Staggered Ladder Spectra

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Karlstads universitet 651 88 Karlstad Tfn 054-700 10 00 Fax 054-700 14 60 Faculty of Technology and Science

Joakim Strömwall

Staggered Ladder Spectra

Physics

Degree Project of 15 ECTS, Bachelor Level

Date/Term: 2012-05-15 Supervisor: Jürgen Fuchs Examiner: Marcus Berg

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Abstract

We discuss aspects of a quantum mechanical system which has a staggered ladder spectrum. The raising and lowering operators of this spectrum are presented and calculated. The system we consider arises from a rewrit- ing of a Fokker-Planck equation, which consists in constructing a generalized form of an Ornstein-Uhlenbeck system. From this Fokker-Planck equation we define a Hamiltonian operator, for which we study the eigenvalue problem. As we solve this eigenvalue problem, even and odd eigenfunctions are obtained with respective even and odd eigenvalues, which for given parity are equidistant, but for different parities are staggered.

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Contents

1 Introduction 4

2 Algebraic properties of quantum mechanics 6

2.1 The quantum harmonic oscillator . . . . 6

2.2 Properties of Hermite polynomials . . . 10

2.3 Raising and lowering operators . . . 11

2.4 The bracket notation . . . 12

3 Ornstein-Uhlenbeck processes and the Fokker-Planck equation 13 3.1 Statistics of Brownian motion . . . 13

3.2 Correlation functions . . . 13

3.3 Ornstein-Uhlenbeck processes . . . 14

3.4 The Fokker-Planck equation . . . 16

4 Staggered ladder spectra 20 4.1 Eigenvalues and eigenfunctions . . . 20

4.2 Propagators . . . 24

4.3 Commutators . . . 27

5 Conclusion 29

I Appendix A 30

II Appendix B 33

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1 Introduction

In the first quarter of the twentieth century, the formulation of quantum mechanics was developed by W. Heisenberg, E. Schr¨odinger, P.A.M. Dirac, among others [5]. In 1925 Dirac derived the transition from classical mechanics of Poisson brackets to quantum mechanics as

[ ˆA, ˆB]_cl → [ ˆA, ˆB]_qm

i~ , (1.1)

for operators ˆA and ˆB. This gave rise to the construction of the quantum mechanical commutator algebra. Related to this, ladder operators where introduced, probably by Dirac. In an article by L.S. Ornstein and G.E. Uhlenbeck [8], published in 1930, an application of the statistical motion of quantum mechanical particles was given, taking the equation of motion as

˙

p = −γp + f (t), (1.2)

for the momentum p, damping coefficient γ and a rapidly fluctuating force f (t).

The description of the processes made in their paper “On the Theory of the Brownian Motion” has later come to be refered to as Ornstein-Uhlenbeck processes. In recent years (2006-2007), articles have been published by Arvedson et al. [1, 6, 9, 10] approaching a generalization of the Ornstein-Uhlenbeck processes for the equation of motion

˙

p = −γp + f (x, t), (1.3)

which eigenvalue problem for certain conditions gives rise to eigenvalues consti- tuting staggered ladder spectra.

In this bachelor-level thesis, we construct a staggered ladder spectrum and calculate transitions between eigenvalues of the same parity (even-even or odd- odd) by commutator algebra.

We first consider the one-dimensional quantum harmonic oscillator, for which an energy spectrum with equidistant energy levels is obtained. Raising and lowering operators for this case is presented. This spectrum with operators is illustrated in Figure 1. Introducing the statistics of Brownian motion, we consider an ensemble of particles for the Ornstein-Uhlenbeck process in its regular form (1.2). In this system, the probability distributions for momentum and position are calculated. Generalizing the Ornstein-Uhlenbeck process as (1.3), the Fokker-Planck equation is introduced, providing a description of the process. First in its general form, then derived into the so called generic case (for which we have a diffusion constant D(p) ∼ |z|⁻¹), with a dimensionless form of the Fokker-Planck operator F , which is given in Hermitian form. This equation is expressed in terms of theˆ momentum probability density P . Taking P₀ ∝ e^−|z|³^/3, we define the Hamiltonian operator ˆH for our generalized system. By solving the eigenvalue problem for this Hamiltonian, we obtain eigenvalues for even and odd parts of the respective eigenfunctions. Their respective eigenvalues constitute a staggered ladder spectrum, which is illustrated in Figure 2. An exact solution is calculated to the Fokker- Planck equation, using the propagator for the obtained Hamiltonian. Raising and

