Solving the classical problem of hydrogen hamiltonian eigenvalues one obtain energies (in atomic units) E_n=-0.5, -0.125... etc
These energies depend only on principal quantum number. How one can obtain orbital splitting in the electromegnetic field by means of numerically searching for eigenvalues of the hamiltonian with matrix elements < n L | H | n1 L1 >?
When I use only < n 0 | H | n1 0 > elements of free hamiltonian, I obtain E_n=-0.5, -0.125... etc (as expected).
If I use the whole matrix < n L | H | n1 L1 >, how can I obtain the correct energy levels in the case of atomic hamiltonian and in the case of the atom in the electromagnetic field by means of hamiltonian diagonalization?
As you certainly know, the hydrogen atom in an electric field and a magnetic field is in quantum mechanics text books normally treated under the headlines 'Stark effect' and 'Zeeman effect' respectively. All you may want to know about the energy values of these systems you can find there.
If you would like to approach the problem by diagonalizing a matrix, you will discover that the matrix to which you are led by straightforward discretisation is prohibitively large. So, although natural in principle, this is not (yet ?) a recommendable method.
Numerical methods, either for computation of eigenvalues or for the time evolution of wave functions can easily be implemented on PC's for one-dimensional systems. Here one replaces continuous 1D space by a lattice of discrete points (say 500) and uses the methods of Numerical Analysis to discretize the equations under consideration. Also in 2D this approach works for simple systems. Nice examples can be found by searching the net for e.g. 'Bernd Thaller' 'Visual Quantum Mechanics'
Thank for your quick reply.
Of course, I know about Zeeman and Stark effects. And I know, that in 1D space the problem can be solved easily, but this is not my case (as far as I can see).
In my case I have to diagonalize the hamiltonian (I am testing a new method of diagonalization). The question is how to deal with 4D matrix (two n indices and two L indices) < n L | H | n1 L1 >.
Methods that I know deal only with 2D matrices.
I can reduce 4D matrix to 2D one (see attachment), but in this case, if I solve the eigensystem for this new matrix, I obtain wrong eigenvalues.
The 4D vs 2D problem, as I see it, does not exist. Simply order the states lexicographically (or in any other systematic way). Explicitely for your example:
e_1 = |1 1>, e_2 =|1 2>, e_3 = |1 3>, e_4 = |2 1>, e_5 = |2 2>, e_6 = |2 3>,
e_7 = |3 1>, e_8 = |3 2>, e_9 = |3 3>
f_1 = |0 0>, f_2 = |0 1>, f_3 = |1 0>, f_4 = |1 1>
Then a 9 times 4 matrix M is given as
M_p_q := (e_p , H f_q ) (where I deliberately mishandled the bra ket formalism a bit )
By completing with valid data or by filling with zeros where appropriate one completes M to a square matrix and diagonalizes by some numerical method.
Lexicographical 2D matrix made from 4D is the method I used (see post #3). I thought that I have had a mistake in the algorithm.
After your post I checked again the program, a misprint was found, and now I get the right results. Thank you!
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