Energy in a finite two dimensional spinless electron gas

Energy in a finite two dimensional spinless electron gas Energy in a finite two dimensional spinless electron gas Orion Ciftja, Bradley Sutton, and Ashley Way Citation AIP Advances 3, 052110 (2013); d[.]

Energy in a finite two-dimensional spinless electron gas

Orion Ciftja, Bradley Sutton, and Ashley Way

Citation: AIP Advances 3, 052110 (2013); doi: 10.1063/1.4804933

View online: http://dx.doi.org/10.1063/1.4804933

View Table of Contents: http://aip.scitation.org/toc/adv/3/5

Published by the American Institute of Physics

Energy in a finite two-dimensional spinless electron gas

Orion Ciftja, Bradley Sutton, and Ashley Way

Department of Physics, Prairie View A&M University, Prairie View, Texas 77446, USA

(Received 13 March 2013; accepted 29 April 2013; published online 8 May 2013)

We study the properties of a finite two-dimensional electron gas system in the Hartree-Fock approximation We obtain exact analytical expressions for the energy in a fi-nite two-dimensional fully spin-polarized (spinless) system of electrons interacting with a Coulomb potential immersed in a finite square region uniformly filled with

a neutralizing positive charge The difficult two-electron integrals over the finite square domain are reduced to simple compact expressions involving analytic auxil-iary functions We provide results for the potential energy of systems with a finite number of electrons and show how the energy slowly converges towards its ther-modynamic limit bulk value C 2013 Author(s) All article content, except where

otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4804933]

I INTRODUCTION

The jellium model in condensed matter physics is a very common model for treating systems of interacting electrons The basic idea behind the model is to consider the whole system as a uniform gas of electrons moving in a background of uniform positive charge (i.e the positive ions) The jellium model has been adopted with great success to study the properties of a three-dimensional electron gas (3DEG), as well as of a two-dimensional electron gas (2DEG) In both cases, the quantum mechanical treatment of the problem starts with a Hamiltonian of the form:

ˆ

H = ˆT + ˆV ee + ˆV eb + ˆV bb , (1) where ˆT is the kinetic energy operator for the electrons, ˆ V eeis the electron-electron Coulomb interac-tion potential, ˆV ebis the electron-background Coulomb interaction term and ˆV bbis the background-background interaction energy, namely, the Coulomb self-energy of the uniform background-background charge

An infinite 3DEG or 2DEG is uniquely characterized by its electron number density,ρ0= N/ which is a single constant number in the thermodynamic limit of N → ∞ and  → ∞ (where

 represents volume or area, respectively, for a 3DEG or a 2DEG) A lot of work for the case of

infinite systems has been dedicated to the calculation of the energy (both kinetic and potential) and its dependence on the density A variety of numerical calculations, mostly based on quantum Monte Carlo methods, have fairly well established the thermodynamic properties of a 3DEG and 2DEG for

a wide range of densities.1 6The most straighforward theoretical treatment of such systems involves use of an anti-symmetrized Slater determinant wave function of plane waves7as starting point Such

an approach constitutes the underlying idea of the Hartree-Fock (HF) approximation (See pg 332 of Ref.8, pg 69 of Ref.9or pg 21 of Ref.10), therefore, we refer to both treatments as the HF method Within this framework, the energy of an infinite system can be calculated exactly and consists of

a kinetic and a potential (exchange) energy term Obviously, the two HF energy terms are just the leading ones in the perturbation series The other higher order energy terms are collectively refered

to as the correlation energy and are much more difficult to obtain.11 – 14

Differently from the earlier focus of research on the properties of an electron gas in the ther-modynamic limit, the rapid growth of nanoscience has stimulated increased interest in the opposite direction, namely, in the quantum treatment of finite system of particles at low dimensionalities (mostly in 2D) Therefore, it is of current interest to study the behavior of a finite 2DEG model

where N electrons move freely in a finite 2D square region Earlier studies of small systems in the

2158-3226/2013/3(5)/052110/9 3, 052110-1 Author(s) 2013

052110-2 Ciftja, Sutton, and Way AIP Advances 3, 052110 (2013)

3D case indicate that even the simplest of such finite systems, namely, a system of N= 2 electrons confined to a box interacting with a Coulomb potential, is quite challenging and provides very inter-esting results.15Based on these considerations, in this work, we study the properties of a finite system

of N electrons confined within a finite uniformly charged 2D square domain, namely, a finite 2DEG.

