For the quartic oscillator ( )

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Quantum Mechanics · Scientific programming for physicists

Scientific Computing for Physicists, Ch Letting the computer do advanced algebra: operators, series and differentiated programs

Problem

For the quartic oscillator

\hat H=\tfrac12p^2+\tfrac12x^2+gx^4

(

\hbar=m=\omega=1

), generate the series for the first excited level

E_1

, whose unperturbed value is

\tfrac32

. ((a)) Find

\langle1|x^4|1\rangle

from the chapter's fourth-moment result. ((b)) From the normal-ordered form of

(a+a^\dagger)^4

, find every state

|j\rangle

with

\langle j|x^4|1\rangle\ne0

j\ne1

, and the value of each matrix element. ((c)) Using the same second-order formula as for the ground state, with

E_j-E_1

in the denominators, write

E_1

to order

g^2

. ((d)) Evaluate your series at

g=0.01

and compare it with a diagonalisation of

\hat H

in the first

40

number states. The third-order coefficient, from carrying the generation one order further, is

\tfrac{3915}{16}

. Is the disagreement what the series predicts?