quantum mechanics course degenerate pt

8/14/2019 Quantum mechanics course Degenerate Pt

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Lecture 21. The Variational Method

* Degenerate perturbation theory

The coupled quantum well

* The Variational method

The triangular well


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Here we discuss the application of DEGENERATE perturbation theory to the problem ofCOUPLED quantum wells

* The system we consider is shown below and consists of a well centered on xL andanother that is centered on xR

This problem CAN be solved exactly using a discussion of T-matrices

* It seems a reasonable GUESS however that the lowest two states of the double wellshould consist of a MIXTURE of the lowest states (L & R) in each of the two wells

We therefore develop a treatment of DEGENERATE perturbation theory in termsof these two states

Degenerate Perturbation Theory

THE PROBLEM OF COUPLED QUANTUM WELLS THAT WESTUDY HERE IS ANALOGOUS TO THAT OF A DIATOMIC

MOLECULE

THECOUPLING STRENGTHBETWEEN THE TWO WELLS

IS DETERMINED BY THE STRENGTH OF THEIRCOMMON

BARRIER

THEGROUND-STATEWAVEFUNCTIONS OF THE TWO

WELLS ARE ALSO SHOWN


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We begin by writing the Schrdinger equations for the left and right wells INDIVIDUALLYas

* The next step is to write the wavefunction for the COUPLED system as a linear sum ofsingle-well eigenfunctions

* With this definition the Schrdinger equation for the coupled system becomes

Where we have defined the following matrix elements as


)1.21()(LLL

VT )2.21()(RRR

VT

n

nna )3.21(

)4.21( n

nmn

n

nmnaSEaH

)6.21(*nmmn

S )5.21(*nmmn

HH


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While we are used to matrix elements of the form of Smn VANISHING when m n due tothe property of orthonormality this is NOTthe cases here

* The point here is that the wavefunctions L and R are instead solutions to SEPARATESchrdinger equations

* The matrix equation to be solved is now a generalized eigenvalue problem


L RL(x) R(x)

)7.21(*mnnmmn

S

)8.21(ESH aa

THE EIGENFUNCTIONSL ANDRARENOTORTHONORMALSINCE THEY ARE SOLUTIONS TOSEPARATESCHRDINGER

EQUATIONS

THE POINT TO NOTE IS THAT THE TWO WELLS ARESHIFTED

FROM EACH OTHER BY A DISTANCExR xL

IF WE CONSIDER ANY GIVEN EIGENSTATE OF THEINDIVIDUAL

WELLS THEN WECANNOTSAY THATL(x) =R(x)


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The Hamiltonian may be solved here may be written in the form

* We begin by calculating the matrix element HLL which may be written as

The additional term c here is referred to as the CRYSTAL FIELD and gives theexpectation value of the added potential VR for the wavefunction L and wedenote this term as

c to remind us that the potential wells are ATTRACTIVE

The other DIAGONAL element is of the same form as Eq. 21.10


)9.21(H

RRRL

LRLL

HH

HH

)10.21()(**

cVVVTHLRLLRLLLL

)11.21()(**

cVVVTHRLRRRLRRR


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Similarly it can be shown that the two OFF-DIAGONAL terms are also EQUAL

* The integral t is referred to as the TRANSFER, TUNNELING or OVERLAP INTEGRAL

* The integral s results from the NON-ORTHOGONALITY of the basis functions L and Rand from its definition the matrix S that appears in Eq. 21.8 may be written as

With these results the matrix form of the Schrdinger equation may be written as


)12.21()(***

LRLRRLRLRLRRLHtsVVVTH

)13.21(1

1S

s

s

)14.21(1

1aa

s

sE

cts

tsc


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Solution of Eq. 21.14 requires

* This yields the following ROOTS

The approximation of neglecting all HIGHER states that we have employed herewill be valid as long as the overlap integral s is SMALL

This corresponds to the case where the coupling between the two wells is WEAKand the expressions above can then be expanded as


)15.21(0det0H)Sdet(

cEtsEs

tsEscEE

)16.21(11

,11 s

t

s

cE

s

t

s

cE

)17.21()( tcstE


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Eq. 21.17 shows that the effect of the tunneling integral between the wells is to SPLIT theenergy levels of the two wells

* It is straightforward to show that the wavefunctions corresponding to these energylevels can be written as

As discussed for molecules these are just the BONDING and ANTI-BONDINGstates where the anti-bonding state is + and has HIGHER energy


)18.21()( tcstE

)19.21()(2

1RL

INDIVIDUAL WELLS AND THEIR GROUND

STATE WAVEFUNCTIONS

BONDINGANDANTI-BONDINGSTATES OF THE

COUPLEDQUANTUM WELL

-

+E-

E+


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The variational method is a specialized approach that may be used to ACCURATELYestimate the GROUND-STATE energy of a system

* Obviously we know that this ground state has an associated eigenfunction 1 thatmust satisfy the Schrdinger equation

* If we multiply both sides of this equation by 1* and integrate we obtain

* Reorganization of this equation then yields

The Variational Method

)20.21(111 H

)21.21(1

*

1111

*

11

*

1 H

)22.21(1

*

1

1

*

1

1

H


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If instead of the eigenfunction 1 we now consider some ARBITRARY wavefunction y thatsatisfies the required boundary conditions then the variational principle asserts that

* To prove the variational principle we start by expressing the wavefunction y in termsof the TRUE eigenfunctions of the Schrdinger equation

* With this substitution the RHS of Eq. 21.23 becomes


)23.21(*

*

1

H

)24.21(n

nna

)25.21(*

*

*

*

n nn

n nnn

aa

aaH


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We thus arrive at the variational principle by noting that the following inequality holds

* The basic idea of the variational method is to GUESS the form of the wavefunction witha small number of ADJUSTABLE parameters

These parameters are then adjusted to MINIMIZE the ground-state energy

* As an APPLICATION of this method we estimate the ground-state energy in aTRIANGULAR well

We assume the so-called FANG-HOWARD form for the ground state wavefunctionin which the parameter b is allowed to vary FREELY


)26.21(*

*

11*

*

1

*

*

H

aa

aa

aa

aa

n nn

n nn

n nn

n nnn

)27.21(2

1exp)(

bxxx


12/14


13/14

To minimize the RHS of Eq. 21.30 we differentiate with respect to b and set theresult to zero

* The variational principle may then be expressed as

This result should be compared with the EXACTone found previously for thetriangular well


)31.21(2

62

smeE

b

)32.21(2

)(4764.2

2

)(

16

2433/1

23/1

23/1

1

m

eE

m

eEss

3/12

12

)(3381.2

m

eEE

s

VARIATIONAL METHODAGREESWITH

THE EXACT RESULT TO WITHIN6%!


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Homework

P21.1 Use variational method for solving the Schrodinger equation for the truncatedharmonic oscillator potential

Use the trial function x)=xexp(-bx) (b is a variational parameter) to calculate anapproximate value for the ground state energy and compare with the exact result.

0for

0for2

1)(

2

x

xkxxV

quantum mechanics course degenerate pt

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