approximation methods to the schrödinger...

Approximation Methods to the Schrödinger Equation

2006 Quantum Mechanics Prof. Y. F. Chen




Approximate solutions can be evaluated for a large variety of systems, and

these results have contributed inestimably to the practical application of

quantum theory to the development of modern physics.

We will describe a systematic general approach to the construction and

interpretation of approximate solutions to the Schrödinger equation.



Suppose we have a system Ho, for which the Schrödinger wave equation

can be solved exactly. Consider a second system H, which is physically

similar to system Ho in many respects but whose wave equation can not

allow an exact solution.

If system H can be imaged to have been formed from Ho by the introduction

of a small, continuous deformation, or perturbation, it appears reasonable

that one could approximate the wave functions of system Ho by applying a

small, continuous mathematical perturbation to the known wave functions of

system H.

Nondegenerate Perturbation Theory



We will assume that system H (called the unperturbed system) has no

degeneracies. There is an unambiguous one-to-one correspondence

between the wave functions of systems Ho and H and that a similar

correspondence exists for the eigenvalues of the two systems.

Let be the Hamiltonian operator of system Ho, and let be a

complete set of orthonormal eigenfunctions of .

The corresponding set of eigenvalues is given by .


oH { })0(nψ

oH

{ })0(nE



The Hamiltonian of system H be given by

(11.1)

where is called the perturbation and is generally thought of as a small

additional term which distorts, or perturbs, the system from Ho and H.

It is mathematically convenient to express the perturbation in terms of

a dimensionless real parameter λ:

(11.2)


po HHH ˆˆˆ +=

pH

pH

VH pˆˆ λ=



The eigenfunctions and eigenvalues of the perturbed system H will be

designated as and , respectively.

Assume that the eigenfunctions and eigenvalues of the perturbed system

can be expressed as power series in the parameter λ.

(11.3)

and

(11.4)


{ }nψ { }nE

∑∞

=

+=1

)()0(

j

jn

jnn ψλψψ

∑∞

=

+=1

)()0(

j

jn

jnn EEE λ



The terms and are called the jth-order perturbations of the

wave function and energy, respectively, of the nth state.

The Schrödinger equation for the nth state of the perturbed system is given

by

(11.5)

Substituting (11.3) and (11.4) into (11.5),

(11.6)


)( jnψ )( j

nE

( ) nnno EVH ψψλ =+ ˆˆ

0ˆˆ1

)()0(

1

)()0( =⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−−+ ∑∑

∞

=

∞

= j

jn

jn

j

jn

jno EEVH ψλψλλ



Expanding the above equation and collecting terms according to ascending

powers of the perturbation parameter λ,

(11.7)

The first term is just the Schrödinger equation for the unperturbed state and

is called the zeroth-order equation. Eq. (11.7) is satisfied for only if

each coefficient of each power of λ vanishes separately.

The general form of the coefficient of is given by

(11.8)

which is called the jth-order perturbation equation.


( ) ( ) 0ˆˆˆ1

1

0

)()()1()()0()0()0( =⎥⎦

⎤⎢⎣

⎡−+−+− ∑ ∑

∞

=

−

=

−−

j

j

k

kn

kjn

jn

jnno

jnno EVEHEH ψψψλψ

0≠λ

jλ

( ) 0ˆˆ1

0

)()()1()()0( =−+− ∑−

=

−−j

k

kn

kjn

jn

jnno EVEH ψψψ



The jth-order energy is obtained by multiplying (11.8) on the left by

and integrating over all configuration space to obtain

(11.9)

choose the wave function of the perturbed state such that they satisfy the

relationship

(11.10)


)0(nψ

( ) 0ˆˆ1

0

)()0()()1()0()()0()0( =−+− ∑−

=

−−j

k

knn

kjn

jnn

jnnon EVEH ψψψψψψ

1)0( =nn ψψ



Substituting (11.3) into (11.10), it can be verified that all the perturbation

functions of the nth state are orthogonal to the wave function of the

corresponding zeroth-order state,

(11.11)

the only term which survives in the summation part of (11.9) is the one for

which .

the general expression for the jth-order energy correction is given by

(11.12)

Eq. (11.12) indicates that it is necessary to know the -order

perturbation correction to the wave function to compute jth-order correction

to the energy.


kk

nn ,0)()0( δψψ =

0=k

)1()0()( ˆ −= jnn

jn VE ψψ

)1( −j



The first-order energy correction is given by

(11.13)

which is simply the perturbation averaged over the unperturbed state of the

system.

It is convenient to incorporate the perturbation parameter λ

into the

definition of , so that the total energy of the nth state of the perturbed

system to a first-order approximation is given by .

