The following is the problem that I'm struggling with from the book of Gasiorowicz:
If we have an electron in a potential well of width $a$ and depth $V_0$, then the kinetic energy is, by the uncertainty principle, larger than ${\hbar^2 \over 2ma^2}$. Thus to get a bound state, the kinetic energy ust not only be negative, but it must also be larger in magnitude than ${\hbar^2 \over 2ma^2}$. On the other hand, there is always a bound state in one dimension, no matter how small $V_0$ is. What is wrong with this argument?
If we have an electron in a potential well of width $a$ and depth $V_0$, then the kinetic energy is, by the uncertainty principle, larger than ${\hbar^2 \over 2ma^2}$. Thus to get a bound state, the kinetic energy must not only be negative, but it must also be larger in magnitude than ${\hbar^2 \over 2ma^2}$. On the other hand, there is always a bound state in one dimension, no matter how small $V_0$ is. What is wrong with this argument?
This leads to my confusion about the the interpretation of the lower bound of the standard deviation of energy followed by the uncertainty principle. Does $\Delta K={\Delta p^2 \over 2m}\ge {\hbar^2 \over 2ma^2}$ really mean the kinetic energy must be larger than ${\hbar^2 \over 2ma^2}$?
The suggested solution for this problem is that the argument is wrong because the electron is not localized in the potential. Can anyone help me understand this?
