Handbook of mathematics for engineers and scienteists part 188 docx

Various potentials U x are considered below and particular solutions of the boundary value problem T8.1.10.1 or the Cauchy problem for Schr¨odinger’s equation are presented.. For solutio

2◦ In the cases where the eigenfunctions ψ n (x) form an orthonormal basis in L2(R), the

solution of the Cauchy problem for Schr¨odinger’s equation with the initial condition

is given by

w (x, t) =

–∞ G (x, ξ, t)f (ξ) dξ, G (x, ξ, t) =

∞



n=0

ψ n (x)ψ n (ξ) exp



–iE n

 t



Various potentials U (x) are considered below and particular solutions of the boundary

value problem (T8.1.10.1) or the Cauchy problem for Schr¨odinger’s equation are presented

T8.1.10-2 Free particle: U (x) =0

The solution of the Cauchy problem with the initial condition (T8.1.10.2) is given by

w (x, t) = 1

2√ iπτ

 ∞

–∞exp



–(x – ξ)

2

4iτ



f (ξ) dξ, τ= t

2m, √

ia=



e πi/4√|

a| if a >0 ,

e–πi/4√

|a| if a <0

T8.1.10-3 Linear potential (motion in a uniform external field): U (x) = ax.

Solution of the Cauchy problem with the initial condition (T8.1.10.2):

w (x, t) = 1

2√ iπτ exp –ibτ x –13ib2τ3  ∞

–∞exp



–(x + bτ

2– ξ)2

4iτ



f (ξ) dξ, τ= t

2m, b= 2am

 2

T8.1.10-4 Linear harmonic oscillator: U (x) = 12mω2x2.

Eigenvalues:

E n=ω n+ 12

, n=0,1,

Normalized eigenfunctions:

π1 4√2n

n ! x0 exp –

1

2ξ2

H n (ξ), ξ = x

x0, x0=



mω,

where H n (ξ) are the Hermite polynomials The functions ψ n (x) form an orthonormal basis

in L2(R)

T8.1.10-5 Isotropic free particle: U (x) = a/x2

Here, the variable x≥ 0plays the role of the radial coordinate, and a >0 The equation with

U (x) = a/x2results from Schr¨odinger’s equation for a free particle with n space coordinates

if one passes to spherical (cylindrical) coordinates and separates the angular variables The solution of Schr¨odinger’s equation satisfying the initial condition (T8.1.10.2) has the form

w (x, t) = exp



–12iπ (μ +1) sign t 2|τ|

0

√

xy exp



i x

2+ y2

4τ



J μ



xy

2|τ|



f (y) dy,

τ = t

2m, μ= 2am

2 +

1

4 ≥ 1,

where J μ (ξ) is the Bessel function.

T8.1.10-6 Morse potential: U (x) = U0(e– 2x/a–2e–x/a).

Eigenvalues:

E n = –U0



1– 1

β (n + 12)

2

, β = a

√2

mU0

 , 0 ≤n < β –2.

Eigenfunctions:

ψ n (x) = ξ s e–ξ/2Φ(–n,2s+1, ξ), ξ =2βe–x/a, s= a √

–2mE n

whereΦ(a, b, ξ) is the degenerate hypergeometric function.

In this case the number of eigenvalues (energy levels) E n and eigenfunctions ψ n is

finite: n =0, 1, , nmax

T8.2 Hyperbolic Equations

T8.2.1 Wave Equation ∂2w

∂t2 = a2∂2w

∂x2

This equation is also known as the equation of vibration of a string It is often encountered

in elasticity, aerodynamics, acoustics, and electrodynamics

T8.2.1-1 General solution Some formulas

1◦ General solution:

w (x, t) = ϕ(x + at) + ψ(x – at), where ϕ(x) and ψ(x) are arbitrary functions.