The jth-order correction to the wave function can be expanded in the

complete set of eigenfunctions of the unperturbed system,

(11.14)


)0()0()1( ˆnnn VE ψψ=

)( jnE

)1()0(nnn EEE +=

∑≠

=nm

jmnm

jn c )(

,)0()( ψψ



Substituting (11.14) into (11.8) for the first-order correction,

(11.15)

Multiplying on the left by ( ) and integrating over all

configuration space, we obtain

(11.16)

Note that only terms survive. The coefficients are then given

by

(11.17)


( ) )0()1()0()1(,

)0()0()0( ˆnnn

nmmnmnm EVcEE ψψψ +−=−∑

≠

)0(iψ ni ≠

( ) )0()0()1(,

)0()0( ˆnmmnnm VcEE ψψ−=−

nmi ≠= )1(,mnc

( ))0()0(

)0()0()1(

,

ˆ

mn

nmmn EE

Vc

−=

ψψ



Using (11.3) and (11.17),

(11.18)

where it is customary to incorporate the parameter λ with the matrix element

.

Substituting (11.18) into (11.12), the second-order energy correction is

found to be

(11.19)


( )∑≠ −

+=nm

mmn

nmnn EE

V)0(

)0()0(

)0()0()0(

ˆψ

ψψψψ

)0()0( ˆnm V ψψ

( )∑≠ −

=nm mn

nmn EE

VE )0()0(

2)0()0()2(

ˆ ψψ



With (11.19), the eigenenergy to second order is given by

(11.20)


( )∑≠ −

++=nm mn

npmnpmnn EE

HHEE )0()0(

2)0()0()0()0()0(

ˆˆ

ψψψψ



For perturbation theory to work, convergence must be achieved with the first

two corrections.

Expressions (11.18) and (11.20) show that the convergence criterion can be

found to be given by

(11.21)

If the unperturbed energy levels and are too close (i.e., nearly

degenerate) then condition (11.21) would break down.

Nearly degenerate energy levels require a method that is different from the

nondegenerate treatment. This issue will be considered in the following

section.


( ) 1ˆ

)0()0(

)0()0(

<<− mn

nm

EEV ψψ

)0(nE )0(

mE



Example

A particle of charge q and mass m in a one-dimensional harmonic potential

of frequency w is subject to a weak electric field E . (a) Find the exact

expression of the eigenenergy. (b) Using the perturbation theory to

calculate the eigenenergy and to compare it with the exact result obtained in

(a).




Solution

The external electric field brings about a term that needs to be

added to the Hamiltonian of the oscillator:

(11.22)

(a) The eigenenergies of (11.22) can be obtained exactly by using a

variable change . Consequently, the Hamiltonian can be

written as

(11.23)


xqH p ˆˆ E=

xqxmxd

dm

HHH p ˆˆ21

2ˆˆˆ 22

2

22

0 E+⎟⎟⎠

⎞⎜⎜⎝

⎛+−=+= ωh

2ωmqxx E+=′

( )2

222

2

22

2ˆ

21

2ˆ

ωω

mqxm

xdd

mH E

−⎟⎟⎠

⎞⎜⎜⎝

⎛′+

′−=h



Eq. (11.23) is the Hamiltonian of a harmonic oscillator from which a

constant, , is subtracted.

Therefore, the exact eigenenergies can be easily found to be

(11.24)

(b)Note that the first-order correction to the energy is zero because

(11.25)

With (11.19), the second-order correction to the energy is given by

(11.26)


( ) 22 2 ωmqE

( )2

2

221

ωω

mqnEnE

−⎟⎠⎞

⎜⎝⎛ += h

0ˆ)1( == nxnqE n E

( ) ∑≠ −

=nm mn

n EEnxm

qE )0()0(

22)2(

ˆE



Using the relations:

(11.27)

and

(11.28)

We can obtain

(11.29)

It can be found that the result in (11.29) agrees fully with the exact energy

obtained in (11.24).


( )1,1,12

ˆ −+ ++= nmnm nnm

nxm δδωh

ωh⎟⎠⎞

⎜⎝⎛ +=

21)0( nE n

( ) ( ) ( ) ( )2

2

)0(1

)0(

2

)0(1

)0(

22)2(

2)2()2(1

ωωω

mq

EEmn

EEmn

qEnnnn

nEE −=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+

−+

=−+

hh



using (11.18), the wave function of the nth state of the perturbed system to

a first approximation is given by

(11.30)

Note that the expression (11.30) is not the same as the exact wave function.


