Shankar Quantum Mechanics, Chapter 5, Page 160-161

[DaddyGriffiths]

TL;DR Summary

So why is it that he says there is 1 more constraint than free parameters instead of 2?

For the case of general potential V(x), what does it mean when he says that there are always one more constraint than free parameters? At each interval, ψ and ψ' must be continuous, so that is 2 constraints at each interval, and I understand that there are 2 parameters of the wavefunction in each interval between the leftmost and rightmost intervals. Getting rid of the rising exponential term at the extremes (x=±∞) so that the wavefunction doesn't blow up, that leaves us with one parameter at each extreme. So why is it that he says there is 1 more constraint than parameters instead of 2?

 

[BvU]

Hello @DaddyGriffiths ,

!​

Can you follow the argument half way on page 160 about finite potential well

A bit of context is also nice, for readers who don't have your book !​

​

This is about three regions with bounds ##\ -\infty, -L/2, L/2, \infty ## .​

The wave function has an A and a B free parameter in each region.​

Two are required to avoid blowup at ##\ \pm \infty## .​

At ##\ \pm L/2\ ## we require continuity in ##\Psi\ ## and ##\ \Psi'\ ## and that's four constraints.​

So no problem ?​

Not so: there is also the normalization constraint $$\int_{-\infty}^{+\infty}\Psi^* \Psi \, dx = 1$$ which determines the over-all scale for all parameters simultaneously.​

It may seem that there are four free parameters ... The over-all scale of ##\ \Psi\ ## is irrelevant in both the eigenvalue equation and the continuity conditions, these being linear in ##\ \Psi\ ## and ##\ \Psi'\ ##

 

[DaddyGriffiths]

Hello @DaddyGriffiths ,

!​

Can you follow the argument half way on page 160 about finite potential well

A bit of context is also nice, for readers who don't have your book !​

​

This is about three regions with bounds ##\ -\infty, -L/2, L/2, \infty ## .​

The wave function has an A and a B free parameter in each region.​

Two are required to avoid blowup at ##\ \pm \infty## .​

At ##\ \pm L/2\ ## we require continuity in ##\Psi\ ## and ##\ \Psi'\ ## and that's four constraints.​

So no problem ?​

Not so: there is also the normalization constraint $$\int_{-\infty}^{+\infty}\Psi^* \Psi \, dx = 1$$ which determines the over-all scale for all parameters simultaneously.​

Thank you! Turns out I forgot the normalization constraint and might've initially misinterpreted the part about the continuity constraints (it is actually 2 at each boundary, not 2 in each interval as I initially thought).