Linear combination of phi(n)

[Kit]

please refer to the attachment

in the last part, it states that...

If solutions phi.n.(r) can be found, for different values of En, then the linear combination is also a solution of the Schroedinger equation, because the Schroedinger equation is a linear equation. However, such a linear combination does not represent a particle with a well defined energy.

if it is not represent a particle with a well defined energy, then what is it represent?

does it represent all the energy states from ground state to En?

thx for answering

 

[RedX]

Let i be the energy representation
|psi>=Sum{|i><i|psi>}
Transforming to the position representation j
|psi>=Sum{Sum{|j><j|i>}<i|psi>}
|psi>=Sum{u_i(x)e^(-(E_i)t/h}<i|psi>}

So a superposition of the states of definite energy represents a state with the probability <i|psi> of having energy E_i.

So the Schrodinger equation (the equation which governs the dynamics of the wave equation in the language of position space) must be linear for it to give a correct representation of the quantum system.

 

[R0man]

The linear combination of phi(n) represents a superposition of different energy states. This means that the particle does not have a well-defined energy, but rather exists in a combination of different energy levels. This can be thought of as the particle being in a state of uncertainty about its energy.

It does not necessarily represent all energy states from ground state to En, as there can be an infinite number of possible energy states. However, it does represent a combination of energy states that are solutions to the Schroedinger equation.

It is important to note that this is a mathematical concept and may not have a direct physical interpretation. It is often used in quantum mechanics to describe the behavior of particles and their energy levels.