Should we think a free particle as a particle in an infinitely big box?

[nathatanu0]

I've found that the momentum expectation of a particle inside an one dimensional finite box of length 'L' is '0'... we can say the probability of going right=that of the left... so sum up to zero. Now I calculate the same(<p>) for a free particle in one dimension if I think that free particle is not exactly free but kept inside an infinitely long box(L-->infinity)... the result should be the same... does that happen... does it make any sense to draw analogy between the two... ?? please help.

atanu

 

[Avodyne]

This is a subtle question. For the finite box, I assume you required the wave function to vanish at the walls; in other words, you imposed the boundary conditions $\psi(0)=0$ and $\psi(L)=0$. But there is another kind of finite box: a circle instead of a line-segment. Then the wave function is required to be periodic: $\psi(x+L)=\psi(x)$. (Actually, you can have periodicity up to a phase instead,$\psi(x+L)=e^{i\phi}\psi(x)$, but I want to ignore that possibility.) In this case, you can have a free-particle wave function that has nonzero $\langle p\rangle$, such as $\psi(x)=L^{-1/2}\exp(2\pi i n x/L)$, where $n$ is an integer (positive, negative, or zero); then $\langle p\rangle=2\pi\hbar n/L$. We could now consider the limit of $L\to\infty$; if we also take $n\to\infty$, we end up with nonzero $\langle p\rangle$. But for the hard-wall box, we would always get $\langle p\rangle=0$. So the results depend on the boundary conditions, even for an infinite box.

 

[sanjibghosh]

it can be think as
for free particle $$\Delta$$X=infinity
therefore, infinitely big box is not so big for free particle

 

[confinement]

There is no reason to suppose that the (discrete set of) energy eigenstates of the square well will tend to the (continuum of) energy eigenstates for a free particle, but it is possible to write a general state as a sum of energy eigenstates and have this sum tend to an integral, and take apart the sines into e^ i k x and e ^ - i k x to see that the particle in an infinitely large box is like a free particle which has reflected off of infinity and so is in a superposition of +k and -k momenta. Such is nonrelativistic QM!