conditions are imposed at one end of the interval, instead of some conditions

120 CHAPTER 5. GREEN FUNCTIONS

at one end and some at the other. The same set of ingredients go into to

constructing the Green function, though.

Consider the problem

dy

’ Q(t)y = F (t), y(0) = 0. (5.22)

dt

We seek a Green function such that

d

Lt G(t, t ) ≡ ’ Q(t) G(t, t ) = δ(t ’ t ) (5.23)

dt

and G(0, t ) = 0.

We need χ(t) = G(t, t ) to satisfy Lt χ = 0, except at t = t and need

G(0, t ) = 0. The unique solution of Lt χ = 0 with χ(0) = 0 is χ(t) ≡ 0. This

means that G(t, 0) = 0 for all t < t . Near t = t we need

G(t + µ, t ) ’ G(t ’ µ, t ) = 1 (5.24)

The unique solution is

t

G(t, t ) = θ(t ’ t ) exp Q(s)ds , (5.25)

t

where θ(t ’ t ) is the Heaviside step function

0, t < 0,

θ(t) = (5.26)

1, t > 0.

G(t,t™)

1

t

t™

The Green function G(t, t ) for the ¬rst-order initial value problem .

5.2. CONSTRUCTING GREEN FUNCTIONS 121

Therefore

∞

y(t) = G(t, t)F (t )dt ,

0

t t

= exp Q(s) ds F (t ) dt

0 t

t t t

exp ’

= exp Q(s) ds Q(s) ds F (t ) dt . (5.27)

0 0 0

In chapter 3 we solved this problem by the method of variation of parameters.

Example: Forced, Damped, Harmonic Oscillator. An oscillator obeys the

equation

x + 2γ x + („¦2 + γ 2 )x = F (t).

¨ ™ (5.28)

Here γ > 0 is the friction coe¬ecient. Assuming that the oscillator is at rest

at the origin at t = 0, we show that

1 t

e’γ(t’„ ) sin „¦(t ’ „ )F („ )d„.

x(t) = (5.29)

„¦ 0

We seek a Green function G(t, „ ) such that χ(t) = G(t, „ ) obeys χ(0) =

χ (0) = 0. Again, the unique solution of the di¬erential equation with this

initial data is χ(t) ≡ 0. The Green function must be continuous at t = „ ,

but its derivative must be discontinuous there, jumping from zero to unity

to provide the delta function. Thereafter, it must satisfy the homogeneous

equation. The unique function satisfying all these requirements is

1

G(t, „ ) = θ(t ’ „ ) e’γ(t’„ ) sin „¦(t ’ „ ). (5.30)

„¦

G(t, „ )

t

„

The Green function G(t, „ ) for the damped oscillator problem .

122 CHAPTER 5. GREEN FUNCTIONS

Both these initial-value Green functions G(t, t ) are identically zero when

t < t . This is because the Green function is the response of the system to

a kick at time t = t , and in physical problems, no e¬ect comes before its

cause. Such Green functions are said to be causal .

Physics Application: Friction without Friction ” The Caldeira-

Leggett Model in Real Time.

This is an application of the initial-value problem Green function we found

in the preceding example.

When studying the quantum mechanics of systems with friction, such as

the viscously damped oscillator of the previous example, we need a tractable

model of the dissipative process. Such a model was introduced by Caldeira

and Leggett1 . They consider the Lagrangian

fi2

1 ™2 12 1

Q ’ „¦2 Q2 ’ Q i 22

Q2 , (5.31)

qi ’ ωi qi ’

L= fi q + ™ 2

2 2 2 ωi

i i i

which describes a macroscopic variable Q(t), linearly coupled to an oscillator

bath of very many simple systems representing the environment. The last sum

in the Lagrangian is a counter-term which is inserted cancel the shift

fi2

122 1

Q2 ,

„¦ Q ≡ V (Q) ’ Ve¬ (Q) = V (Q) ’ (5.32)

2

2 2 ωi

i

caused by the bath. The shift arises because a slowly varying Q gives fi qi =

’(fi2 /ωi )Q, and substituting these values for the qi , we have

2

fi2

1 22 1

i

Q2 .

fi q + ωi qi = ’

Q (5.33)

2

2 2 ωi

i

We will denote the counter-term by 1 ∆„¦2 Q2 .

2