Marder M.P. Condensed Matter Physics

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Variational Techniques

B.l Functionals and Functional Derivatives

A functional is a rule for obtaining a number from a function. For example, the rule

might be "integrate the function

i\)

over all space," in which case the functional is

F{ip} = f df

ip(r

(B.l)

Other examples might involve multiplying the function ip by exp(—r

/rg) or by

itself before integrating.

A functional derivative describes how a functional changes when the function

placed into it changes by a small amount. Functional derivatives can be defined

formally by a natural extension of the definition of ordinary differentiation. Given

the functional F {'(/>}, then the functional derivative describing how F responds to

small alterations of

xj)

in the vicinity of r is

SF{M FU(r')+e6(r'-r))-F^(r>))

T7PT =

lim

n — " •

Alter

by adding a small function peaked around r, and divide through by the

integral of the added function.

Example. If

F{^(r')} = Jdr'g(^(r')), (B.3)

where g is an ordinary function, then

6FM I dr'gU(r') + eS(r'-r))-gN>(T>

„ \Z! = lim

(B.4)

6il>(r) e^o e

gU(7'))+e

'U(r'))ö(r'

lim

- (B.5)

_> \ Courage is needed to perform a Taylor expansion in powers of a ,

f) j

■

delta function, which is infinitely large where its argument vanishes. (B.Ô)

More mathematically compelling accounts of this procedure can be

found, for example, in Zeidler (1995) or Hassani (1999), Chapter

30,

pp. 973-1002.

903

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

904

Appendix B. Variational Techniques

B.2 Time-Independent Schrödinger Equation

The time-independent Schrödinger equation is equivalent to a variational principle,

a point of view that is particularly valuable in suggesting approximation schemes.

To derive the variational rule, consider the functional

F'u{lb\

(ibl^Kllb) A functional is a rule for obtaining a number

(B.7)

from

function, in this case from the wave

function \ip).

and find its extrema subject to the constraint that

<V#>

(B.8)

An extremum

of a

functional

F is a

function ip causing functional derivatives

6F/ôip(r) to vanish for all r.

function that minimizes

will be an extremum,

but so will be a function which maximizes F, or one for which

is at

saddle

point, such as sketched in Figure 7.4.

There are two ways to enforce the constraint (B.8).

• Divide the functional Fx through by

(ip\ip)

and take functional derivatives of

<^|Ä|V>

<</#>

(B.9)

•

\I/J)

is multiplied by any overall scale factor, (B.9) does not change, because

the factor cancels between numerator and denominator. An extremum of (B.9)

is therefore sensitive to the shape of

ip,

but cannot depend upon its normaliza-

tion. After finding an extremum of (B.9), one is free to set

(ip\ip)

— 1.

Use the method of Lagrange multipliers. In this method, one takes the con-

straint that is supposed to be imposed, multiplies it by an arbitrary constant,

the Lagrange multiplier, and subtracts the product from the original func-

tional. In the present case, that means finding extrema of

£ is the Lagrange multiplier.

(B.10)

The simplest way to find extrema of

FH;

is by treating

(ip\

and

\tp)

as indepen-

dent variables, and requiring the variation of

F<x

to vanish simply by writing

wKh/>>_w>

^,jmw

(B11)

d(ip\

(V#)

<V#>

' '

<V#>

■Mlp) =

S.\lp)

With £= V,, r

The method of Lagrange

(ß.12)

(lp\w)

multipliers gives the same result.

Therefore the variational procedure is equivalent to Schrödinger's equation.

Time-Dependent Schrödinger Equation 905

If the steps leading to Eq. (B.12) seem unjustified, it is easy to employ the defi-

nition of functional differentiation (B.2) on the functional /

dr'ip*

(?') \K

—

E^ipif)

and show that

0 =

WÏ7)

I^V)^-

]^)

)

=> 0 =

\0~C

— £.\lb(~r) Anyone uncomfortable with treating ip and ip" as inde- (B.14)

pendent functions is free to write everything out in terms

of the real and imaginary parts of ip, differentiate with

respect to them, and verify that the same results are ob-

tained again.

B.3 Time-Dependent Schrödinger Equation

The equivalence of the time-independent Schrödinger equation to extrema of the

mean value of the Hamiltonian is well known. It is less commonly appreciated that

the time-dependent Schrödinger equation can be obtained from a variational prin-

ciple. Like so many other formal relations in quantum mechanics, the observation

is due to Dirac.