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lowering operators are introduced and their acting in the spectrum is calculated by commutator algebra.

Section 2 consists of the basic algebraic properties of quantum mechanics given in standard books on quantum mechanics, e.g. [2, 3, 5]. Section 3 introduces non- standard quantum mechanics of some special systems presented in e.g. [1, 6, 7, 8, 9, 10]. In Section 4 a special solution is presented to one of these systems. This is described in [1, 6, 9]. Conclusions and recommended further studies are given in Section 5. Finally, commutator algebraic calculations are presented in Appendix A and some additional calculations are given in Appendix B.

In this bachelor-level thesis, I have not added any new information to the field of research. My contribution in this paper has been to fill in calculations and performing derivations, verifying already existing results. The main goal has been to perform the derivations and calculations validating the staggered ladder spectra obtained in [1], for the generic case. I also mention here that some special polynomials and functions will be used in deriving some results. These results do not require any explicit calculations of these polynomials/functions, so they have not been described in detail. These are: the Hermite polynomials, the associated Laguerre polynomials and the modified Bessel functions.

I would like to thank my supervisor J¨urgen Fuchs for interesting discussions throughout the making of this paper.

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2 Algebraic properties of quantum mechanics

2.1 The quantum harmonic oscillator

Here, the one-dimensional quantum harmonic oscillator will be considered, to- gether with solutions of the form of hypergeometric functions, which are expressed in series. This section will mainly follow [3], with some parts taken from [2].

Consider the one-dimensional harmonic oscillator, calling the coordinate x. For a mass point with mass m moving in a central force potential described by Hooke’s law F = −kx², where k = mω² is the constant of the force and ω is the oscillation frequency, the potential is given by

V (x) = 1

2kx² = 1

2mω²x². (2.1)

The total energy can be described by the Hamilton operator, H = T + V , with the kinetic energy T given by T = _2m^p^ˆ² for the quantum mechanical momentum operator ˆp = −i~_dx^d. Thus, we get the Hamiltonian

H = −ˆ ~² 2m

d² dx² +1

2mω²x², (2.2)

which allows us to express the Schr¨odinger eigenvalue equation as

− ~² 2m

d²ψ(x) dx² +1

2mω²x²ψ(x) = Eψ(x), (2.3) where ψ(x) is the wave eigenfunction for the mass point and E is the total energy.

To solve this differential equation, we begin by putting all terms on the same side of the equality:

~² 2m

d²ψ(x)

dx² + Eψ(x) − 1

2mω²x²ψ(x) = 0. (2.4) Then, multiplying by ^2m

~² gives the second order derivative-term by itself, d²ψ(x)

dx² + 2mE

~² − m²ω²x²

~²

ψ(x) = 0. (2.5)

Now, we introduce the notations

α² = 2mE

~²

(2.6) and

β = mω

~ . (2.7)

Then the Schr¨odinger equation becomes d²ψ(x)

dx² + (α²− β²x²)ψ(x) = 0. (2.8)

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Continue by making the transformation z = βx². Then we get dψ(z)

dz = dψ(z) dz

dz

dx = dψ(z)

dz 2βx (2.9)

and d dx

dψ(z) dz 2βx

= 2βdψ(z)

dz + (2βx)² d²ψ(z)

dz² = 2βdψ(z)

dz + 4βzd²ψ(z)

dz² . (2.10) Now, (2.8) can be expressed as

4βzd²ψ(z)

dz² + 2βdψ(z)

dz + (α²− βz)ψ(z) = 0. (2.11) Dividing by 4β gives

zd²ψ(z) dz² +1

2 dψ(z)

dz −

c + 1

4z

ψ(z) = 0, c = −α²

4β. (2.12) For z → ±∞, we get terms in the differential equation which tend to infinity.

