How is the wave function of a free particle represented mathematically?
The wave function of a free particle in quantum mechanics is mathematically represented by a complex-valued function known as the plane wave. The plane wave is a solution to Schrödinger's equation for a one-dimensional free particle, which describes the behavior of quantum systems.
To understand the mathematical representation of the wave function, let's consider a one-dimensional free particle moving along the x-axis. The wave function, denoted by Ψ(x, t), describes the probability amplitude of finding the particle at position x and time t.
The plane wave solution for a free particle is given by:
Ψ(x, t) = A * exp(i(kx – ωt))
In this equation, A is the amplitude of the wave, k is the wave number, ω is the angular frequency, i is the imaginary unit, x is the position, and t is the time.
The wave number k is related to the momentum p of the particle by the equation:
k = p / ħ
Here, ħ is the reduced Planck's constant, which is equal to h / (2π), where h is the Planck's constant.
The angular frequency ω is related to the energy E of the particle by the equation:
ω = E / ħ
The wave function Ψ(x, t) represents a plane wave that extends infinitely in both positive and negative x-directions. The exponential term in the wave function describes the spatial and temporal variation of the wave.
The modulus squared of the wave function, |Ψ(x, t)|^2, gives the probability density of finding the particle at position x and time t. The probability density is a real-valued function and is normalized such that the integral of |Ψ(x, t)|^2 over all space is equal to 1.
The plane wave solution represents a free particle because it does not experience any external forces or potentials. In this case, the particle's energy and momentum are well-defined, and the wave function describes the particle's motion in a uniform manner.
It is important to note that the plane wave solution represents an idealized scenario of a truly free particle. In reality, particles are often subject to external forces and potentials, which require more complex wave functions to describe their behavior accurately.
The wave function of a free particle is mathematically represented by a plane wave solution to Schrödinger's equation. The plane wave has a complex exponential form and describes the probability amplitude of finding the particle at a given position and time. The wave function is related to the particle's energy and momentum and provides valuable information about its behavior.
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