In quantum mechanics, the position of a particle in a free state is described by a probability distribution rather than a definite position. The probability distribution is given by the square modulus of the wave function, denoted as |ψ(x)|^2, where ψ(x) is the wave function of the particle.
To find the position of a particle in a free state, you need to determine the probability of finding the particle at a particular position x. This can be done by evaluating the square modulus of the wave function at that position:
P(x) = |ψ(x)|^2
Here, P(x) represents the probability density function, which gives the probability of finding the particle in an infinitesimally small interval around position x.
To obtain the average or expected position of the particle in a free state, you need to calculate the expectation value of the position operator, as I mentioned in a previous response. The expectation value of the position operator, denoted as ⟨X⟩, is given by:
⟨X⟩ = ∫x |ψ(x)|^2 dx
This means you integrate the product of the position x and the probability density |ψ(x)|^2 over the entire range of x.
It's important to note that in quantum mechanics, the position of a particle in a free state is not determined with certainty. Instead, it is described by a probability distribution. The wave function provides information about the likelihood of finding the particle at different positions. The expectation value gives the average position that you would obtain if you made repeated measurements on identical systems prepared in the same state.