We take care of these asymptotic parts, by making the ansatz

ψ(z) = e^−z/2φ(z), (2.13)

leading to

dψ(z) dz =

−1

2φ(z) +dφ(z) dz

e^−z/2 (2.14)

and

d²ψ(z) dz² = 1

4φ(z) − dφ(z)

dz +d²φ(z) dz²

e^−z/2. (2.15) Putting this into (2.12) and dividing the resulting equation by the exponential factor e^−z/2, one then gets

zd²φ(z) dz² + 1

2 − z dφ(z) dz −

c + 1

4

φ(z) = 0. (2.16)

The general form of this equation is expressed as zd²φ(z)

dz² + (b − z)dφ(z)

dz − aφ(z) = 0, (2.17)

with constants a, b ∈ R and can be recognized as Kummer’s hypergeometric differential equation (by e.g. [3] and [4]).

Now, the general solution to (2.17) is given by

φ(z) = A ·₁F₁(a; b; z) + Bz^1−b·₁F₁(a − b + 1; 2 − b; z), (2.18) with constants A, B ∈ C. The the first term on the right hand side of the equality corresponds to the even part of the solution and the second term stands for the odd part of the solution. We have

b = 1

2 (2.19)

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in (2.17) (by (2.14)), that results in the solution

φ(z) = A ·₁F₁ a;¹₂; z + Bz^1/2·₁F₁ a +¹₂;³₂; z , (2.20) where the so called Kummer’s hypergeometric function₁F₁ is expressed as

1F₁(a; b; z) =

∞

X

n=0

(a)_n (b)_n

zⁿ

n! = 1 + a b

z

1!+ a(a + 1) b(b + 1)

z²

2! + . . . . (2.21) In this summation, we have the Pochhammer symbol

(a)_n= a(a + 1) · · · (a + n − 1) = (a + n − 1)!

(a − 1)! , (2.22)

for any parameter a.

Remark 2.1 Throughout the text, Kummer’s hypergeometric function will be expressed as ₁F₁, or in terms of the Gamma function [11]. In this case we have the relations

1F₁(a; b; z) ∝ Γ(b)

Γ(a)e^zz^a−b (2.23)

and also

1F₁(a; b; z) ∝ Γ(b)

Γ(b − a)(−z)^−a, (2.24)

where the Gamma function can be defined (cf. [4]) by Γ(z) :=

Z ∞ 0

e^−tt^z−1dt, <(z) > 0. (2.25) Inserting z = ¹₂ into this definition, the Gamma function in this case becomes the integral due to Euler (derived from the well-known Gaussian integralR∞

−∞e^−x²dx =

√π , by the substitution x =√ t ), Γ ¹₂ =

Z ∞ 0

e^−tt^−1/2dt =√

π . (2.26)

The Pochhammer symbol can also be expressed as (a)_n= Γ(n + a)

Γ(a) , (2.27)

since

Γ(n) = (n − 1)! . (2.28)

The energy spectrum can be obtained by analyzing the following two cases for (2.20) (in the variable βz² obtained in our transformation):

Case I (even solutions): B = 0 and a = −n, n ∈ Z≥0. The eigenfunctions are given by

ψ_n(z) = N_ne^−(βz²^)/2·₁F₁ −n;¹₂; βz² , (2.29)

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with normalization constants (given by [3])

N_n= s

β²

√π 2ⁿn! (2.30)

and the energy eigenvalues are obtained from a in (2.17), which by (2.16) is equal to c + ¹₄, which by (2.12) is equal to −^α_4β² + ¹₄ and finally by (2.6) and (2.7) we obtain:

a = −1 2

E

~ω +1

4. (2.31)

