Time ind. S.E., qualitative solution

[jalalmalo]

In Robert Scherrer´s "Quantum Mechanics, an accessible intorduction", starting from page 53 the author gives qualitative solutions to the time independent S E with a definite energy:
d^2psi/dx^2=2m/hbar^2(V(x)-E) by studying the sign of the function and the second derivative for different values of E and V, which leads some graphs for function which behave well and the author makes some deductions from that. What buzzles me is the fact the equation is easily solved analytically, and the function will have two dinstict cases. one exponential if V>E and the second is complex when E<V.

Why would the author go through all this trouble instead of just solving the equation and in the case of complex function why even bother abouth trying to figure out the behaviour of the function psi, it have no physical meaning after all. anny comments would be appreciated

 

[alxm]

Well I can't speak for the author, but my guess is that his intention is to try to introduce this way of thinking about the problem. ("What can we say about the solution? What properties will it have?")

Once you start looking at real systems, the wave equation will not have an analytical solution, and you're left with trying to figure out what you can say about it. Indeed, this is essentially how the work is done; not by empirically studying the solution, but analytically characterizing it. Using symmetry arguments for instance.

 

[jalalmalo]

In Robert Scherrer´s "Quantum Mechanics, an accessible intorduction", starting from page 53 the author gives qualitative solutions to the time independent S E with a definite energy:
d^2psi/dx^2=2m/hbar^2(V(x)-E) by studying the sign of the function and the second derivative for different values of E and V, which leads some graphs for function which behave well and the author makes some deductions from that. What buzzles me is the fact the equation is easily solved analytically, and the function will have two dinstict cases. one exponential if V>E and the second is complex when E<V.

Let me refraise. If u would have this S E could u conclude something physical from it when V<E, i e when psi gives a complex value? Besides taking the absolut square and integrating to get a probabilistic interpretation.

 

[alexgs]

You are right, when V>E the wave function is a dying exponential. This situation is never happens classically but in quantum mechanics this usually manifests itself as the particle being able to tunnel into a region where V>E. The dying exponential indicates that you are less and less likely to find the particle the deeper into the barrier (region where V>E) you look.

When V<E the particle is free(ish): the wavefunction is periodic and this corresponds to the particle being able to move through space (the only possibility in the classical case).

I'm guessing the author is talking about the distinction between a propagating particle (V<E) and a tunneling one (V>E).

You should work out an example with a simple potential to get the idea. Try V(x)=0 when x<0 and V(x)=V (some constant value) when x>0. What will the wavefunction look like in the regions x<0 and x>0 and what does this mean physically?

Hope that helps.