[ ]

[ ]1112

12

)0(1

)0(1

)0(

++−−+=

+−+= +−

nnnnm

qn

nnm

qnnnn

ωω

ψψωω

ψψ

h

h

h

h

E

E



We employ perturbation theory to find the energy spectrum and the states

of a system whose unperturbed Hamiltonian has numerous nearly

degenerate states:

(11.31)

If there exists a set of m different eigenstates , where ,

that have eigenenergies to be nearly degenerate:

(11.32)

Perturbation Theory for Nearly degenerate States

oH

( ) nnnon EHHH ψψψ =′+= ˆˆˆ

{ }kn,φ mk ,.....,2,1=)0(

,knE

knknkno EH ,)0(

,,ˆ φφ =



In the zeroth-order approximation the eigenfunction can be

expressed as a linear combination in terms of :

(11.33)

Note that the states are orthonormal with respect to the index k.

Substituting (11.33) into (11.31), we obtain

(11.34)


nψ

{ }kn,φ

∑=

=m

kknkn c

1,φψ

{ }kn,φ

( ) ∑∑==

=′+m

kknkn

m

kknko cEcHH

1,

1,

ˆˆ φφ



Multiplying on the left by and integrating over all configuration space,

we obtain

(11.35)

where .

Eq. (11.35) can be written as

(11.36)

where .


in,φ

[ ] ni

m

kkninkiknk EcHEc =′+∑

=1,,,

)0(,

ˆ φφδ

mi ,.....,2,1=

( )[ ] 0ˆ1

,,,)0(

, =−′+∑=

m

kkinkikiknk EHEc δδ

kiki HH φφ ′=′ ˆˆ,



More explicitly, the representation of (11.36) yields the following matrix

eigenvalue equation:

(11.37)

This is an mth order equation in ; its solutions yield the energy

spectrum of the system: .

Knowing the set of eigenvalues , we can

determine the coefficients by substituting the eigenvalues into (11.35)

and then solve the resulting expression.


( )( )

( )0

ˆˆˆ

ˆˆˆˆˆˆ

,)0(

,2,1,

,22,2)0(2,1,2

,12,11,1)0(

1,

=

−′+′′

′−′+′′′−′+

nmmmnmm

mnn

mnn

EHEHH

HEHEHHHEHE

L

MOMM

L

L

nE

mnnnn EEEE ,3,2,1, ,,, K

mnnnn EEEE ,3,2,1, ,,, K

kc



One of the most significant applications of perturbation theory is to find the

energy corrections for the hydrogen atom, especially the corrections due to

the fine structure and the Zeeman effect.

In the rest frame of the electron the proton appears to orbit electron and this

moving positive charge produces a magnetic field, which interacts with the

intrinsic magnetic moment of the electron.

Applications of Perturbation Theory: Fine structure



This adds a term of the form to the potential energy, where B is

the magnetic field due to the orbital motion of the proton relative to the

electron and is the intrinsic magnetic moment of the electron.

To find the energy correction, we use the fact that if the electron’s velocity is

; then the proton’s velocity from the electron’s rest-frame point of view

is .

Also, if the radius vector is from the proton to the electron, from the

electron to proton it becomes .


sB BV μvv⋅−=

sμv

vv

vv−

rv

rv−



From the Biot-Savart law, the magnetic field experienced by the electron is

given by

(11.38)

where is the orbital angular momentum of the electron.

The interaction of the electron’s spin dipole momentum with the orbital

magnetic field of the nucleus brings about the interaction energy

.

However, this energy turns out to be twice the observed spin-orbit

interaction because this result is based on the rest frame of the electron.


33 rL

cme

rvr

ceB

e

vvvv=

×=

prL vvv×=

sμv

Bv

BH sso

vv ⋅−= μˆ



For a correct treatment, we need to transform the rest frame of the nucleus.

This transformation gives rise to an additional motion resulting from the

precession of ; it is known as the Thomas precession.

The transformation back to the rest frame of the nucleus leads to a

reduction of the interaction energy by a factor 2.

Therefore, the spin-orbit interaction energy for the hydrogen atom is given

by

(11.39)


sμv

( )LS

re

cmrL

cmeH

ee

sso

vrvv

⋅=⋅= 3

2

23 21

2ˆ μ



The perturbation theory can be used to calculate the contribution of the

spin-orbit interaction energy for the hydrogen atom.

With (11.39), the Hamiltonian for the hydrogen atom, including spin-orbit

coupling, is given by

(11.40)

where is the unperturbed Hamiltonian and is the perturbation

term.


( )LS

re

cmreHHH

eso

ˆˆ2

12

ˆˆˆ3

2

2

22

2

0

vrh⋅+−∇−=+=

μ

0H soH

approximation methods to the schrödinger...

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