We also have a = −n, so

−n = −1 2

E_n

~ω +1

4, (2.32)

giving the energy eigenvalues

E_n=

2n +1 2

~ω. (2.33)

Case II (odd solutions): A = 0 and a + ¹₂ = −n, n ∈ Z≥0. The eigenfunctions are given by

ψ_n(z) = N_ne^−(βz²^)/2z ·₁F₁ −n;³₂; βz² , (2.34) with the energy eigenvalues

E_n =

(2n + 1) + 1 2

~ω. (2.35)

Summarizing the two cases, the entire discrete energy spectrum is given by E_n =

n +1

2

~ω, n ∈ Z^≥0. (2.36)

The energy eigenvalues (2.36) are bound states with evenly spaced energy levels, that are non-degenerate. The ground state energy E0 = ¹₂~ω is called the zero-point energy, which is the lowest possible energy state. The energy spectrum we obtained (from our Hamiltonian (2.2)) is illustrated in Figure 1 (where the operators ˆα^† and ˆα will be explained in Section 2.3).

We introduce Hermite polynomials H_n for the even and the odd cases respectively:

(H_2n(√

β z) = (−1)^{n (2n)!}_n! ·₁F₁ −n;¹₂; βz² ; H_2n−1(√

β z) = (−1)^{n 2(2n+1)!}_n! √

β z ·₁F₁ −n;³₂; βz² . (2.37) In terms of these relations, the wave function is expressed as

ψ_n(z) = N_ne^−(βz²^)/2H_n(p

β z). (2.38)

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Figure 1: Energy spectrum for the one-dimensional harmonic oscillator.

2.2 Properties of Hermite polynomials

Here we will present some results regarding Hermite polynomials. For detailed derivations cf. [3].

We shall consider the normalized eigenfunctions of the harmonic oscillator.

The condition fulfilled for the eigenfunctions to be normalized is given by Z

|ψ(z)|² dz = 1. (2.39)

The normalized eigenfunctions are expressed as

ψ_n(z) = s

β²

√π 2ⁿn! e^−βz²^/2H_n(p

β z), (2.40)

in accordance with (2.38). Some relations can then be derived for the Hermite polynomials. Under the condition that

α² = 2β n + ¹₂ , (2.41)

the Schr¨odinger equation (2.8) is fulfilled for functions ψ(z) = e^−βz²^/2Hn(√ β z).

Make this substitution for the wave functions, and then make the substitution ξ =√

β z. These derivations are presented in [3], so we skip to the results directly (since they are needed when introducing raising and lowering operators). One uses the relation of Hermite polynomials

ξHn(ξ) = nHn−1(ξ) + ¹₂Hn+1(ξ), (2.42) to get the relation for wave functions,

ξψ_n(ξ) =r n

2 ψ_n−1+

rn + 1

2 ψ_n+1. (2.43)

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By using another relation of the Hermite polynomials,

∂H_n(ξ)

∂ξ = 2nHn−1(ξ), (2.44)

one gets for the wave functions that

∂

∂ξψ_n(ξ) =

s √

√ β

π 2ⁿn! e^−ξ²^/2H_n(ξ) ∂H_n(ξ)

∂ξ − ξ

(2.45)

=

s √

√ β

π 2ⁿn! e^−ξ²^/2H_n(ξ)(2nH_n−1(ξ) − ξ).

Expressing the above equation in wave functions, one also gets

∂

∂ξψ_n(ξ) = 2r n

2 ψ_n−1(ξ) − ξψ_n(ξ) (2.46) and finally, by inserting (2.43) into (2.46), one gets the expression

∂

∂ξψ_n(ξ) =r n

2 ψ_n−1(ξ) −

rn + 1

2 ψ_n+1(ξ). (2.47)

2.3 Raising and lowering operators

The material here comes from [3].

