

ð4Þ

n¼1

where

C kn ðtÞ ¼

Z

LðtÞ

/k ðx; tÞ/_ n ðx; tÞdx:

ð5Þ

0

We will be interested in the case where the initial wave function is the instantaneous ground state, where the initial condition is given by Ak ð0Þ ¼ dk;1 . For convenience, we introduce the quantities

gðtÞ ¼

_ LðtÞ ; LðtÞ

f ðtÞ ¼

Z

t

du L2 ðuÞ

0

In order to eliminate the term with

Bk ðtÞ ¼ ð1Þk Ak ðtÞe

ip2 k2 f ðtÞ

ð6Þ

:

k ðtÞ in Eq. (4), let ð7Þ

;

yielding

wðx; tÞ ¼

1 X 2 2 ð1Þn Bn ðtÞein p f ðtÞ /n ðx; tÞ

ð8Þ

n¼1

with

B_ k ðtÞ ¼

1 X

Dkn ðtÞBn ðtÞ;

n¼1

and

Dkn ðtÞ ¼ 2gðtÞ

kn 2

k n2

eip

2 ðk2 n2 Þf ðtÞ

ð1 dk;n Þ:

ð9Þ

We must therefore deal with the family of homogeneous linear first-order coupled differential equations, which, in obvious vector notation, adopt the form

ðIot 2gðtÞDÞ~ B ¼ 0;

ð10Þ

where the matrix D is

2

0

6 2 3p2 if ðtÞ 6 3e 6 6 3 8p2 if ðtÞ e D¼6 6 8 6 4 15p2 if ðtÞ 6 15 e 4 .. .

2 3p2 if ðtÞ e 3

3 8p2 if ðtÞ e 8

4 15p2 if ðtÞ e 15

0

6 5p2 if ðtÞ e 5

2 12p2 if ðtÞ e 3

65 e 23 e

5p2 if ðtÞ

12p2 if ðtÞ

.. .

0 7p2 if ðtÞ

12 e 7 .. .

12 e 7

7p2 if ðtÞ

0 .. .

3

7 7 7 7 7 7 7 7 5 .. .

ð11Þ

Before looking at the solution, we note that the wave function normalization is

NðtÞ ¼

1 X n¼1

jBn ðtÞj2

ð12Þ

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and that 1 X X ðB_ k Bk þ c:cÞ ¼ ðDkn þ Dnk ÞB_ k Bn ;

_ NðtÞ ¼

k¼1

ð13Þ

k;n

which vanishes, since D is antihermitian. Hence, the normalization of w is independent of time. On the other hand, the energy

EðtÞ ¼

1 p2 X 2

L ðtÞ

n2 jBn ðtÞj2

ð14Þ

n¼0

is time dependent. Other quantities of interest are the phase after a period

UðTÞ ¼

Z

T

dt

0

¼ p2

Z

LðtÞ

dx w ðx; tÞiot wðx; tÞ

0 1 X

n2

Z 0

n¼1

T

jBn ðtÞj2 2

L ðtÞ

dt þ 2i

1 X X 2

n¼1 k–n

Z

nk k

n2

T

gðtÞ½1 eip

2 ðk2 n2 Þf ðtÞ

0

Bn ðtÞBn ðtÞdt

ð15Þ

and the mean coordinate

XðtÞ ¼

Z

LðtÞ

xjwðx; tÞj2 dx ¼

0

0 1 4LðtÞ X mn 2 2 2 Re½eiðn m Þp f ðtÞ Bm ðtÞBn ðtÞ; LðtÞ þ 2 2 p m–n ðn2 m2 Þ2

ð16Þ

where the sum is conditioned by m þ n ¼ odd. 3. Examples In this paper, we will consider an approximation to the type of problems we are dealing with based on (8) and the truncation of the infinite matrix (11) to a finite order. In a certain sense, this is equivalent to substituting the initial periodic system by another one with only a finite number of modes. Let us consider now some specific examples. 3.1. A contracting wall In order to check our approximation, first we look briefly at the case of a uniformly contracting wall, LðtÞ ¼ 1 t, 0 6 t 6 1, for which the exact wave function is known [3]. For our calculations, we have used a 12 12 truncated matrix (11). The modulus of the first four coefficients Bn ðtÞ are shown in the two plots of Fig. 1. In addition, in the same figure, we have compared our results (solid lines) with those of Pinder [3] (dotted lines). The agreement on these coefficients is complete, indicating the excellent convergence in (8). 3.2. Simply periodic moving wall As a second example we consider a harmonic moving wall of the form

LðtÞ ¼ 2 þ sinðtÞ;

ð17Þ

for which

gðtÞ ¼

cosðtÞ ; 2 þ sinðtÞ

ð18Þ

1.0

0.02

B1

B3

0.6 0.01

0.2

B2 0.2

0.4

0.6

t 0.8

0

B4

t 0.1

0.2

0.3

0.4

0.5

Fig. 1. Values of jB1 ðtÞj to jB4 ðtÞj for LðtÞ ¼ 1 t. The solid lines correspond to the calculation carried out in the present work based on a truncation of (11) to a 12 12 matrix; the dots have been obtained after working out the exact solution of [3] with our initial condition Bk ð0Þ ¼ dk;1 .

