A question about the adding of potential in Schrodinger equation

In summary, the conversation discusses the relation between different functions (f1_i(x), f2_j(x), f_k(x)) in the context of the Schrödinger picture of quantum mechanics. It is mentioned that the operators must form a complete set of commuting observables (CSCO) for the "machinery" of QM to be applicable. The conversation ends with Daniel discussing the importance of CSCO in QM.
  • #1
wenty
20
0
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)?
 
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  • #2
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.
 
  • #3


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 Schrodinger 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 Schrodinger 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.
 

1. What is the Schrodinger equation?

The Schrodinger equation is a mathematical formula used in quantum mechanics to describe the behavior of particles in a quantum system.

2. What is potential in the context of the Schrodinger equation?

Potential in the Schrodinger equation refers to the energy associated with a particle in a particular position within a system. It can be thought of as the force that affects the movement of the particle in the system.

3. How is potential added in the Schrodinger equation?

In the Schrodinger equation, potential is added as a term in the equation, typically denoted as V(x) or V(r), depending on the variables used in the equation. This term represents the potential energy of the particle in a specific position within the system.

4. Why is the adding of potential important in the Schrodinger equation?

The addition of potential in the Schrodinger equation allows us to accurately describe the behavior of particles in a quantum system. It takes into account the varying energy levels and forces within the system, which are crucial in understanding the behavior of particles at the quantum level.

5. What are some examples of potential in the Schrodinger equation?

Some common examples of potential in the Schrodinger equation include the potential energy of an electron in an atom, the potential energy of a particle in a potential well, and the potential energy of a particle in a harmonic oscillator. Essentially, any system with varying energy levels can be described using potential in the Schrodinger equation.

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