The main objective of this work is to provide exact analytical expressions in the HF approximation

for the energy of a finite 2DEG at arbitrary values of N Since expressions for the kinetic energy

are easy to obtain for both finite and infinite systems, the main thrust of the work is to obtain exact

analytic expressions for the potential energy of a finite 2DEG of N of electrons For simplicity, we

consider only a fully spin-polarized (spinless) system of electrons since the generalization to a two species of spins is obvious

II A FINITE FULLY SPIN-POLARIZED 2DEG

A finite 2DEG model consists of N electrons moving in a uniform background charge that fills

a finite 2D square region with area A = L2 The electron number density in this case is written as:

ρ0 = N

We assume that we are dealing with a fully spin-polarized (spinless) system of electrons described

by the Hamiltonian:

ˆ

where

ˆ

T =

N



i=1

ˆ

p i2

is the kinetic energy operator and

ˆ

is the potential energy operator that represents, respectively, the electron, electron-background and electron-background-electron-background interaction energies The electron-electron interaction potential is written as:

ˆ

V ee= 1 2

N



i=1

N



j =i

ˆv(|r i − r j |), (6)

where ˆv( |r i − r j |) = k e e2/r i j is the Coulomb interaction potential between two electrons with charge,−e (e > 0) separated by distance r i j = |r i − r j | and k eis Coulomb’s electric constant The other two potential energy terms are, respectively:

ˆ

V eb = −ρ0

N



i=1



A

d2rˆv(|r i − r|), (7) and

ˆ

V bb= ρ2 2



A

d2r1



A

d2r2 ˆv( |r1− r2|). (8) The vector r in Eq. (7) represents the continuous background coordinate while vector r i is the

position of the i-th electron In a short-hand notation O = | ˆO|/| represents the quantum

expectation value of operator ˆO with respect to wave function, | One can prove that the expectation

value of ˆV eb , namely V eb, can be written as:

V eb = −ρ0



d2r1ρ(r1)



whereρ(r1) in the above expression is the one-particle density function quite generally defined as:

ρ(r1)= N



d2r2· · ·d2r N ||2

where | is the given wave function In the HF approximation, a normalized N-particle wave

function that is properly antisymmetric and is an eigenstate of the many-particle free Hamiltonian

is given by a Slater determinant wave function:

 = √1

N ! Det {ψ α1(r1), , ψ α N(r N)}, (11) where the single-particle states are assumed to be ortho-normalized Note that the prefactor multi-plying the Slater determinant is suitably chosen to normalize the overall wave function,| = 1.

In this aproach, the electrons are seen as free particles moving in a 2D square region If one imposes periodic boundary conditions (PBC) in both directions, the single-particle states of electrons are 2D plane waves:

ψ k r) = √1

where the allowed values of wave vector k = (k x , k y) are:

⎧

⎨

⎩

k x =2π

L x n x; n x = 0, ±1, ±2,

k y=2π

L y n y; n y = 0, ±1, ±2, (13)

If all k-s satisfy the PBC, then clearly the plane wave functions satisfy the ortho-normality condition.