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and the modified time coordinate f ðtÞ, given in Eq. (6), is shown in Fig. 2. If the particle is initially in the ground state, then we must solve (10) and (11) for the initial condition Bk ð0Þ ¼ dk;1 . If we truncate D to an 12 12 matrix, we find the values in Table 1. It thus appears that retaining only four terms in (8) will adequately represent the wave function, at least over a few periods of the wall motion. The behavior of the first four coefficients is shown in Fig. 3. We display the time dependence of the energy (14) in Fig. 4 and in the same plot show the ratio EðtÞ=1 ðtÞ. In spite of the fact that the numerator and denominator fluctuate over a wide range, their ratio varies very slowly. It seems that the system keeps mainly to the instantaneous ground state in an adiabatic way. we will comment later on this behavior. In Fig. 5, we display the modulus of the wave function, as a function of both the time t and the spatial coordinate x, and in Fig. 6, we plot the mean coordinate XðtÞ, as given in Eq. (16), and for comparison, the value of LðtÞ=2.

f(t)

7

5

3

1 π

3π

5π

7π

t

Fig. 2. Effective time coordinate f ðtÞ versus time t, for the simply periodic moving wall LðtÞ ¼ 2 þ sinðtÞ.

Table 1 Square of the modulus of the first coefficients Bn ðtÞ for the harmonic well (17) at two different times: t ¼ 0:7 and t ¼ 2p jBn ð0:7Þj2

n

jBn ð2pÞj2 6

1 2 3 4 5 6 7 8 9 10 11 12

971811:0 106 26726:5 106 13836:7 107 465:8 107 231:6 107 18:4 107 43:0 107 20:4 107 0:8 107 2:0 107 0:1 107 1:2 107

965134:0 10 33448:6 106 12787:2 107 1251:0 107 18:4 107 117:9 107 0:2 107 1:5 107 14:5 107 9:2 107 1:3 107 2:9 107

1.0

B1

0.6

B2

0.2

B3

π

2π

3π

4π

t

Fig. 3. From top to bottom, the first three wave function coefficients jBn ðtÞj versus t for the harmonic well (17).

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10

E(t)

6

2

π

2π

3π

4π

t

Fig. 4. In the upper curve the energy EðtÞ is plotted versus the time t; the lower curve is the ratio of the energy to

1 ðtÞ.

0 π 2π 2 1 0 0

t

1

x

2 3 2

Fig. 5. Wave function modulus jwðx; tÞj for the simply periodic moving wall.

1.6

1.2

0 .8 π

2π

4π

t

Fig. 6. The mean position of the particle XðtÞ (the wiggling curve) in the simply periodic moving wall as well as the instantaneous half-length LðtÞ=2 of the well.

3.3. An oscillating well with adjustable parameters There are three main features for an oscillating wall: minimum well-width, frequency and oscillation amplitude. By varying these one can examine the variety of behavior that the system can adopt. A convenient model, which exhibits the same sort of behavior as that in the preceding section and at the same time is analytically more tractable, is

A LðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ B cosðxtÞ

ð19Þ

with

pffiffiffi ab 2 A ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 a2 þ b

2

B¼

b a2 2

b þ a2

;

ð20Þ

where, by varying a and b, one can easily control the oscillation range ½a; b and amplitude. In this case, the function gðtÞ is given by

gðtÞ ¼

Bx sinðxtÞ ; 2 1 þ B cosðxtÞ

ð21Þ

2072


and the modified time coordinate f ðtÞ, given in Eq. (6), is simply

f ðtÞ ¼

xt þ b sinðxtÞ : x A2

ð22Þ

The system behavior changes drastically with frequency and shows three different regions. The frequency at which a shift between regions occurs depends also on the minimum well-width and the amplitude. In this paper, we have chosen the specific oscillation range of the wall to be ½10; 13. The frequencies selected are x1 ¼ 0:01, x2 ¼ 1, and x3 ¼ 1000, since the differences between them are so large that we only need to plot a few periods to enable a proper comparison. By plotting the first coefficients of the series approximation B1 ðtÞ; . . . ; B5 ðtÞ versus time, for the three chosen frequencies, we find the three types of behavior shown in Figs. 7–9. The plots in Fig. 7 shows a behavior, which we call adiabatic, characteristic of low frequencies, We call the behavior shown in Fig. 8, where periodicity is lost, chaotic, and note that for the high frequency region behavior on Fig. 9, regularity is recovered. We have observed that the range of frequencies leading to chaotic behavior starts when it is comparable to the frequency range corresponding to a stationary state. Notice that the terms B2 ðtÞ; . . . ; B5 ðtÞ are three or more orders of magnitude smaller than the first term B1 ðtÞ in the adiabatic region, which shows that the system remains essentially in the instantaneous ground eigenstate. In the other two regions, B2 ðtÞ and B3 ðtÞ are comparable to B1 ðtÞ and transitions between instantaneous eigenstates are possible. The transition

500

B1

1000

1500

2000

t

0.008

B2

0.999995 0.006 0.999985 0.004 0.999975

B3

0.002

B4

0.999965 500

1000

t

1500

2000

15

20

t

0.02

t

Fig. 7. jB1 ðtÞj (left) and jBn ðtÞj; n ¼ 2; 3; 4; 5 (right) versus t for x1 ¼ 0:01.