Inserting ψ_n−1 from (2.43) into (2.47) gives us

∂

∂ξψn = ξψn−p

2(n + 1) ψn+1. (2.48)

This can be put in a nicer form by collecting ψ_n−terms and dividing both sides by√

2 :

√1 2

ξ − ∂

∂ξ

ψn=√

n + 1 ψn+1. (2.49)

Now, inserting ψ_n+1 from (2.43) into (2.47) yields

∂

∂ξψ_n =r n

2 ψ_n−1+r n

2 ψ_n−1− ξψ_n, (2.50) where we again collect ψ_n−terms and divide both sides by √

2 :

√1 2

ξ + ξ

∂ξ

ψ_n=√

n ψ_n−1. (2.51)

Here we define the “raising” and “lowering” operators respectively:

ˆ

α^†:= 1

√2

ξ − ∂

∂ξ

(2.52)

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and

ˆ α := 1

√2

ξ + ∂

∂ξ

, (2.53)

which implies that

ˆ

α^†ψ_n =√

n + 1 ψ_n+1 (2.54)

and

ˆ

αψ_n=√

n ψ_n−1. (2.55)

By (2.54) and (2.55), we can make each ψ_n an eigenfunction of the operator product ˆα^†α asˆ

ˆ

α^†αψˆ _n=√

n ˆα^†ψ_n−1 = nψ_n. (2.56) Thus, we can define a number operator ˆN ,

N := ˆˆ α^†α,ˆ (2.57)

where

N ψˆ _n = nψ_n. (2.58)

So, the operator ˆN has eigenvalues n and eigenfunctions ψ_n, n ∈ Z^≥0. For an illustration of how the raising and lowering operators act, cf. Figure 1.

2.4 The bracket notation

Here we will follow [2].

Consider a system in one dimension, say in z-direction. Let ψ₁(z) and ψ₂(z) be two square integrable functions. Then the scalar product is by definition:

hψ₁|ψ₂i ≡ Z

ψ^∗₁(z)ψ₂(z) dz, (2.59) where ∗ denotes complex conjugation. The following algebraic relations are satis- fied by the scalar product, with ψ₃(z) a square integrable function and c ∈ C an arbitrary number:

hψ_i|ψ_ji = δ_ij =

(0 if i 6= j;

1 if i = j, 0 ≤ i, j ≤ n. (2.64)

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3 Ornstein-Uhlenbeck processes and the Fokker- Planck equation

3.1 Statistics of Brownian motion

The material here comes from [7] and [8].

The standard notation of angular brackets will be adapted for average values throughout the text.

Consider a force f (t) which is randomly fluctuating. The main principle of Brownian motion is that the average value of this force becomes zero, i.e.

hf (t)i = 0. (3.1)

We introduce the definition of variance, or diffusion of our force [7]:

σ² :=(f (t) − hf (t)i)² , (3.2) which becomes, for systems fulfilling the condition (3.1),

σ² =f²(t) . (3.3)

I.e. for such systems, the variance becomes the average of the square. We will only consider systems for which (3.1) holds, so we will simply denote the variance as the average of the square throughout the text. This is also referred to as the distribution of the force.

Proceeding with the statistics, we now introduce the so-called correlation functions.

3.2 Correlation functions

The notations that will be introduced here follow [1].

Consider the function f (t) at an initial time t₁ = t and at a final time t₂ = t⁰. We are interested in their correlation. Assuming that f (t) satisfy the stochastic statistics (3.1), we require that it also satisfies

hf (t)f (t⁰)i = C(t − t⁰), (3.4) where the two-force correlation function C is a function which depends only on the difference between the functions that are correlated (in this case it can be referred to as the time correlation function, since it only depends on time).

We will now introduce a coefficient, called the correlation length which will be denoted by τ . This factor is a measure of the difference in the range of fluctuations between correlations. A typical behaviour of C(t − t⁰) for an Ornstein-Uhlenbeck process (which we will soon describe) is

C(t − t⁰) ∼ e^−|t−t⁰^|/τ, (3.5) where |t − t⁰| < τ (one can say that the correlation length is exponentially small).

In the case of (3.9), it is sensible to require that τ is < _γ¹.

