The wave function of an infinite square well with time dependence included is a mathematical representation of the probability amplitude of a particle confined within an infinitely deep potential well. It takes into account the time evolution of the system and is described by the time-dependent Schrödinger equation.
Time dependence introduces a time factor into the wave function, which causes it to oscillate between positive and negative values. This time factor is known as the phase factor and is given by e^(-iEt/ħ), where E is the energy of the particle and t is time. This means that the wave function will change over time and the particle's probability of being found in a certain position will also change.
The boundary conditions for the wave function of an infinite square well with time dependence included are that the wave function must be continuous and differentiable everywhere within the well. This means that the wave function must have the same value at the boundaries of the well and cannot have any sudden changes or discontinuities.
The energy of the particle is directly related to the frequency of the oscillations in the wave function. As the energy increases, the frequency of the oscillations also increases, resulting in a shorter wavelength for the wave function. This means that the particle has a higher probability of being found in the center of the well, where the wavelength is the shortest.
Yes, the wave function of an infinite square well with time dependence included can be used to determine the probability of the particle being in a specific energy state. The probability is given by the square of the amplitude of the wave function at that energy state, which is known as the probability density. This allows us to calculate the probability of the particle being in any energy state within the well at any given time.