B1

5

10

15

20

B2

t 0.3

0.98

0.2 0.94 0.1

0.9

B3 B4 5

10

Fig. 8. jB1 ðtÞj (left) and jBn ðtÞj; n ¼ 2; 3; 4; 5 (right) versus t for x2 ¼ 1.

0.005

B1 0.98

0.96

0.01

0.015

0.02

t 0.25

B2

0.2 0.15

B3

0.1 0.94

B4

0.05 0.92

0.005

0.01

Fig. 9. jB1 ðtÞj (left) and jBn ðtÞj; n ¼ 2; 3; 4; 5 (right) versus t for x3 ¼ 1000.

0.015

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probabilities are proportional to the width of the oscillating spatial range. This is due to the fact that an appreciable variation of the well width modifies the instantaneous ground state energy and the minimum width energy can be big enough to reach the excited states of the maximum width situation. Figs. 10–12 display the instantaneous kinetic energy of the particle (left) and the mean position of the particle (right) for the frequencies representing the three regimes. In the high and low frequency cases, the mean energy is conserved as one expects for adiabatic behavior. However, in the chaotic regime, at least over the first few periods we have examined, the mean kinetic energy increases steadily. The mean position of the particle XðtÞ is very interesting because in the adiabatic case there is no appreciable difference with the mean value of the well width, LðtÞ=2, whereas the chaotic case shows a slight global periodicity and the fast frequency region varies much less with well width and with frequency. The probability amplitude plotted in Fig. 13 also exhibits how, in the adiabatic regime, the particle follows wall movement while for high frequencies the particle seems not to feel the boundary being confined to the minimum width region; for several periods the probability increases a little in the region of variation. Chaotic behavior becomes quickly apparent.

500

E(t)

1000

1500

t

2000

X(t)

L(t)/2

6.4

0.09

6 0.08

5.6 0.07

5.2 0.06

500

1000

1500

t

2000

Fig. 10. Kinetic energy EðtÞ (left) and mean position XðtÞ and LðtÞ=2 (right) for x1 ¼ 0:01.

0.6

E(t)

6.4

L(t)/2

6.2

0.5

6

0.4

5.8 0.3

5.6 0.2

5.4

0.1

5.2 5

10

15

20

X(t)

t

5

10

15

20

t

0.02

t

Fig. 11. Kinetic energy EðtÞ (left) and mean position XðtÞ and LðtÞ=2 (right) for x2 ¼ 1.

0.099

E(t)

L(t)/2 5.3

0.097

5.2

5.1 0.095

X(t) 0.005

0.01

0.015

0.02

0.005

0.01

0.015

Fig. 12. Kinetic energy EðtÞ (left) and mean position XðtÞ and LðtÞ=2 (right) for x3 ¼ 1000.

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0.4

0.4

0.4 2000

0.2

20 0.2

1500

0 0

1000 5

x

500 10 0

15 0 0

t

10

t

5 5

x 10 0

0.02

0.2

0.015

0 0

0.01

t

5

x

0.005 10 0

Fig. 13. Wave function modulus jwðx; tÞj vs t with t 2 f0; 3:5 T i g and x width for (i) x1 ¼ 0:01; (ii) x2 ¼ 1 and (iii) x3 ¼ 1000. Range of oscillation: LðtÞ 2 ½10; 13.

4. Conclusions In this note, we have set up a determinantal equation for determining the time-dependent wave function of a particle in an infinite square well with one moving wall. The particle is assumed to be in one of the stationary states at t ¼ 0. We have examined three cases: (i) A linearly contracting well, for which the exact solution is known [3] which provided a check on our approximation procedure; (ii) a harmonically oscillating well, whose solution identifies the features of the particle motion which we wish to describe; and (iii) a mathematically more tractable, for the relevant functions f ðtÞ and gðtÞ, non-harmonically oscillating well for which these features can be examined as the important parameters (minimum well-width, amplitude and frequency) are varied. Here, we examined variations of the frequency fixing the other parameters. The instantaneous ground state has an associated natural frequency x0 given by that of a nonrelativistic particle with the same mass and ground level energy. The results suggest that in the low frequency regime (compared to the frequency associated with the lowest stationary eigenstate, x x0 ) the motion can be described as adiabatic. As the frequency is increased, the system enters a chaotic regime, and at high frequencies ðx x0 Þ the motion again becomes periodic, but not adiabatic (i.e., it does not follow the wall motion). In a future work, we will approach this behavior using analytical approximations. Acknowledgement This work is supported by the Spanish MEC (project MTM2005–09183 and MLG Grant SAB2003–0117) and Junta de Castilla y León (Excellence Project GR224). MLG thanks the Universidad de Valladolid for hospitality and the NSF (USA) for partial support (DMR–0121146). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

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