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The Brownian motion we will consider throughout the text will be confined to one spatial dimension. We will call this coordinate x. For the randomly fluctuating force depending on position x, as well as on time t, we get the correlation of the force f (x, t) with f (x⁰, t⁰) expressed as

hf (x, t)f (x⁰, t⁰)i = C(x − x⁰, t − t⁰). (3.6)

3.3 Ornstein-Uhlenbeck processes

We will now consider stochastic processes satisfying Brownian motion. The material here follows from [7] and [8].

Consider an ensemble of heavy particles moving in a viscous medium (a fluid or a gas) consisting of light molecules. While the particles are moving in the medium, they are randomly subjected to forces from the molecules, such that the path of movement of each particle becomes completely random. The forces of which each of the heavy particles are subjected can be described by two parts. One damping force, which depends on the velocity of the particle, as well as its mass. That is, the momentum p = mv, as well as a coefficient of the damping, which we will denote by γ, gives the first part of the two forces,

F₁ = −pγ. (3.7)

The second part is a rapidly fluctuating force, which we shall denote by f (t), for the time variable t. This is called a random function and since its evolution is time-dependent, it is refered to as a stochastic process. We have

F₂ = f (t). (3.8)

The equation of motion for each particle in the ensemble is given by Newton’s second law, which takes the form

˙

p = F₁+ F₂ = −pγ + f (t), (3.9) where ˙p is the time-derivative of the momentum.

Ornstein-Uhlenbeck processes are stationary stochastic processes that follow the statistics of the Langevin approach [7]:

The Langevin approach

We will now analyze the momentum of Brownian particles. For short time intervals |t − t⁰| < τ , the equation of motion (3.9) was integrated by Langevin, resulting in the momentum equation

p(t) = p(0)e^−γt+ e^−γt Z t

0

dt⁰e^γt⁰f (t⁰), (3.10) where p(0) is determined by an initial condition. We will assume each particle in the ensemble to be initially at rest, i.e. p(0) = 0. The average value of the momentum for the above equation, keeping (3.1) in mind, becomes

hp(t)i = e^−γt Z t

0

dt⁰e^γt⁰hf (t⁰)i = 0. (3.11)

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The distribution of the momentum is given by p²(t) = e^−2γt

Z t 0

dt⁰ Z t

0

dt⁰⁰e^γ(t⁰^+t⁰⁰⁾hf (t⁰)f (t⁰⁰)i . (3.12) Using the notation (3.4) and making the substitution u = t⁰+ t⁰⁰ and v = t⁰− t⁰⁰, we get the distribution as

p²(t) = 1 2e^−2γt

Z 2t 0

du e^γu Z t

−t

dv C(v) = 1

2γe^−2γt(e^2γt− 1) Z t

−t

dv C(v). (3.13) The second integral has values (significantly) different from zero only for short time intervals (since as we stated earlier, the correlation length is exponentially small), so there will be no loss of generality to extend the integration limits to infinity. I.e. the process is stationary. At equilibrium, we get t⁰ = 0 and t⁰⁰ = t such that

p²(t) = D₀

γ (1 − e^−2γt), (3.14)

where the (time) diffusion constant D₀ is given by D0 = 1

2 Z ∞

−∞

dt hf (t)f (0)i = 1 2

Z ∞

−∞

dt C(t). (3.15)

The distribution (3.14) is called a momentum probability distribution (or probability density).

Proceeding after the Langevin approach, we require an equation describing the position x of the particle. Similar to the momentum approach, by integrating (3.10) (and dividing both sides by m), the equation

x(t) = 1 m

Z t 0

dt⁰e^−γt⁰ Z t⁰

0

dt⁰⁰e^γt⁰⁰f (t⁰⁰), (3.16) was obtained and integrated by parts by Ornstein and Uhlenbeck [8] resulting in:

x(t) = 1 mγ

Z t 0

dt⁰f (t⁰) − e^−γt Z t

0

dt⁰e^γt⁰f (t⁰)

, (3.17)

where the boundary condition has been chosen as x(0) = 0, which is valid, since we can always choose our coordinate system arbitrarily without the loss of generality.

