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ScienceDirect Materials Today: Proceedings 5 (2018) 9094–9101

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NCNN 2017

Quantum bound states with interaction potential having linear combination of exponential functions and its possible applications Vinay Kumar Shuklaa*, and Ambrish Kumar Srivastavab a

Department of Physics, Indian Institute of Technology Kanpur, Kanpur-208016 b Department of Physics, University of Lucknow, Lucknow-226007

Abstract We study the bound states formed in a potential which can be expressed as linear combination of exponential functions. The potential described here features to transform from single well into double just by adjusting control parameters. We solve the Schrodinger wave equation by two different strategies viz. variational technique as well as numerically using shooting method and obtained a close agreement between calculated ground state energies. We extended our discussion to its possible applications in matter-wave interferometry and generation of matter waves. This potential is also found to be useful in the study of interaction between atoms in heterostructures. Finally, we suggest that one can produce finite number of matter waves as desired by using this potential. © 2017 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of 6th NATIONAL CONFERENCE ON NANOMATERIALS AND NANOTECHNOLOGY (NCNN VI - 2017 ).

Keywords: Bound states, Schrodinger equation, Variational method, Numerical solution.

*Email address: [email protected]

2214-7853 © 2017 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of 6th NATIONAL CONFERENCE ON NANOMATERIALS AND NANOTECHNOLOGY (NCNN VI - 2017 )

Vinay Kumar Shukla and Ambrish Kumar Srivastava / Materials Today: Proceedings 5 (2018) 9094–9101

1.

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Introduction

In the present era of technological advancement, various techniques have been developed to synthesize the thin films of materials at nanoscale. With reducing the dimensionality of the materials their various properties of such as magnetic, transport etc. gets modified significantly. However, in order to understand the physical properties of these systems, one has understand the experimental as well as theoretical aspects of materials with reduced dimensionality. In the present work, we suggest a new potential well which is a linear combination of exponential functions and that can be utilized in understanding the interface of thin films, heterostructures, metal-insulator-metal (M-I-M) junctions etc. We further estimate the ground state of this potential by conventional variational approach as well as the numerical techniques. The heart of Schrodinger wave equation (SWE) lies in a potential energy term. We are familiar with various interaction potentials such as harmonic oscillator, infinite well, finite well [1-3] etc. for which the SWE is exactly solvable. But for more complicated potentials exact solution of SWE is not easy and we have to invoke numerical techniques to solve it. Recently, the studies of the Gaussian wells and barriers employing the variational principle have been reported [4, 5]. Choice of the potential decides whether it gives bound states or scattering states. Interaction potentials having bound state solutions are interesting due to their strong practical applications in real systems involved in atomic, nuclear as well as condensed matter physics. In the present study, we have chosen a new type of potential which is linear combination of exponential functions described by-

V ( x)  V1 exp( x 2 )  V2 exp( x 2 )

(1)

Here V1 and V2 are real numbers. These two parameters provide us freedom to play with depth and shape of the potential. Consequently, it offers bound states which are of great interest in many physical phenomena. The SWE [1] for this potential takes the form-



1 d 2  [V1 exp( x 2 ) V 2exp( x 2 )]  E 2 dx 2

(2)

For simplicity, we have set that  = m = 1. We have solved above equation by two different methods viz. variational principle and numerical solution of SWE using shooting method. We obtained ground states for this potential and discussed some of its possible applications. 2.

Estimation of ground states by variational method

The straight forward approach to get the ground state energy for the potential given by eq. (1) is by solving eq. (2) through variational technique. The most suitable candidate for ground state trial wave function is simple node free Gaussian function of type-

 ( x)  A exp(bx 2 )

(3)

Where A is real normalization constant and b is the variational parameter. But since, single guassian trial wave function gives poor results for V2 3.0, we have bound states satisfying the condition, Vmin < Eg < Vmax. We can obtain corresponding ground state wavefunctions just by putting different values of variational parameter (b) for different sets of V1 and V2 in eq. (8). Figure 1 shows the nature of potential V(x) for different sets of V1 and V2. Apparently by changing control parameters V1 and V2, one can switch from harmonic oscillator like potential (fig. 1(a)) to double well like potential with extended boundaries (fig. 1(b), 1(c), 1(d)). In figure 2, we have shown maximum/minimum values of potential V(x) as well as ground state energy (Eg or Emin) versus b. One can clearly observe that with decreasing b, Eg switches from unbound or scattering state (Vmin < Vmax < Eg) to the bound state (Vmin < Eg < Vmax) at b ~ 0.66. It is also evident that Eg is close to Vmax indicating that the bound states are in just bound configuration. . Table1. Ground state energy Eg and control parameters in bound states V1