For the case of a Slater determinant wave function of ortho-normalized plane wave orbitals, the one-particle density function reads:

ρ(r) =

N



j=1

The electron-electron potential energy term can be written in compact form as:

V ee = 1 2

N



i=1

N



j =i

[i j|ˆv|i j − i j|ˆv| j i], (15) where in “bra-ket” notation:

i j|ˆv|i j =



A

d2r1



A

d2r2ψ i(r1) ψ j(r2) ˆv(r12)ψ i(r1)ψ j(r2) (16) The two integrals,i j|ˆv|i j and i j|ˆv| j i in Eq.(15)are called, respectively, the Coulomb and

exchange integrals Note that since the Coulomb and exchange integrals cancel each other for i = j the restriction j = i can be dropped from Eq.(15)and, thus, we can write:

V ee =1 2

N



i=1

N



j=1

[i j|ˆv|i j − i j|ˆv| j i]. (17)

By putting together all terms, one can write the total potential energy of a finite fully spin-polarized 2DEG as:

U = 1 2



A

d2r1



A

d2r2



ρ(r1)− ρ0

ˆv(r12)



ρ(r2)− ρ0

−1 2

N



i=1

N



j=1

052110-4 Ciftja, Sutton, and Way AIP Advances 3, 052110 (2013)

TABLE I Roman-numbered shells in increasing order of energy, E n x ,n y =  2

2 m

2π L

2

(n2+ n2), where N sis number of

states for each shell (degeneracy), N is the total number of particles (for a spin-polarized system) and (n x , n y) are the corresponding quantum states.

Shell n2+ n2 N s N (n x , n y)

II 1 4 5 ( − 1, 0), (0, −1), (0, 1), (1, 0)

III 2 4 9 ( − 1, −1), ( − 1, 1), (1, −1), (1, 1)

IV 4 4 13 ( − 2, 0), (0, −2), (0, 2), (2, 0)

V 5 8 21 ( − 2, −1), ( − 2, 1), ( − 1, −2), ( − 1, 2), (1, −2), (1, 2), (2, −1), (2, 1)

VI 8 4 25 ( − 2, −2), ( − 2, 2), (2, −2), (2, 2)

VII 9 4 29 ( − 3, 0), (0, −3), (0, 3), (3, 0)

VIII 10 8 37 ( − 3, −1), ( − 3, 1), ( − 1, −3), ( − 1, 3), (1, −3), (1, 3), (3, −1), (3, 1)

IX 13 8 45 ( − 3, −2), ( − 3, 2), ( − 2, −3), ( − 2, 3), (2, −3), (2, 3), (3, −2), (3, 2)

X 16 4 49 ( − 4, 0), (0, −4), (0, 4), (4, 0)

XI 17 8 57 ( − 4, −1), ( − 4, 1), ( − 1, −4), ( − 1, 4), (1, −4), (1, 4), (4, −1), (4, 1) XII 18 4 61 ( − 3, −3), ( − 3, 3), (3, −3), (3, 3)

XIII 20 8 69 ( − 4, −2), ( − 4, 2), ( − 2, −4), ( − 2, 4), (2, −4), (2, 4), (4, −2), (4, 2) XIV 25 12 81 ( − 5, 0), (0, −5), (0, 5), (5, 0), (−4, −3), (− 4, 3), (− 3, −4), (−3, 4), (3,

−4), (3, 4), (4, −3), (4, 3)

XV 26 8 89 ( − 5, −1), ( − 5, 1), ( − 1, −5), ( − 1, 5), (1, −5), (1, 5), (5, −1), (5, 1) XVI 29 8 97 ( − 5, −2), ( − 5, 2), ( − 2, −5), ( − 2, 5), (2, −5), (2, 5), (5, −2), (5, 2) XVII 32 4 101 ( − 4, −4), ( − 4, 4), (4, −4), (4, 4)

XVIII 34 8 109 ( − 5, −3), ( − 5, 3), ( − 3, −5), ( − 3, 5), (3, −5), (3, 5), (5, −3), (5, 3) XIX 36 4 113 ( − 6, 0), (0, −6), (0, 6), (6, 0)

XX 37 8 121 ( − 6, −1), ( − 6, 1), ( − 1, −6), ( − 1, 6), (1, −6), (1, 6), (6, −1), (6, 1)

At this juncture, we point out the main difficulty faced when one has to deal with Eq.(18), namely, the mathematical challenge of calculating difficult four-dimensional integrals over a finite domain

