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

In summary, the momentum expectation of a particle inside an one dimensional finite box of length L is 0. The same applies for a free particle in one dimension if the particle is not exactly free but kept inside an infinitely long box.
  • #1
nathatanu0
3
0
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
 
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  • #2
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 [itex]\psi(0)=0[/itex] and [itex]\psi(L)=0[/itex]. But there is another kind of finite box: a circle instead of a line-segment. Then the wave function is required to be periodic: [itex]\psi(x+L)=\psi(x)[/itex]. (Actually, you can have periodicity up to a phase instead,[itex]\psi(x+L)=e^{i\phi}\psi(x)[/itex], but I want to ignore that possibility.) In this case, you can have a free-particle wave function that has nonzero [itex]\langle p\rangle[/itex], such as [itex]\psi(x)=L^{-1/2}\exp(2\pi i n x/L)[/itex], where [itex]n[/itex] is an integer (positive, negative, or zero); then [itex]\langle p\rangle=2\pi\hbar n/L[/itex]. We could now consider the limit of [itex]L\to\infty[/itex]; if we also take [itex]n\to\infty[/itex], we end up with nonzero [itex]\langle p\rangle[/itex]. But for the hard-wall box, we would always get [itex]\langle p\rangle=0[/itex]. So the results depend on the boundary conditions, even for an infinite box.
 
  • #3
it can be think as
for free particle [tex]\Delta[/tex]X=infinity
therefore, infinitely big box is not so big for free particle
 
  • #4
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!
 

1. What is a free particle?

A free particle is an object or entity that is not subject to any external forces or constraints. In other words, it is not affected by any outside influences and can move freely without any restrictions.

2. What is an infinitely big box?

An infinitely big box, also known as an infinite potential well, is a theoretical concept often used in physics to model the behavior of a free particle. It is a hypothetical space with walls that extend infinitely in all directions, creating an unbounded environment.

3. How is a free particle related to an infinitely big box?

A free particle can be thought of as a particle confined within an infinitely big box. This means that the particle is free to move within the box, but it cannot escape beyond the boundaries of the box.

4. What is the significance of considering a free particle in an infinitely big box?

Studying a free particle in an infinitely big box allows us to better understand the behavior and properties of a free particle in an unbounded environment. It also serves as a useful mathematical model for many real-world systems, such as an electron in a semiconductor material.

5. Is it possible for a free particle to exist in an infinitely big box in reality?

No, an infinitely big box is a theoretical concept and cannot exist in reality. However, it is a useful tool for understanding the behavior of free particles in various scenarios.

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