Time dependance of a particle in an infinite square well
Posted on
In this post, we'll see how the wavefunction of a particle in a 1D infinite square well evolves with time. We'll first have a look at the time evolution of energy eigenfunctions. Then, we'll examine the evolution of superposition states.
Time evolution of energy eigenfunctions
As we derived in this post, the energy eigenvalues \(E_n\) and energy eigenfunctions \(\psi_n\) of a particle trapped in a box of length \(L\) are the following:
To find how time affects eigenfunctions, we simply append the usual time-dependant coefficient \(e^{-i\frac{E_n}{\hbar}t}\):
An important thing to notice is that, by appending this time-dependant coefficient, the energy eigenfunctions become complex functions.
The graph below shows how the real and imaginary parts of energy eigenfunctions of the particle in a 1D box of length L change with time.
The probability density function associated with energy eigenfunctions is given by the following:
Where \(\psi^*\) represents the complex conjugate of \(\psi\). As you can see, the time dependance has vanished!
The probability density functions associated with superposition states, however, does change with time! We'll have a look now.
Time evolution of superposition states
The following graph shows how the probability density functions of superposistion states changes with time.
Try setting two of the three sliders on 0. The function then represents a pure state, which's probability density function is time-independant! However, as soon as you set multiple sliders to non-zero values, the function starts wiggling: the probability function associated with superposition states is time-dependant!