The property (3.1) holds and the distribution of the position (i.e. its probability distribution) becomes

x²(t) = 1 m²γ²

Z t 0

dt⁰ Z t

0

dt⁰⁰hf (t⁰)f (t⁰⁰)i −2e^−γt m²γ²

Z t 0

dt⁰ Z t

0

dt⁰⁰e^γt⁰hf (t⁰)f (t⁰⁰)i + e^−2γt

m²γ² Z t

0

dt⁰ Z t

0

dt⁰⁰e^γ(t⁰^+t⁰⁰⁾hf (t⁰)f (t⁰⁰)i . (3.18)

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We again use the notation (3.4) and we make the following substitutions: in the first and the third term, u = t⁰+ t⁰⁰ and v = t⁰− t⁰⁰ and in the second term, u⁰ = t⁰ and v = t⁰− t⁰⁰. Then,

x²(t) = 1 2m²γ²

Z 2t 0

du Z t

−t

dv C(v) − 2e^−γt m²γ²

Z t 0

du⁰e^γu⁰ Z t

−t

dv C(v) (3.19) + e^−2γt

2m²γ² Z 2t

0

du e^γu Z t

−t

dv C(v).

By the same reasonings as before, we can extend the integration limits of the second integral in each term to infinity. At the same time evaluating the first integral of each term, we get

x²(t) = 1 2m²γ²

2t

Z ∞

−∞

dv C(v) − 4e^−γt

γ (e^γt− 1) Z ∞

−∞

dv C(v) (3.20)

+e^−2γt

γ (e^2γt− 1) Z ∞

−∞

dv C(v)

= D_x

2t + (4 − e^−γt)e^−γt− 3 γ

, with the space diffusion constant

D_x = D₀

m²γ². (3.21)

We will now consider a generalization of the standard Ornstein-Uhlenbeck process. The generalization consists in extending the force f (t) to be dependent on position as well. Then, the equation of motion (3.9) is generalized to:

˙

p = −γp + f (x, t), (3.22)

where f (x, t) follow the statistics of (3.1) and (3.6). For the study of this generalized equation of motion, we introduce the Fokker-Planck equation.

3.4 The Fokker-Planck equation

Here the material in [1] and [9] will be used.

The Fokker-Planck equation provides an expression for the generalized momentum probability density P (p, t), dependent of the momentum p and the time t. It is obtained by first approximating the equation of motion (3.22) as the correlation length approaches zero. Arvedson et al. [1, 9] showed that this could be done by a Langevin equation:

dp = −γp dt + dw, (3.23)

where dw is assumed to satisfy the statistics hdw²i = 2D₀dt (and as usual hdwi = 0), where the diffusion constant D₀ is given by (3.15).

Introducing the general form of the diffusion constant, dependent of the momentum,

D(p) = 1 2

Z ∞

−∞

dt C(pt/m, t), (3.24)

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we can express the general form of the Fokker-Planck equation [9] of the momentum probability density P (p, t) as

∂P (p, t)

∂t = −∂

∂p(v(p)P (p, t)) + ∂²

∂p²(D(p)P (p, t)). (3.25) In terms of the statistics of the Langevin equation (3.23), we can express the functions

v(p) = hdpi

dt , (3.26)

which corresponds to a force, and

D(p) = hdp²i

2 dt , (3.27)

corresponding to diffusion. Evaluating the distribution hdp²i, we get

dp² = γ²p²dt²− 2γp dt hdwi +dw² . (3.28) It is argued in Arvedson et al. [1, 9] that, considering the equation of motion (3.23) as the forces depend principally on the small changes in position rather than the small changes in time, the force dw can be approximated as an impulse δw. The variance of the impulse was obtained as

δw² = 2D(p) δt, (3.29)

for a small time interval δt. The approximation also implies that the dt−term in (3.23) becomes very small. Thus, for the variance (3.28), the dt²−term can be neglected. The diffusion constant, using this approximation, becomes

D(p) = hdw²i

2 dt = 2D(p)dt

2 dt = D(p), (3.30)

with D(p) given by (3.24).

