Length of an infinite square well?

*melinda*

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:

[tex] \Psi(x,0) = Ax[/tex] if [tex](0 \leq x \leq \frac{a}{2})[/tex], and [tex] =A(a-x)[/tex] if [tex](\frac{a}{2} \leq x \leq a) [/tex]

And of course [tex]\Psi(0,0) = \Psi(a,0) = 0[/tex]

I was asked to find [tex]\Psi(x,t)[/tex] , which I did, and I was also asked to find the probability that "a measurement of the energy would yield the value [tex] E_1[/tex]", 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 [tex] E_1[/tex]" 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*

borgwal

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.