The action L giving the time-dependent equation is simply

L= f dt £, (B.15)

where

£ = (#-ftj^)-M%>). (B.16)

Lagrange's equation are

d dL dL

0 (B.17)

(B.18)

dtd(ip\ d(ip\

=r- 'Ä— \ip) —

J~C\lp).

Because there is no dependence upon (ip|, ev- (B.19)

Ot erything comes from the right hand side.

A more detailed derivation of Eq. (B.19) can be obtained by using the methods of

functional differentiation described at the beginning of this appendix. The approxi-

mation scheme suggested by this calculation is to substitute for

\ip)

some restricted

set of wave functions parameterized by a number of variables, and then to use

Eqs.

(B.16) and (B.18) to find equations of motion.

This variational principle has great advantages when one's goal is to obtain ef-

fective equations of motion for collective coordinates. Unlike an effective Hamil-

tonian, an effective Lagrangian

makes few demands. There is no requirement

that one identify canonical momenta conjugate to the coordinates. One can take

any parameterization of

that seems handy, insert it into the Lagrangian, and ob-

tain equations of motion for the parameters. This method is employed in Section

15.5.5 to find the dispersion relation for superfluid helium, and in Section 16.4 to

find the equation of motion for electron wave packets.

906 Appendix B.

Variational

Techniques

B.4 Method of Steepest Descent

Many intractable integrals can be approximated well by the method of steepest

descent. The integrals are of the form

I dxe~

\ (B.20)

and the approximation becomes good in the limit where the parameter ß becomes

infinite. Find the point xo in the complex plane where

H(XQ)

is minimized, and

deform the integration contour so that instead of passing necessarily along the real

axis,

the contour heads off into the complex plane and passes through

XQ.

If H(x)

is singular off the real axis and deforming the contour involves picking up large

numbers of poles, it may be advisable to settle for the point x\ on the real axis

where H(x) is minimized. In any event, write

H(x) « H(xo) +

X0)

H"(x

) +. . . (B.21)

e"^

« e~^

^ . \ Do the Gaussian integral. (B 22Ï

y\ßH"(x

)\'

}

This method is employed in Section 16.3.1 on Zener tunneling.

References

S. Hassani (1999), Mathematical Physics: A Modern Introduction to Its Foundations, Springer-

Verlag, New York.

E. Zeidler (1995), Applied Functional Analysis: Applications to Mathematical Physics, Springer-

Verlag, New York.

Second Quantization

C.l Rules

C.1.1 States

Begin with a complete orthonormal set of basis functions ipi. Any collection of

identical particles can be described by sums of products of these functions. In the

formalism of second quantization, one focuses upon many-body basis functions,

which describe how many particles are in each state. For example,

|0,2,3,10,...) (C.l)

means that no particles are in state ip\, two particles are in state

V>2,

three are in

state fa, and so on. The integers describing the numbers of particles are called

occupation numbers.

C.1.2 Operators

The operators of second quantization change the numbers of particles in these

quantum states. There is a creation operator with index / that adds one particle

to state / and an annihilation operator with index / that takes one particle away

from state /.

Fermions. The Pauli principle prohibits more than one electron from occupying

any given quantum state, so the occupation numbers all are zero or one. The cre-

ation and annihilation operators are usually denoted by cj and c/ respectively. The

way they operate is

ci\n\ri2 ... ni ...) =

. „ , .

, (L.za)

' \ |m«2 • • . 0 . . .) if«/ = 1

ti \_/° ifn/ = l

c\\n\ni ... ni ...) =

. , > .

„ (C.2b)

" ' \ \n\n

. . . 1 . . .) if

= 0.

The operators anticommute:

c]c},+c],c] = 0 (C.3a)

C[Ci>

+ CfCi = 0

(C.3b)

cic],

c],ci

Su'

■

(C.3c)

907

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

908

Appendix C. Second Quantization

Bosons. Bosons can inhabit any quantum state as often as they please, so the

occupation numbers range over all non-negative integers. The creation and an-

nihilation operators are usually denoted by a[ and

<3/

respectively. The way they

operate is

âi\n\H2 . . . ni . . .} =

y/h~i\nin

...«/-

.. .) (C.4a)

âj|ni«2 ...«/...) = \A*/ + l|«i«2 • •

•

ni + l . . .). (C.4b)

The operators commute:

â\â], -

â],â]

= 0 (C.5a)

âiâi'

—

âi'âi = 0 (C.5b)

âiâj,

—

aj,âi =

Ou'

•

(C.5c)

C.1.3 Hamiltonians

A Hamiltonian that is given as a sum of operators on single particles can be rewrit-

ten in second quantized notation as

J-C

:= \ f: fj means an operator such as f(7j) that acts (C.6)

^—? in some identical fashion upon each particle

J j in turn.

