- #1
vbrasic
- 73
- 3
Homework Statement
I have ##H'=ax^3+bx^4##, and wish to find the general perturbed wave-functions.
Homework Equations
First-order correction to the wave-function is given by, $$\psi_n^{(1)}=\Sigma_{m\neq n}\frac{\langle\psi_m^{(0)}|H'|\psi_n^{(0)}\rangle}{n-m}|\psi_m^{(0)}\rangle.$$
The Attempt at a Solution
The formula becomes (focusing exclusively on the cubic term), $$\psi_n^{(1)}=\Sigma_{m\neq n}\frac{\langle\psi_m^{(0)}|ax^3|\psi_n^{(0)}\rangle}{n-m}|\psi_m^{(0)}\rangle.$$
(I shall denote, ##|\psi_{n,m}^{(0)}\rangle## as ##|n,m\rangle## to avoid typesetting overly complicated formulas. Using the ladder operator definition of ##x## then, I have, $$|n^{(1)}\rangle=\frac{a(\frac{\hbar}{2m\omega}^{3/2})}{\hbar\omega}\Sigma_{m\neq n}\frac{\langle m|(a+a^+)^3|n\rangle}{n-m}|m\rangle.$$
Now the trouble is, according to my textbook, the cubic term contributes a first-order correction to the wavefunction, but I'm not seeing it. There would never be an equal number of raising and lowering operators, such that by orthogonality all the coefficients would become ##0##.