2◦ If w(x, t) is a solution of the wave equation, then the functions

w1= Aw( λx + C1, λt + C2) + B,

w2= Aw



x – vt

1– (v/a)2,

t – va– 2x

1– (v/a)2



,

w3= Aw



x

x2– a2t2,

t

x2– a2t2



are also solutions of the equation everywhere these functions are defined (A, B, C1,

C2, v, and λ are arbitrary constants) The signs at λ’s in the formula for w1 are taken

arbitrarily The function w2 results from the invariance of the wave equation under the

Lorentz transformations.

T8.2.1-2 Domain: –∞ < x < ∞ Cauchy problem.

Initial conditions are prescribed:

w = f (x) at t =0, ∂ t w = g(x) at t=0

Solution (D’Alembert’s formula):

w (x, t) = 1

2[f (x + at) + f (x – at)] +

1

2a

 x+at

x–at g (ξ) dξ.

T8.2.1-3 Domain: 0 ≤x<∞ First boundary value problem.

The following two initial and one boundary conditions are prescribed:

w = f (x) at t=0, ∂ t w = g(x) at t =0, w = h(t) at x =0 Solution:

w (x, t) =

⎧

⎪

⎨

⎪

1

2[f (x + at) + f (x – at)] +

1

2a

 x+at

x

a,

1

2[f (x + at) – f (at – x)] +

1

2a

 x+at

at–x g (ξ) dξ + h



t– x

a



for t > x

a

In the domain t < x/a the boundary conditions have no effect on the solution and the expression of w(x, t) coincides with D’Alembert’s solution for an infinite line (see

Para-graph T8.2.1-2)

T8.2.1-4 Domain: 0 ≤x<∞ Second boundary value problem.

The following two initial and one boundary conditions are prescribed:

w = f (x) at t=0, ∂ t w = g(x) at t =0, ∂ x w = h(t) at x=0 Solution:

w (x, t) =

⎧

⎪

⎪

1

2[f (x + at) + f (x – at)] +

1

2a [G(x + at) – G(x – at)] for t < x

a,

1

2[f (x + at) + f (at – x)] +

1

2a [G(x + at) + G(at – x)] – aH



t–x

a



for t > x

a,

where G(z) =  z

0 g (ξ) dξ and H(z) =

 z

0 h (ξ) dξ.

T8.2.1-5 Domain: 0 ≤x≤l Boundary value problems

For solutions of various boundary value problems, see Subsection T8.2.2 forΦ(x, t)≡ 0

T8.2.2 Equation of the Form ∂2w

∂t2 = a

2∂2w

∂x2 +Φ(x, t)

T8.2.2-1 Solutions of boundary value problems in terms of the Green’s function

We consider boundary value problems on an interval 0 ≤ x ≤ l with the general initial conditions

w = f (x) at t=0, ∂ t w = g(x) at t =0 (T8.2.2.1) and various homogeneous boundary conditions The solution can be represented in terms

of the Green’s function as

w (x, t) = ∂

∂t

 l

0 f (ξ)G(x, ξ, t) dξ +

 l

0 g (ξ)G(x, ξ, t) dξ +

 t

0

 l

0 Φ(ξ, τ)G(x, ξ, t–τ) dξ dτ.

(T8.2.2.2)

Here, the upper limit l can assume any finite values.

Paragraphs T8.2.2-2 through T8.2.2-4 present the Green’s functions for various types

of homogeneous boundary conditions

Remark Formulas from Subsections 14.8.1–14.8.2 should be used to obtain solutions to corresponding nonhomogeneous boundary value problems.

T8.2.2-2 Domain: 0 ≤x≤l First boundary value problem.

Boundary conditions are prescribed:

w=0 at x=0, w=0 at x = l.

Green’s function:

G (x, ξ, t) = 2

aπ

∞



n=1

1

nsin

nπx

l



sin

nπξ

l



sin

nπat

l



T8.2.2-3 Domain: 0 ≤x≤l Second boundary value problem

Boundary conditions are prescribed:

∂ x w=0 at x=0, ∂ x w=0 at x = l.

Green’s function:

G (x, ξ, t) = t

l + 2

aπ

∞



n=1

1

n cosnπx

l



cosnπξ

l



sinnπat

l



T8.2.2-4 Domain: 0 ≤x≤l Third boundary value problem (k1 >0, k2>0) Boundary conditions are prescribed:

∂ x w – k1w=0 at x=0, ∂ x w + k2w=0 at x = l.