III ENERGY OF A FINITE FULLY SPIN-POLARIZED 2DEG

For a Slater determinant wave function of plane waves, the allowed kinetic energies of an

electron in a 2D square box with length L are:

E n x ,n y = 2

2 m

2π

L

2

n2x + n2

y

; n x ,y = 0, ±1, ±2, (19) Almost all of the quantum states (with the exception of the ground state) are degenerate Therefore, different quantum states have the same energy values and, thus, belong to the same energy shell

The number of different states, N sfor a given energy shell represents the degeneracy of that energy value We list in Table.Isome shells in increasing order of energy (roman-numbered), with their

degeneracies and the respective quantum states, (n x , n y) that belong to that shell For a homogeneous electron gas, the electron density of plane-wave basis orbitals is uniform throughout the system and equals the density of the positive background, ρ(r) = ρ0 Since these equal and opposite charge densities cancel out each other exactly it is easy to see that:

1 2



A

d2r1



A

d2r2



ρ(r1)− ρ0

ˆv(r12)



ρ(r2)− ρ0

As a result, Eq.(18)reduces to the term:

U = −1 2

N



i=1

N



j=1

that represents the exchange interaction energy alone One can explicitely write the quantity in Eq (21)as:

U= −1 2



A

d2r1



A

d2r2|ρ(r1, r2)|2ˆv(r12), (22) where

ρ(r1, r2)=

N



j=1

ψ k j(r1) ψ k j(r2)= 1

A

N



j=1

e −i k j(r1−r2 ), (23) represents the one-particle density matrix for this case For 2D plane wave orbitals, one has:

|ρ(r1, r2)|2= 1

A2

N



i=1

N



j=1

e i ( k i −k j) (r 1−r2 ). (24)

By substituting Eq.(24)into Eq.(22)one obtains:

U = −k e e2

2

1

A2

N



i=1

N



j=1



A

d2r1



A

d2r2

e i ( k i −k j) (r 1−r2 )

|r1− r2| . (25)

A quick look of the above expression indicates the main challenge to the calculation represented by

a finite 2D square integration domain with area A = L2 An exact analytic calculation of the potential energy in Eq.(25)hinges upon the calculation of the following difficult two-electron integral:

I ( k)=



A =L2

d2r1



A =L2

d2r2

e i  k ( r1−r2 )

|r1− r2|, (26) where for simplicity we denoted k = k i − k j Obviously, an exact calculation of I ( k) by direct

integration techniques is not possible and poses great challenges Nevertheless, we show in this work that a suitable mathematical transformation to new variables, does allow one to simplify considerably all the calculations An important result that we will derive is that all expressions can

be reduced to one-dimensional closed-form integrals that can be readily handled The details of the

mathematical approach used to calculate I ( k) are shown in the Appendix.

IV RESULTS AND DISCUSSION

We consider a finite fully spin-polarized (spinless) 2DEG system of N electrons where the

number of electrons corresponds to closed shells (thus, the system has zero total linear momentum):

N = 5, 9, 13, 21, 25, (27)

Even though one can handle any value of N, in the interest of brevity, we present data only for a selection of up to N= 45 electrons By starting from Eq.(25)and using the results of the Appendix, one can can write the total potential energy of the system as:

U = −k e e2√1

π

1

L

N



i=1

N



j=1

R(q i x − q j x , q i y − q j y), (28)

where R(a, b) is an auxiliary function defined as:

R(a, b) =

 ∞

0

The functions, F(t, a) and F(t, b) are explicitely given in the Appendix Because of quantum

considerations (sinceq = k L), the values of a and b are: a = 0, ±2 π, ±4 π, and b = 0, ±2 π,

±4 π, Obviously, the total potential energy of the system is an intensive property which scales exactly with the area (thus, it scales with the number N of electrons for fixed density ρ0) Therefore, it

is convenient to calculate the potential energy per electron, U/N even for a finite number of electrons.