V2

Vmax

xmin

Vmin

b

Eg

Enum

0.2

2.8

3.0

±1.149

1.497

0.638

2.845

2.824

0.2

3.8

4.0

±1.213

1.743

0.621

3.592

3.408

0.2

4.8

5.0

±1.261

1.960

0.608

4.334

3.947

0.2

5.8

6.0

±1.298

2.153

0.597

5.073

4.442

0.2

6.8

7.0

±1.328

2.332

0.589

5.810

4.894


Fig. 1. The shape of interaction potential, V(x) for different values of V1 and V2. All quantities are in arbitrary units.

Fig. 2. The variation in V(x) or E with the variational parameter (b).

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3.

Numerical solution of Schrodinger wave equation by shootings method

The solutions associated with the one-dimensional SWE are of great importance in understanding quantum processes. For example, the energy levels and transport properties of electrons in nanostructures such as quantum wells, dots, and wires are crucial in the development of the next generation of electronic devices. We have employed so called the ‘shooting method’ in order to solve the eq. (2) numerically. This method is extensively used for eigenvalue and boundary value problems and provides very accurate results [6]. To apply the method, first we choose two turning points xl and xr such that E = V(x) as shown in Fig. 3. The basic idea is to start from region 1 (or region 3) and integrate through turning point xl (or xr) to region 2 (x = 0) which results in a wavefunction  l ( x) (or

 r ( x) ). The wavefunction and its derivative must be continuous, demanding in region 2,  l ( x)   r ( x)

(10)

 'l ( x)   'r ( x)

(11)

Combining Eq. (10) and Eq. (11), we have,

 'l ( x)  'r ( x)   l ( x)  r ( x)

(12)

Fig. 3. The turning points and various regions chosen for applying the shooting method. All quantities are in arbitrary units.

In shooting method, one has to adjust values of E for which above condition (eq. (12)) is satisfied. This can be done efficiently by any root finding algorithm. We have solved the SWE by this method using a java script described by Belloni et al [7]. The numerically calculated energy eigen-values (Enum) are also listed in table 1 and table 2. We have found that for scattering states, shooting method slightly overestimates energy eigen-values (table 1) while for bound states, numerically calculated values are smaller than those estimated by variational techniques (table 2), as expected due


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to the fact that the variational method provides only an upper bound of ground state energies (eq. (8)). Figure 4 plots energy eigen-functions in all four bound states for V1 = 0.2 and V2 = 18.8. The ground state eigen-function is symmetric with respect to centre of potential well and has maxima on either side with a node at the centre (fig. 4(a)). The first excited state, which is antisymmetric (fig. 4(b)), is very closely spaced with a small energy difference of 0.034 units with respect to the ground state. Similarly, the third excited state is only 0.962 units higher in energy than the second. On the other hand, it is interesting to note the large energy gap between first two excited states which is 7.739 eV. It implies that it can be possible for a particle to go to the first excited state by a slight perturbation in the well but for moving to the next higher excited state, it would require a sufficiently large amount of energy by any external source.

Fig. 4. Energy eigen-functions and corresponding eigen-values in four bound states for V1 = 0.2 and V2 = 18.8. All quantities are in arbitrary units.

4. Applications In this section, we shall discuss two important applications of interaction potential defined by eq. (1). 4.1. Matter-wave interferometry Matter wave interferometry(MWI) involves the interference phenomena which arises due to the wave nature of atoms. Since, atoms are strongly affected by the gravitational forces therefore MWI creates a possibility of detecting the gravitational waves. In the present time, MWI is widely used in atomic molecular physics, metrology etc. One can ask what is basic criterion to obtain MWI? The answer to this is that the quantum evolution of matter waves should be phase preserving. Let me correlate this with the interaction potential which we have chosenConsider the case for which V1 =V2 =Vo that means V(x) looks like harmonic oscillator with extended boundaries as mentioned earlier, see fig. 1(a). Suppose, we fill some Bose-Einstein condensate (BEC) and boundaries of this potential is subjected to external frequency source, say radio or microwave, then vibrations or waves which arises in BEC are matter waves whose origin is wave nature of atoms. This is similar to the case discussed by Schumm et al [8]. Now, suppose at this stage we start varying the control parameters V1 and V2 as shown in fig. 1(b), 1(c), 1(d)) then BEC inside the well gets distributed into two, without exciting. That means the harmonic oscillator type potential changes to double well shaped potential making BEC to distribute itself in the two branches of the well.

