The Particle in a Box
Also called the infinite square well problem, the "particle in a box" is actually one of the very few problems in quantum mechanics that can be solved without approximations, or analytically.
Recall the Schrödinger equation:
The potential function, V, is time independent, while the wave function itself is time dependent.
The infinite square well is defined by a potential function in which the value V(x) is 0 for values of x between 0 and L where L is the length of the box, and infinite in all other positions . In this case, classical and quantum physics are essentially describing the same thing: the impossibility of a particle "leaping" over an infinitely high boundary.
Separating the variables reduces the problem to one of simply solving the spatial part of the equation:
E represents the possible energies that can describe the system.
Inside the box, no forces act on the particle, which means that the part of the wave function inside the box oscillates through space and time with the same form as a free particle:
B and A are arbitrary complex numbers. The frequency of the oscillations through space and time are given by the wave number and the angular frequency, respectively. These are related to the energy of the particle by:
The size or amplitude of the wavefunction at any point is given by the probability of finding the particle there by:
Therefore, the wavefunction must vanish everywhere beyond the edges of the box.
Also, the amplitude of the wavefunction may not "jump" abruptly from one point to the next.
These two conditions are only satisfied by wave functions with the form:
Normalizing, we get
A may be any complex number with absolute value
When treated as a probability function, the wavefunction describes the probability of finding the particle at a given point and at a given time. Four conditions, proposed by Max Born, must be met for this to be true:
- The wavefunction must be single-valued
- The wavefunction must be "square integrable"
- The wave unction must be continuous everywhere
- The first derivative of the wavefunction must be continuous.