For the evaluation of hdpi = −γp dt + hdwi, the approximation resulted in the average of the impulse as

hδwi = δt d

dpD(p), (3.31)

which used in (3.26) becomes

v(p) = −γp + d

dpD(p). (3.32)

Now, by (3.24) and (3.32), the Fokker-Planck equation (3.25) can be expressed as

∂P (p, t)

∂t = ∂

∂p

γp + D(p) ∂

∂p

P (p, t). (3.33)

We will now consider the so called generic case, for which D(p) = D₀_|p|¹ (such that D(p) ∼ _|p|¹ ). For this generic case, the Fokker-Planck equation becomes

∂P (p, t)

∂t = ∂

∂p

γp + D₀ 1

|p|

∂

∂p

P (p, t). (3.34)

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Introducing the dimensionless variable

t⁰ = γt, (3.35)

we get

∂P (p, t⁰)

∂t⁰ = ∂P (p, t⁰)

∂t⁰

∂t = ∂P (p, t⁰)

∂t⁰ γ. (3.36)

This alters (3.34) slightly into

∂P (p, t⁰)

∂t⁰ = ∂

∂p

p + D₀ γ

1

|p|

∂

∂p

P (p, t⁰). (3.37) Then, introducing the dimensionless variable (suggested in [9])

z = γ D₀

1/3

p, (3.38)

we get

∂P (z, t⁰)

∂z = ∂P (z, t⁰)

∂z

∂p = ∂P (z, t⁰)

∂z

γ D₀

1/3

(3.39) and

∂

∂p

"

∂P (z, t⁰)

∂z

γ D₀

1/3#

= 0 + γ D₀

2/3

∂²P (z, t⁰)

∂z² . (3.40)

Inserting this into (3.37), the Fokker-Planck equation becomes

∂P (z, t⁰)

∂t⁰ = ∂

∂z

"

γ D₀

1/3

p + γ D₀

2/3

D0

γ 1

|p|

∂

∂z

#

P (z, t⁰) (3.41)

= ∂

∂z

z + 1

|z|

∂

∂z

P (z, t⁰),

which we denote in terms of the so called Fokker-Planck operator ˆF as

∂P (z, t⁰)

∂t⁰ = ∂

∂z

z + 1

|z|

∂

∂z

P (z, t⁰) ≡ ˆF P (z, t⁰). (3.42) We define the stationary state P₀(z) := P (z, 0) and use the proportionality relation P0(z) ∝ e^−|z|³^/3, also suggested in [9]. Then we get

∂

∂z

z + 1

|z|

∂

∂z

e^−|z|³^/3= ∂

∂zze^−|z|³^/3+ ∂

∂z 1

|z|

∂

∂ze^−|z|³^/3 (3.43)

= e^−|z|³^/3− z³e^−|z|³^/3− 1 z²

∂

∂ze^−|z|³^/3+ 1

|z|

∂²

∂z²e^−|z|³^/3

= 2e^−|z|³^/3− |z|³e^−|z|³^/3− 1

|z|

∂

∂zz²e^−|z|³^/3

= 2e^−|z|³^/3− |z|³e^−|z|³^/3− 2e^−|z|³^/3+ |z|³e^−|z|³^/3 = 0,

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i.e. ˆF P₀ = 0 and the Fokker-Planck equation can thus be put in the Hermitian form

P₀^−1/2F Pˆ ₀^1/2 = 1 2 − |z|³

4 + ∂

∂z 1

|z|

∂

∂z. (3.44)

We will refer to this as the Hamiltonian operator for our generalized Ornstein- Uhlenbeck system. We define our dimensionless Hamiltonian as

H :=ˆ ∂

∂z 1

|z|

∂

∂z +1 2− |z|³

4 . (3.45)