I'm self-learning quantum mechanics and I'm reading the famous Shankar book (Principles of Quantum Mechanics - second edittion). At page 161 of the book, I don't understand the following part of this page:(adsbygoogle = window.adsbygoogle || []).push({});

"Let us restate the origin of energy quantization in another way. Consider the search for acceptable energy eigen-functions, taking the finite well as an example. If we start with some arbitrary values ##\psi(x_0)## and ##\psi'(x_0)##, at some point ##x_0## to the right of the well, we can integrate Schrodinger's equation numerically. (Recall the analogy with the problem of finding the trajectory of a particle given its initial position and velocity and the force on it.) As we integrate out to ##x \to \infty##, ##\psi## will surely blow up since ##\psi_{III}## contains a growing exponential. Since ##\psi(x_0)## merely fixes theoverall scale, we vary ##\psi'(x_0)## until the growing exponential is killed. [Since we can solve problem analytically in regionIII, we can even say what the desired value of ##\psi'(x_0)## is: it is given by ##\psi'(x_0) = -\kappa \psi(x_0)##. Verify, starting with Eq. (5.2.4), that this implies ##B=0##.] We are now out of the fix as ##x \to \infty##, but we are committed to whatever comes out as we integrate to the left of ##x_0##. We will find that ##\psi## grows exponential till we reach the well, whereupon it will oscillate. When we cross the well, ##\psi## will again start to grow exponentially, for ##\psi_I## also contains a growing exponentially in general. Thus there will be no acceptable solution at some randomly chosen energy. It can, however, happen that for certain values of energy, ##\psi## will be exponentially damped in both regionsIandIII. [At any point ##x_0'## in regionI, there is a ratio ##\psi'(x_0')/\psi(x_0')## for which only the damped exponential survives. The ##\psi## we get integrating from regionIIIwill not generally have this feature. At special energies, however, this can happen.] These are the allowed energies and the corresponding functions are the allowed eigen-functions. Having found them, we can choose ##\psi(x_0)## such that they are normalized to unity. For a nice numerical analysis of this problem see the book by Eisberg and Resnick.$".

I need the formula details to understand it. Another question: "overall scale", what does it mean?

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# I Energy quantization of the particle in a box

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