Required math: calculus
Required physics: Schrödinger equation
References: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.39.
Styer, Daniel F. (2000), Quantum revivals versus classical periodicity in the infinite square well. (Online paper).
Suppose a particle is in an infinite square well in the interval . We’re not specifying which state it’s in, so in general, its wave function is
where the coefficients would be determined in the usual way by looking at .
Any wave function in the infinite square well has what is known as a revival time, which is a time after which the wave function returns to its initial state. We can find this time by looking at the time-dependent term above.
In order for this to be equal to , we must have
The first for which this is true is found from
The infinite square well energies are
We want a value of which will result in a return to the initial state for all energies, so we need to pick the longest time, which occurs at . This value of is a multiple of all the times calculated using higher values of , so the argument of the complex exponential will be a multiple of for all these times, ensuring a return to the initial state. Thus the revival time is
Classically, a particle in an infinite square well bounces back and forth between the walls. If its energy is , its velocity is and the classical revival time is the time taken to traverse the well twice:
This is equal to the quantum time if
Since this is smaller than the ground state energy in the quantum system, this energy cannot be realized in the quantum system.
In fact, there seems to be a greater problem than this: the classical revival time depends on the energy, while the quantum time does not; it depends only on the particle’s mass and the width of the well. In Styer’s paper (reference at the top), he argues that the full wave function is not measurable, so what we should be looking at are measurable quantities, such as the expectation value of position . He shows that for a particle made up of a superposition of stationary states centred at quantum number , the period of is
Given that the quantum energy of state is , we can write this as
Using we get
That is, the revival time for the position in the quantum case is equal to the classical revival time.