Changing the Hamiltonian without affecting the wave function

[Isaac.Wang88]

How many ways can we change the Hamiltonian without affecting the wave functions (eigenvectors) of it.
Like multiply all the elements in the matrix by a constant.
I'm facing a very difficult Hamiltonian, I want to simplify it, so the wave function will be much easier to derive.
Thanks in advance.

 

[UltrafastPED]

Try Canonical transformations.

Or revert to the Lagrangian and work it over - much more freedom there - then convert it back. This is equivalent to the canonical transformations.