Green’s function:

G (x, ξ, t) = 1

a

∞



n=1

1

λ n u n 2 sin(λ n x + ϕ n ) sin(λ n ξ + ϕ n ) sin(λ n at),

ϕ n= arctanλ k n

1, u n 2 = l

2 +

(λ2n + k1k2)(k1+ k2)

2(λ2n + k12)(λ2n + k22);

the λ n are positive roots of the transcendental equation cot(λl) = λ2– k1k2

λ (k1+ k2).

T8.2.3 Klein–Gordon Equation ∂2w

∂t2 = a2∂2w

∂x2 –bw

This equation is encountered in quantum field theory and a number of applications

T8.2.3-1 Particular solutions.

w (x, t) = cos(λx)[A cos(μt) + B sin(μt)], b = –a2λ2+ μ2,

w (x, t) = sin(λx)[A cos(μt) + B sin(μt)], b = –a2λ2+ μ2,

w (x, t) = exp( μt)[A cos(λx) + B sin(λx)], b = –a2λ2– μ2,

w (x, t) = exp( λx)[A cos(μt) + B sin(μt)], b = a2λ2+ μ2,

w (x, t) = exp( λx)[A exp(μt) + B exp(–μt)], b = a2λ2– μ2,

w (x, t) = AJ0(ξ) + BY0(ξ), ξ=

√ b a

a2(t + C1)2– (x + C2)2, b>0,

w (x, t) = AI0(ξ) + BK0(ξ), ξ =

√

–b

a

a2(t + C1)2– (x + C2)2, b<0,

where A, B, C1, and C2are arbitrary constants, J0(ξ) and Y0(ξ) are Bessel functions, and

I0(ξ) and K0(ξ) are modified Bessel functions.

T8.2.3-2 Formulas allowing the construction of particular solutions

Suppose w = w(x, t) is a solution of the Klein–Gordon equation Then the functions

w1= Aw( x + C1, t + C2) + B,

w2= Aw



x – vt

1– (v/a)2,

t – va–2x

1– (v/a)2



,

where A, B, C1, C2, and v are arbitrary constants, are also solutions of this equation The signs in the formula for w1are taken arbitrarily

T8.2.3-3 Domain: 0 ≤x≤l Boundary value problems

For solutions of the first and second boundary value problems, see Subsection T8.2.4 for

Φ(x, t)≡ 0

T8.2.4 Equation of the Form ∂2w

∂t2 = a2∂2w

∂x2 –bw +Φ(x, t)

T8.2.4-1 Solutions of boundary value problems in terms of the Green’s function Solutions to boundary value problems on an interval 0 ≤ x ≤ l with the general initial conditions (T8.2.2.1) and various homogeneous boundary conditions are expressed via the Green’s function by formula (T8.2.2.2)

T8.2.4-2 Domain: 0 ≤x≤l First boundary value problem

Boundary conditions are prescribed:

w=0 at x=0, w=0 at x = l.

Green’s function for b >0:

G (x, ξ, t) = 2

l

∞



n=1

sin(λ n x ) sin(λ n ξ)sin t

a2λ2

n + b

a2λ2

n + b

, λ n= πn l .

T8.2.4-3 Domain: 0 ≤x≤l Second boundary value problem.

Boundary conditions are prescribed:

∂ x w=0 at x=0, ∂ x w=0 at x = l.

Green’s function for b >0:

G (x, ξ, t) = 1

l √

bsin t √

b

+ 2

l

∞



n=1

cos(λ n x ) cos(λ n ξ)sin t

a2λ2

n + b

a2λ2

n + b

, λ n= πn l .

T8.2.5 Equation of the Form ∂2w

∂t2 = a

2

∂2w

∂r2 +

1

r

∂w

∂r



+Φ(r, t)

This is the wave equation with axial symmetry, where r =

x2+ y2is the radial coordinate.