052110-6 Ciftja, Sutton, and Way AIP Advances 3, 052110 (2013)

0.8 0.85

0.9 0.95

1 1.05

1.1

N

FIG 1 Potential energy per particle measured relative to its bulk (thermodynamic) value, u(N)/u(N→ ∞) as a function of

N for a finite fully spin-polarized 2DEG system.

Since one may want to compare the potential energy per electron for a finite system to the bulk value (in the thermodynamic limit) one uses the fact that 1/L =√ρ0/N to write the potential energy per

particle as:

u(N )= U

N = −k e e2 √ρ0

√

π N3/2

N



i=1

N



j=1

R(q i x − q j x , q i y − q j y) (30)

This result is to be compared with the exact thermodynamic (bulk) value of the potential energy per particle:

u(N → ∞) = lim

N→∞

U

N = −k e e2√ρ

0

8

3√

which implies that, for a sufficiently large system, we should have:

lim

N→∞

1

N3/2

N



i=1

N



j=1

R(q i x − q j x , q i y − q j y)=8

In Fig.1we show the dependence of the finite potential energy per electron, u(N) as a function of

N for finite systems of N fully spin-polarized electrons The finite potential energies are measured relative to the corresponding thermodynamic value, u(N→ ∞) given in Eq.(31) When N is finite,

one expects the potential energy, u(N) to be different from the bulk value and match this value exactly only in the thermodynamic limit (N→ ∞) Our results show that the difference in energy between

finite and bulk values is only about 5% even for systems of N= 45 electrons This fact indicates that

u(N) approaches rather quickly its u(N→ ∞) asymptotic value One also notes that the dependence

of u(N) as a function of N is not a monotonic one A hint of such non-monotonic dependence also

comes from calculations of the kinetic energy for a finite fully spin-polarized 2DEG system In such

a case, we obtained:

(N)

where (N) = T/N is the kinetic energy per particle for a finite system with N particles while

(N → ∞) =  2

2π ρ0 is the kinetic energy per particle for an infinite 2DEG system (in

0.9 0.925

0.95 0.975

1 1.025

1.05 1.075

1.1

N

FIG 2 Kinetic energy per particle measured relative to its bulk (thermodynamic) value,(N)/(N → ∞) as a function of N

for a finite fully spin-polarized 2DEG with N= 5, 9, , 69 electrons.

the thermodynamic limit) The parameter C(N) can be calculated for any given N A plot of

(N)/(N → ∞) as a function of N is shown in Fig. 2 Clearly, one cannot fail to notice the

non-monotonic variation of kinetic energy as a function of N with C(N) that changes from values larger than 1 (for N = 5) to values smaller than 1 (for N = 9), and so on The non-monotonic dependence of both potential and kinetic energies as a function of N for a finite fully spin-polarized

2DEG system is in stark contrast with what happens to a 2DEG of electrons in a perpendicular

magnetic field For electrons in a 2D disk geometry, the dependence of U/N as a function of N is

monotonic for both the integer16and the fractional quantum Hall regime.17 , 18

To conclude, in this work we considered a finite fully spin-polarized 2DEG system where a small number of electrons is confined in a square jellium background We calculated exactly the potential energy for finite 2D systems of spinless electrons by using a Slater determinant wave function

of plane waves, namely, we applied a treatment equivalent to HF theory for a finite system We developed closed-form expressions for the potential energy in terms of analytic auxiliary functions Exact analytic expressions like the one developed here will allow us to gauge and predict the properties of larger though still finite systems of electrons As number of electrons in the system increases, both the potential and kinetic energy per particle slowly converge toward the limiting bulk value, though such a convergence is not a monotonic one Since the convergence toward the bulk value is non-monotonic, one must be very careful to extrapolate bulk results from calculations of finite systems The results derived here can be useful to gauge the accuracy of various numerical and computational techniques used in studies of electronic systems

ACKNOWLEDGMENT

This research was supported in part by NSF Grant No DMR-1104795

APPENDIX: CALCULATION OF I(  k )

