Vitamin C

Hello, In the last post I demonstrated how to compute a perturbation on the harmonic oscillator up to fourth order. In this post we'll do the same thing but using creation and annihilation operators. The eigenvalues obtained with this method are based on diagonalizing an n by n matrix where n is the number of solutions. Since there are an infinite number of solutions the eigenvalues will converge to the true values as the dimension of the matrix increases to infinity. Since we're not going to be diagonalizing the infinite matrix we can only obtain approximate solutions to the perturbed hamiltonian. A really great resource for this and more examples/examinations of the method can be found here The creation operator takes the form: The annihilation operator is defined as: We generated the above matrices with the following code: We can then redefine the x operator in terms of our new operators, and use those to form a matrix that represents ou

Hello, I've wanted to implement perturbation theory in mathematica for some time now. Here, I've done so for a non-degenerate time independent case. Specifically for the one dimensional harmonic oscillator (not quantum because symbolic constants just make the computation take longer). I tested it on some problems from Shankar's quantum mechanics book and it works. The things you'll need are: 1. Eigenvalues and eigenfunctions for a solved case (in our case the harmonic oscillator). $$H0=\frac{\text{$\#$1} x^2}{2}-\frac{\nabla _{\{x\}}^{}\text{$\#$1}}{2}\&$$ \[ScriptCapitalH]0 = -(1/2) Laplacian[#1,{x}] + 1/2 (x^2) #1 &; \[ScriptCapitalH]1 = x^2 #1 &; \[ScriptCapitalH] = \[ScriptCapitalH]0[#1] + \[ScriptCapitalH]1[#1] &; {eigList, wfnList} = DEigensystem[\[ScriptCapitalH]0[u[x]],u[x],{x}\[Element] FullRegion[1], nmax, Method -> "Normalize"]; where nmax is the maximum number of eigenvalues/eigenfunctions to compute.