= y^ c] {ipi^lfillpi'

(\))ci/.

The wave functions tpi and operator / all act (C.7)

,,, on particle 1. The expression for bose opera-

tors is identical.

The notation

\ipi>(l))

means that particle number

is in state ipy. For example, if

/ is the kinetic energy operator and

tpi

is the product of a Wannier function wi and

a spin function Xh then

<V>/(i)l/ihMi)> = <W /^i ™;(n)

Mn) (C.8)

The leading delta function requires the spins of the two states to be the same.

The Laplacian V^ acts on variable T\.

A Hamiltonian that is given as a sum of operators on pairs of particles can be

rewritten in second quantized notation as

3"C = y fj:/ fjji means an operator such as /(?/, 7y) that (C.9)

M/'

acts in some identical fashion upon pairs of

particles.

= Y, eJ4e/'"e/"(^/(l)#(2)|/i2|#'(l)#"(2)) (C.10)

For example, if fu is the Coulomb interaction and

ipi

is the product of some spatial

wave function

</>/

and a spin function xi, then

(V/(l)#(2)|/i2|#'(l)#"(2)>

f e

= l

xtXi"

x,'Xi'" / àridr

4>*(ri)4>Hr

)7z

—><t>i"(?\)<l>i>"(?i)■

(Cll)

J \

\ —>"2\

Often one does not write down spin sums or spin delta functions explicitly and just

multiplies the final answer by appropriate factors of two.

Derivations

909

C.2 Derivations

C.2.1 Bosons

A collection of Bose particles can be described by a wave function of the form

i"i"2*3...)=J

L, £ ni^M- (ci«

V Permutations Sj y'=l

The function

gives some permutation of the integers

and by summing over

all permutations the wave function is guaranteed to be symmetric under interchange

of all indices.

The function /(_/) is some function into the positive

integers.

The idea is that the

states

ipi

are numbered in a way that may be quite arbitrary. Suppose one decides

to build a many-body state with one particle in state

and two particles in state 3.

The function l(j) could then be

/(1) = 3, /(2) =

/(3) = 3. (C.13)

Notation of the form |?/>2(6)) means that particle number 6 is in state

ip2-

The number of times a certain integer l(j) appears as j ranges from 1 to N is

ni, so m gives the number of particles in state /. The factors of

n\\ri2\

■ ■ ■

account

for the fact that any given term in the sum where n\ particles are in state

appears

n\\

times. To illustrate that the factorials are correctly employed, suppose first of

all that there is only one particle in each distinct state. Then there are N\ distinct

orthogonal functions appearing in the sum (C.12), and the normalization must be

\/y/N\.

On the other hand, suppose all particles are in state ip\. Then all the N\

terms in the sum are identical, and the sum must be divided by

to produce a

normalized wave function.

To study the behavior of this wave function, it is helpful to define the operator

iz-/'

El^(i))(#(;)l- (c.14)

The effect of this operator is to search one at a time for each particle in state

ipi>

and move it to state ipi.

To use this operator, consider a Hamiltonian of the form (C.6),

E fj

E \MJ))(W)\fj\MJ)){MJ)\

(c.15)

= \ £/<_/' (lbi(\)

f\ |l/>;/(l)). The matrix elements of the one-particle oper- (C.16)

.,, ator / do not depend upon which particle is

involved, so the label

can be used instead of

Let £/<_// act upon |«i«2 • • •)• If state

ipi>

is not occupied, the result is zero. If

it is occupied, then in every term of (C.12), there will be precisely

ni>

values of j

910

Appendix C. Second Quantization

for which there is a nonzero result, with the population of state /' being reduced by

1 and the population of state / being increased by 1. The result will not be properly

normalized because \fn[i\n\\ is in the denominator rather than

y/(ni>

—

l)!(n/

1)!,

and a factor of n^ has been acquired along the way. So

£/<-/'l

i«2 • • •)

y «/'(«/

1)|

. . .

«//

. .

. n/

. .

.). (C.17)

For this reason, define

a]\n\

â[\n\

■

■)

A/«Z

. . .

+ l

■

.) =

y/h~i\

. . . ni -

. . .)

£■

£S/^/'

a, a/i.

(Cl 8a)

(C.18b)

so that

(C.19)

It is easy from Eq. (C.18) to check the commutation relations (C.5) by allowing

the creation and annihilation operators to act in various orders upon general states

\n\n

. .

