To find the location of a particle in a free state, you can use the Schrödinger equation and its associated wave function. However, it's important to note that the Schrödinger equation describes the time evolution of a particle's wave function, and the wave function itself contains information about the probability distribution of finding the particle at different positions.
The Schrödinger equation for a free particle in one dimension (ignoring potential energy) is given by:
iħ ∂ψ/∂t = - (ħ^2/2m) ∂^2ψ/∂x^2
where ħ is the reduced Planck's constant, ψ is the wave function, t is time, m is the mass of the particle, and x is the position.
If you're interested in finding the particle's location at a specific time, you can solve the time-independent Schrödinger equation:
-(ħ^2/2m) ∂^2ψ/∂x^2 = Eψ
where E is the energy of the particle. The solutions to this equation will give you the allowed energy levels and corresponding wave functions. However, the time-independent Schrödinger equation does not provide direct information about the particle's location at a specific time.
To extract information about the particle's location from the wave function, you can use operators. The position operator in quantum mechanics is represented by the operator x, which acts on the wave function ψ(x) as follows:
xψ(x) = xψ
The expectation value of the position operator ⟨x⟩ can be calculated by integrating the product of the wave function and the position operator over all space:
⟨x⟩ = ∫ x|ψ(x)|^2 dx
This gives you the average position of the particle based on the probability distribution described by the wave function.
In summary, to find the location of a particle in a free state, you would typically use the Schrödinger equation to obtain the wave function and then calculate the expectation value of the position operator using the wave function.