Eigenfunctions of combined Hamiltonian with separate potentials

[wenty]

If
H1=P^2/2m+V1(x), H2=P^2/2m+V2(x), H=P^2/2m+V1(x)+V2(x)
and
H1 f1_i(x)=E1_i*f1_i(x),
H2 f2_j(x)=E2_j*f2_j(x),
H f_k(x)=E_k*f_k(x)

Is there any relation between f1_i(x),f2_j(x),f_k(x)?Can we express f_k(x) in terms of f1_i(x) and f2_j(x)?

 

[dextercioby]

Not really.U've actually given a very abstract problem.In a QM description in the Schrödinger picture,it is essential to know the potential.Just then,after having specified the physical system (its interactions),u can apply the "machinery" of QM.

One of the most important mathematical theorems in QM regards CSCO-s.If your operators form a CSCO,then u can apply it.

Daniel.

 

[R0man]

Yes, there is a relation between f1_i(x), f2_j(x), and f_k(x). In fact, f_k(x) can be expressed as a linear combination of f1_i(x) and f2_j(x). This is because the Schrödinger equation is a linear equation, meaning that the solutions can be combined in a linear manner.

To understand this better, let's look at the general form of the Schrödinger equation:

HΨ(x) = EΨ(x)

Where H is the Hamiltonian operator, Ψ(x) is the wave function, and E is the energy of the system.

In your case, you have two separate Hamiltonians, H1 and H2, which correspond to two different potentials, V1(x) and V2(x). When these two potentials are added together to form H, the resulting wave function, f_k(x), will also be a combination of the individual wave functions, f1_i(x) and f2_j(x).

So, to answer your question, yes, f_k(x) can be expressed in terms of f1_i(x) and f2_j(x), with the coefficients of the linear combination depending on the specific potentials and energies involved.