.).

C.2.2 Fermions

The wave function describing a collection of fermions must be antisymmetric under

interchange of arguments, and it consists of sums of terms of the form

* = h«2

• - •)

= v^ï

)" II Mœto)>. (c.20)

Permutations Sj

j=\

where the sum is over all permutations Sj of

= 1

. .

. N, with 5 the sign of the

permutation. In order for

not to equal zero, no more than one electron is allowed

to inhabit each individual state.

an electron

is in

state

then «/

one and

otherwise it is zero.

Given the occupation numbers «/ for each state

ipi,

the wave function that can

be formed from the collection is almost

unique.

There is only one ambiguity, which

has to do with the overall sign of the wave function. The ambiguity is avoided by

requiring that /(_/) be an increasing function of

Consider again

Hamiltonian

the form (C.15), acting on antisymmetric

wave functions

as in Eq. (C.20). It is sufficient to examine the behavior of a

single term in the sum (C.15). For example, look at

(*a\Ml))(Ml)\*b).

(C21)

(C.21) is nonzero only

in \^b) ipi'

occupied, ipi unoccupied, while in |^>

)

ijj[i

is unoccupied, ipi is occupied, and otherwise

and

^>b

are identical. To be

explicitly, let

l^>=£(-l)'

^lVM(si)>hMs2))|V>3(s3)) (C.22)

|^>

E(-

r-Ll^(*i)>l^(^2))|V'4(^)>

(C23)

Derivations 911

and look at

|^(l))(V

(l)|**).

The parts of the wave functions that survive are

1 r

(C.24)

(^(2)|(Vi(3)|-(Vi(2)|(^(3)|

'<V>2(l)hMl)>

|^(2))|^(3)) -|^3(2))|Vi(3))

(C.25)

(C.26)

The general lesson to learn from this example is that one must permute

ipi

past all

the states below it in the ordering scheme to produce the term |Vv(l)), obtaining a

factor of

(_!)£;=! »^ (C.27)

where «/ is

if state / is occupied in ty

and zero otherwise. One also has a factor

(-1)

v-V-l

(C.28)

similarly, so that

,/'-!

e^lVvOm'U)!**) =

(-1)^=«

^(-1)^=. "' (C.29)

7=1

if it is not zero.

Therefore, one can again define the operator

£/«_/>

from

Eq.

(C.

14).

Write wave

functions in the occupation number representation

|*> = |«1«2«3

 

>, (C30)

where each

n,-

can be either zero or one. In the example above,

) = |1110000. . .)

) =

1011000. . .).

In acting on such a wave function

£/<_//|«in

n3 • • •}

(-i)EU' ">(-l)^=i

nill

o\n

in2

m . . . n

. . . ni +

. . .).

The creation and annihilation operators are defined so that

(C.31)

(C.32)

(C.33)

£/<_/>

=0%,.

(C.34)

912 AppendixC. Second Quantization

More explicitly,

Q|«in2«3 • •

■)

= 5i,n,(-l)

j=1 n

'\n\n

m . . . rc

;

_i 0 «

/+]

. . .) (C.35a)

cj|«in

«3 • • ^^o.nX-

)

i=l y

|"i"2"3 • •

•

"/-i 1 "/+1 • • •)• (C.35b)

The anti-commutation relations in Eq. (C.3) can be verified explicitly from this

definition.

A final relation that should be verified is Eq. (C.IO). The special ordering of

the creation and annihilation operators results from the condition j ^ / in the sum

over particle numbers. Write

E fjr

= E IV'/O0)I^K/))(V'/a)#(/)l/i/l#K;>/-(/))(V'/»(i)K#»(/)l

;//

(C.36)

= E

[\MJ)){^i"U)\]

[l^'(/)><#»(/)l]

{MJ)MJ')\fjjWi>iJ)rPi>»(j'))

E <W' [\Mj)H^i'"U)\] (MJ)MJ)\fjjWi»UWi>"U))

(C.37)

ll>l"t"'

,,1,11,111

'{"V"

- Y. 5/'/"^/"'^/(l)#(2)|/i2|#'(l)#»(2)> (C38)

E c]"c

//c

,c/»'(^/(l)#(2)|/i

|V'/"(l)#''(2))

E <*/'/» £/V (^(1)#(2)|/

|^/»(1)^»(2)> (C.39)

E c

///c

//(V/(l)V/'(2)|/,

|#'(l)#»(2)>. (C40)

//'/"/

//'/"/"

//'/"/'•