Quantum Mechanics of wave function and probability rationals

[xxxxxx]

Homework Statement

The wave function in state n=2 is given: W2(x)=(2/L)^(1/2)sin(2pix/L) with boundaries x=0 and x=L
at x=L/2, W2(L/2)=0, which means that the probability of finding the particle in a small region about x=L/2 is zero. Nevertheless, there is equal probability to find the particle in the left half of the box as in the right half.
How is this possible if the particle has no way of passing through the point x=L/2? is this an unresolvable paradox in quantum mechanics?

Homework Equations

P(x)=integral of (w(x))^2 dx

The Attempt at a Solution

I approached this question more in the mathematic path.
here is what I wrote
Probability is defined such that dx cannot equal to 0. therefore, even though W2(L/2)=0, it is only the amplitude at x=L/2 which implies x=0. However, dx cannot be 0 if we are looking for the probability, and therefore a non-zero valued dx would have P(x) yield a value that is close to but never 0. therefore, there is a probability of particle passing through point x=L/2 even though it is likely to be very small.

Can anyone check my answer?
Thanks alot

 

[jambaugh]

If I put a ping-pong ball in a box (length L). Shake it around and then put a divider through a slot (at L/2) to split the box into two halves...

...which means that the probability of finding the particle in a small region about x=L/2 is zero. Nevertheless, there is equal probability to find the particle in the left half of the box as in the right half.
How is this possible if the particle has no way of passing through the point x=L/2?

Granted my example actually prevents the ball from crossing the midpoint whereas in the quantum version you quote this is not impossible. Nonetheless the logic of the question is the same. Whether it can cross a point has nothing to do with the probability of it being found on either side.

Furthermore, in the quantum case the statement "the particle has no way of passing through the point at x=L/2" is not in fact correct. A quantum can cross a point and yet have zero probability of being found at that point (see for example tunneling).

 

[tomcue]

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Your response seems to be on the right track. The concept of probability in quantum mechanics is different from classical mechanics, where particles can be thought of as having definite positions and velocities. In quantum mechanics, particles are described by a wave function, which represents the probability of finding the particle at a certain position. This means that the particle does not have a definite position until it is measured.

In this case, the wave function W2(x) describes the probability of finding the particle in state n=2 at different positions along the x-axis. The fact that W2(L/2) = 0 simply means that the amplitude of the wave function at x=L/2 is zero, but this does not mean that the probability of finding the particle at x=L/2 is also zero. As you mentioned, the integral of (W2(x))^2 over a small region around x=L/2 will still yield a non-zero probability.

As for the paradox of the particle having no way of passing through x=L/2, this is a common misunderstanding in quantum mechanics. The particle is not a classical object that needs to physically pass through a point in space. It exists in a superposition of states and has a probability of being found at different positions. This is a fundamental aspect of quantum mechanics and has been confirmed through numerous experiments. So, while it may seem paradoxical, it is a fundamental principle of the theory and not a contradiction.