(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm working on problem 3 on page 136 in Landau and Lifgarbagez Quantum mechanics nonrelativistic theory.

The problem is to determine the energy levels of an anharmonic linear oscillator with

H=H_{0}+ a*x^{3}+ b*x^{4}

(where H_{0}is the hamiltonian of the harmonic oscillator).

To solve this using pertubation theory, I need to find the matrix elements of x^{3}and x^{4}. I have tried many times, but I don't get the same answer as Landau.

2. Relevant equations

Equation (23.4) gives the matrix elements of x: x_{n,n-1}=x_{n-1,n}=sqrt(nh/2mw).

I have no problem getting the pre-factors right, so I simplify this expression by not writing explicitely the the constant term and the square root, so I get

x_{n,n-1}=x_{n-1,n}= n

I use the standard rule for matrix multiplication:

(AB)_{n,m}= SUM_{r}(A_{n,r}B_{r,m})

3. The attempt at a solution

I get the following non-zero elements of x^{2}using the above formulas

x^{2}_{n,n}=x_{m,m-1}x_{n-1,n}+x_{n,n+1}x_{n+1,n}= n^{2}+ (n+1)^{2}

x^{2}_{n,n-2}= x^{2}_{n-2,n}= x_{n,n-1}x_{n-1,n-2}= n(n-1)

This gives next

x^{3}_{n,n-1}=x_{n,n-1}x^{2}_{n-1,n-1}+ x_{n,n+1}x^{2}_{n+1,n-1}= 3n^{3}+2n

I can't find any flaw in this, yet in Landau the solution is given as 9n^{3}

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# Homework Help: Matrix elements of x^3

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