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This phenomenon is analogous to the beam splitting phenomenon of optics [9]. Now, similar matter waves can be generated in two branches of the double well type potential. One can generate in-phase as well as out of phase waves in these two branches. These matter waves can interfere constructively or destructively depending upon the nature of waves (in-phase or out of phase) through the phenomenon of quantum mechanical tunnelling as shown in fig.5. But to achieve MWI the splitting of single well into double should not affect the phase of the matter waves and one needs the constructive interference of these waves. This demand can be fulfilled by our chosen interaction potential whose control parameters can be tuned to get the desired phase and splitting distance. To measure the phase accurately, one can construct MWI on atom chips as discussed by Schumm et al.[8].Thus, this creates a possibility to construct a matter wave interferometer invoking this potential. However, the potential we discussed is one dimensional in nature but it can be easily generalized to three dimensions as, V (r) = V1 exp (r2) + V2 exp (-r2) which can be treated in similar fashion but calculations involved here becomes more complex. One can generate finite or desired number of matter waves by choosing appropriate values of control parameters V1 and V2.

Fig. 5. Oscillatory wave propagation in two branches A and C of the double well. Region B indicates quantum mechanical tunnelling of waves from branch A to C of the double well. All quantities are in arbitrary units.

4.2. Study of heterostructures When we talk about heterostructures in which a layer of material B is sandwiched between two layers of different material A as shown in fig.6, intermediate layer offers a slightly different potential than two side layers depending on the choice of materials. For some materials, there is possibility that resultant potential across the layers of heterostructure can be approximated by our chosen potential V(x). If one closely observes the shape of this potential then it can be interpreted as front-back combination of Lennard-Jones potential [10]. In more simple words, one can consider that there is a mirror at x = 0 plane, left side of which has Lennard-Jones potential and right side is the mirror image of left as shown in fig.6. The x = 0 plane in that case could correspond to heterostructure interface or bilayer interface. We know that Lennard-Jones potential is widely used to study the interaction between the two atoms. Therefore, this potential could be useful to study the interaction between atoms in heterostructures having slightly different potential at the interface.


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Fig. 6. Heterostructure formed by two different materials A and B. Layer of B is sandwiched between two layers of A, leading to different interaction potentials whose resultant is approximated by our chosen potential V(x)). All quantities are in arbitrary units.

5.

Conclusions

In summary, we have solved the Schrodinger equation variationally as well as numerically which provides beautiful results and fruitful applications with suitable choice of the potential. The interaction potential, we chosen, has finite number of bound states. Apart from its applications in matter wave interferometry and study of heterostructures, it can also be used as an analogue to the double well potential. Inverting this potential well will give rise to our known harmonic oscillator type potential but will possess only finite number of bound states. Finite or desired number of bound states can open new opportunities and find new applications. Acknowledgements Authors acknowledge Council of Scientific and Industrial Research (CSIR), New Delhi (India) for financial support in form of Junior Research Fellowships. Authors are also thankful to Prof. Neeraj Misra for valuable suggestions. References: [1] D.J. Griffiths Introduction to Quantum Mechanics 2nd ed. (Prentice Hall, USA), 2004. [2] E. Merzbacher Quantum Mechanics 3rd ed. (Wiley, NY), 1998. [3] R. Shankar Principles of Quantum Mechanics 2nd ed. (Springer, USA), 1994. [4] S. Nandi, Am. J. Phys. 78 (2010), 1341-1345. [5] F.M. Fernández Am. J. Phys. 79 (2011), 752. [6] T. Pang An Introduction to Computational Physics (Cambridge, UK), 2006. [7] M. Belloni, W. Christian and A.J. Cox, Physlet Quantum Physics- An Interactive Introduction (Pearson, USA), 2006. [8] T. Schumm , S.Hofferberth , L.M. Andersson, S. Wildermuth, S. Groth , I.B. Joseph, J. Schmiedmayer and P. Krüger, Nature Physics 1 (2005), 57-62. [9] R.H. Brown and R.Q. Twiss Nature 177(1956), 27-29. [10] J.E. Lennard-Jones 1931 Proc. Phys. Soc. 43 (1931), 461.