T8.2.5-1 Solutions of boundary value problems in terms of the Green’s function

We consider boundary value problems in domain0 ≤r≤Rwith the general initial conditions

w = f (r) at t =0, ∂ t w = g(r) at t =0, (T8.2.5.1)

and various homogeneous boundary conditions at r = R (the solutions bounded at r =0are sought) The solution can be represented in terms of the Green’s function as

w (r, t) = ∂

∂t

 R

0 f (ξ)G(r, ξ, t) dξ

+

 R

0 g (ξ)G(r, ξ, t) dξ +

 t

0

 R

0 Φ(ξ, τ)G(r, ξ, t – τ) dξ dτ (T8.2.5.2)

T8.2.5-2 Domain: 0 ≤r≤R First boundary value problem

A boundary condition is prescribed:

w=0 at r = R.

Green’s function:

G (r, ξ, t) = 2ξ

aR

∞



n=1

1

λ n J2

1(λ n)J0



λ n r R



J0



λ n ξ R



sin



λ n at R



,

where λ n are positive zeros of the Bessel function, J0(λ) =0 The numerical values of the

first ten λ nare specified in Paragraph T8.1.5-2

T8.2.5-3 Domain: 0 ≤r≤R Second boundary value problem

A boundary condition is prescribed:

∂ r w=0 at r = R.

Green’s function:

G (r, ξ, t) = 2tξ

R2 +

2ξ aR

∞



n=1

1

λ n J02(λ n)J0



λ n r R



J0



λ n ξ R



sin



λ n at R



,

where λ n are positive zeros of the first-order Bessel function, J1(λ) = 0 The numerical

values of the first ten roots λ nare specified in Paragraph T8.1.5-3

T8.2.6 Equation of the Form ∂2w

∂t2 = a

2 ∂2w

∂r2 +

2

r

∂w

∂r



+Φ(r, t)

This is the equation of vibration of a gas with central symmetry, where r =

x2+ y2+ z2

is the radial coordinate

T8.2.6-1 General solution forΦ(r, t)≡ 0

w (t, r) = ϕ (r + at) + ψ(r – at)

where ϕ(r1) and ψ(r2) are arbitrary functions

T8.2.6-2 Reduction to a constant coefficient equation

The substitution u(r, t) = rw(r, t) leads to the nonhomogeneous constant coefficient

equa-tion

∂2u

∂t2 = a2

∂2u

∂r2 + r Φ(r, t),

which is discussed in Subsection T8.2.1

T8.2.6-3 Solutions of boundary value problems in terms of the Green’s function Solutions to boundary value problems on an interval 0 ≤ x ≤ R with the general initial conditions (T8.2.5.1) and various homogeneous boundary conditions are expressed via the Green’s function by formula (T8.2.5.2)

T8.2.6-4 Domain: 0 ≤r≤R First boundary value problem

A boundary condition is prescribed:

w=0 at r = R.

Green’s function:

G (r, ξ, t) = 2ξ

πar

∞



n=1

1

nsin



nπr R



sin



nπξ R



sin



anπt R



T8.2.6-5 Domain: 0 ≤r≤R Second boundary value problem

A boundary condition is prescribed:

∂ r w=0 at r = R.

Green’s function:

G (r, ξ, t) = 3tξ2

R3 +

2ξ ar

∞



n=1

μ2

n+1

μ3

n sin



μ n r R



sin



μ n ξ R



sin



μ n at R



,

where μ n are positive roots of the transcendental equation tan μ – μ =0 The numerical

values of the first five roots μ nare specified in Paragraph T8.1.7-3

For solutions of the first and second boundary value problems, see Subsection T8.2.4 for

Φ(x, t)≡

T8.2.4 Equation of the Form ∂2w... ≤x≤l Boundary value problems

For solutions of various boundary value problems, see Subsection T8.2.2 for< i>Φ(x, t)≡

T8.2.2 Equation of the Form ∂2w... C1, and C2are arbitrary constants, J0(ξ) and Y0(ξ) are Bessel functions, and< /i>

I0(ξ) and K0(ξ)