Why is the Time Independent Schrodinger Equation so important? [closed]

Question

The main equation of Quantum Mechanics (QM) is the Schrodinger Equation (SE): $$i\hbar\frac{\partial \psi (x,t)}{\partial t}=H(x,t)\psi(x,t)$$ Why is this equation so important? It's important because if we know the state (the wavefunction) of a particle at a time $t_0$ SE allows us to predict, to know, the state (wavefunction) of the particle at any other time $t$.

There is also another fundamental thing to know about this topic:

• if the Hamiltonian is time independent
• and if the wavefunction at the time $t_0$ is an eigenfunction of the Hamiltonian ($H\psi(x,t_0)=E\psi(x,t_0)$)

then finding the wavefunction at a time $t$ get's a lot easier! In fact, in this case, we can prove that:

1. The wavefunction is factorizable: $\psi (x,t)=\phi(x)g(t)$
2. $g(t)$ must equate: $$g(t)=\exp{\left[\frac{1}{i\hbar}E(t-t_0)\right]}$$
3. $\phi(x)$ is constant (this is obvious given 1.) and is indeed the eigenfunction of $H$ (also obvious given 1.): $$\phi(x) \ | \ H\phi(x)=E\phi(x)$$

The equation $$H\phi(x)=E\phi(x) \tag{1}$$ is of course the eigenfunction equation for $H$, but it's also called: time independent Schrodinger equation.

My question is: is that it? Am I missing something? If this is it I don't really understand why (1) deserves a name of its own, why it's called "Schrodinger equation".. (1) seems to be simply a part of a mathematical trick to find the solutions to a differential equation under some special initial conditions.

If, in a QM problem/exercise, the wavefunction $\psi(x , t_0)$ is not an eigenfunction of $H$ then the time independent Schrodinger equation is useless, am I right? It is only useful in problems when the initial state of the particle happens to be a stationary state (synonym for eigenfunction of H), right?

Are there other uses for the time independent Schrodinger equation?

Answer 1

Suppose the system starts with a wavefunction that is not an eigenfunction of $H$: is there any convenient way of calculating its time evolution? Yes there is: you can expand it in terms of eigenfunctions, and (sometimes) easily calculate future states.

If the eigenfunctions are $\psi_n(x)$ where $n$ indexes the eigenvalues, then you can express $\psi(x,t)$ as $$\psi(x,t)=\sum_n a_n \psi_n(x) e^{iE_n t/\hbar}$$ where $a_n = \langle \psi_n(x) | \psi(x,t_0)\rangle$.

This is a good reason to use the time independent equation.

Answer 2

It is a matter of terminology: we could call it Schrödinger equation, and not have a special name for the equation used in the eigenvalue problem afte the separation of variables. However, since we often have to resort to one or to the other (time dependent and time independent) it is practical to have two different names.

An important general point to stress regarding physics terminology is that there is no in existence any international committee, giving names to equations, models, approximations, etc. These arise naturaly in scientific discussion and publications and become established names, if many people use them and approximately agree on what is meant by the name.

The agreed names exist in some rather special cases, such as measurement units (e.g., SI system), chemical compounds, biological species, etc. Also, the teachers, particularly in school or early university years, tend to insist on rigorous definitions - a practice which is really meant to avoid students getting too confused about the subject, but which often becomes an end in itself and certainly does not prepare one to the "real life", where the definitions are mostly ambiguous (just like the human langauge in general).