The calculation of:

I ( k)=



A =L2

d2r1



A =L2

d2r2

e i  k ( r1−r2 )

|r1− r2|, (A1)

052110-8 Ciftja, Sutton, and Way AIP Advances 3, 052110 (2013)

involves a non-trivial four-dimensional integration over a finite 2D square domain with area

A = L2 If we choose a Cartesian system of coordinates with origin at the center of the square,

the domain of integration is A : −L

2 ≤ x i ≤ +L

2 ; −L

2 ≤ y i ≤ +L

2 where i= 1, 2 The following transformation19is introduced to simplify the calculations:

1

|r1− r2| =

2

√

π

 ∞

0

du e −u2(r1−r2 ) 2

By substituting Eq.(A2)into Eq.(A1)one obtains:

I ( k)= √2

π

 ∞

0

du



A =L2

d2r1



A =L2

d2r2 e i  k ( r1−r2 )−u2 (r1−r2 ) 2

The next step in the calculation is to group the variables x1and x2 together but separately from y1

and y2 We also introduce dimensionless variables:

X i = x i

L; Y i = y i

L; t = u L; q = k L. (A4) After some algebra, one writes the quantity in Eq (A3)as a one-dimensional integral involving auxiliary functions:

I ( q) = 2 L√π3

 ∞

0

dt F (t , q x ) F (t , q y), (A5) where

F (t, q x)=

 +1/2

−1/2 d X1

 +1/2

−1/2 d X2e

i q x (X1−X2 )−t 2(X1−X2 ) 2

and

F (t, q y)=

 +1/2

−1/2 dY1

 +1/2

−1/2 dY2e

i q y (Y1−Y2 )−t 2(Y1−Y2 ) 2

At this point, one sees clearly that the following function:

F (t, a) =

 +1/2

−1/2 d x

 +1/2

−1/2 d y e

i a (x −y)−t2(x−y)2

is a key player in all calculations When considering the function F(t, a) one must remember that the parameter a stands for either q x or q y Therefore, a should be treated as a real parameter (positive, negative or zero) Similarly, t is also a real but non-negative variable One can calculate that for

a= 0 one has:

F (t , a = 0) = −1 + e −t

2 +√π t er f (t)

where

er f (z)=√2

π

 z

0

is an error function After some lengthy and tedious calculations one can also derive an exact

expression for F(t, a) which reads:

F (t , a) = −1

t2 +e −i a−t

2

+ e i a −t2

2 t2 −a

√

π e−a2

4 t2

4 t3

×er f i

a

2 t − i t+ er f ia

2 t + i t

+i

√

π e−4 t2 a2

2 t ×er f i

a

2 t − i t− er f i a

2 t + i t

+a

√

π e−a2

4 t2

2 t3 er f i

a

2 t



er f i (z)= √2π

 z

0

is an imaginary error function, erfi (z) = erf (i z)/i By looking at the above expressions, one easily sees that F(t, a) is an even function of a:

Several interesting exact results can be obtained by using Eq.(A5)as a starting basis For example, one can calculate that:

I (q = 0) = 2 L3



2 (1−√2)

3 + 2 ln1+√2



At this point, we recall thatρ2

0e2L3 = Q2/L, where Q = N e is the total charge of the background.

Given that the Coulomb self-energy of the background can be written as:

V bb=k e ρ2

0e2

one immediately obtains the following exact result for the Coulomb self-energy of a uniformly charged 2D square:

V bb= k e Q2

L



2 (1−√2)

3 + 2 ln1+√2



in agreement with earlier calculations.20

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a finite 2D square integration domain with area A = L2 An exact analytic calculation of the potential energy. .. Ciftja, Sutton, and Way AIP Advances 3, 052110 (2013)

involves a non-trivial four -dimensional integration over a finite 2D square domain with area

A. .. number of electrons is confined in a square jellium background We calculated exactly the potential energy for finite 2D systems of spinless electrons by using a Slater determinant wave function