# Length of an infinite square well?

by *melinda*
Tags: infinite, length, square
 P: 86 Actually, this is more of a general question relating to a homework problem I already did. I was given the initial wavefunction of a particle in an infinite square well: $$\Psi(x,0) = Ax$$ if $$(0 \leq x \leq \frac{a}{2})$$, and $$=A(a-x)$$ if $$(\frac{a}{2} \leq x \leq a)$$ And of course $$\Psi(0,0) = \Psi(a,0) = 0$$ I was asked to find $$\Psi(x,t)$$ , which I did, and I was also asked to find the probability that "a measurement of the energy would yield the value $$E_1$$", the ground state energy, which I also did. However, this probability is dependent on the length of the well, given by 'a'. I was curious about this, and I found that the probability that "a measurement of the energy would yield the value $$E_1$$" increases as the value of 'a' increases. Taking the limit as 'a' approaches infinity gives a probability of finding the energy in the ground state to be approx. 0.986, which means that there is a non-zero probability of finding the particle in another energy level. OK, as far as I can tell for a given wavefunction, increasing the length of the infinite square well increases the probability of finding the particle in the lowest energy state. Mathematically I understand this, but I am still lacking physical intuition about what is actually happening when the length of the well increases. So, for a given wavefunction, WHY does increasing the length of the infinite square well increases the probability of finding the particle in the lowest energy state? Thanks, I hope this question makes sense! *melinda*
 P: 370 Are you sure? Did you normalize the wavefunction properly? The probability is a dimensionless number: but a is a length, and it is the only length in the problem here: there is no way to make a dimensionless number that depends on a. So I'd think, without actually calculating anything, that the probability to find E_1 should be independent of a.