Changing the Hamiltonian without affecting the wave function

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Changing the Hamiltonian can be achieved without altering the wave functions by methods such as multiplying all matrix elements by a constant. Simplifying a complex Hamiltonian can facilitate easier derivation of wave functions. Canonical transformations are suggested as a viable approach for this simplification. Alternatively, reverting to the Lagrangian formulation offers more flexibility before converting back to the Hamiltonian. These strategies can effectively maintain the integrity of the eigenvectors while simplifying the Hamiltonian.
Isaac.Wang88
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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,:cry: I want to simplify it, so the wave function will be much easier to derive.
Thanks in advance.
 
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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.
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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