Particle in a Box

In quantum physics a particle moving in free space may be of any energy. Its energy spectrum is continouos. A particle moving in a force field holding it in a bounded space region, has the discrete spectrum of the energy eigenvalues. An example is the finite motion of the hydrogen electron in the Coulomb field of the nucleus. The discreteness of the energy levels of particles trapped in a bounded region is a consequence of a dual nature of particles to be the fundamental difference between quantum physics and classical physics.

A simple physical model of the finite motion is the motion of a particle in a one-dimensional ?potential well? with infinitely high walls. The particle cannot leave a region of size L. It moves in this region undergoing multiple reflections from the walls. In the view of the wave theory, two de Broglie waves move in opposite directions between the walls. This resembles two counter-traveling waves in a vibrating string with fixed ends. As with the string, to the stationary states there correspond standing waves which are formed if there is an integral number of half-waves in L:

L = n ? (λ / 2) (n = 1, 2, 3, ...)

Thus to the stationary states of a particle trapped in a potential well there corresponds a discrete set of the wavelengths. Since in quantum-mechanical case the wavelength λ of a particle is a single-valued function of its momentum: λ = h / p, and the particle momentum determines the energy of its motion: E = p2 / (2m) (the nonrelativistic approximation), then the particle energy is quantized too. The quantum-mechanical calculation gives:

 En = n2 h2 = n2E1 8mL2

Here m is the particle mass, h ? Planck's constant, E1 = h2 / (8mL2) ? the lowest-state energy.

It should be noted that unlike the classical particle, the quantum-mechanical particle cannot be at rest at the potential well bottom, that is, it cannot be of energy E1 = 0 because of a contradiction to the uncertainty relation

Δx ? Δpx ≥ h

Actually, the momentum of a particle at rest is strictly zero, therefore, Δpx = 0. However, the coordinate uncertainty of the particle is Δx ≈ L. For this reason the product Δx ? Δpx for a particle being at rest at the potential well bottom, should be zero.

The uncertainty relation allows one to estimate the lowest energy E1 of the particle. By setting px ≈ Δpx for the lowest-energy state, for the lowest energy E1 we have the following:

 E1= px2 ≈ Δpx2 ≥ h2 2m 2m 2mL2

This crude estimation gives a correct order of E1.

Standing de Broglie waves produced by a particle moving in a potential well are the wave functions, or the psi-functions, by means of which quantum mechanics describes the stationary states of micro-objects. The square of the wave function modulus |Ψ|2 is defined as the probability of finding a particle in different space points.

The computer model allows you to change the potential well width L and the mass m of a particle trapped in the well. In the left window there are displayed the drawings of wave functions Ψ(x) or the squares of their moduli |Ψ|2 for a few stationary states (n = 1?5). In the right window there is displayed the energy spectrum of the particle, that is, the spectrum of possible values of its energy. Notice that the energy levels shift down with increasing the potential well width L and trapped particle mass m.

The model gives the particle mass in terms of proton mass mp = 1.67?10?27kg. Therefore, it simulates the states of relatively heavy particles (heavy nuclei) being in a potential well whose width is on the order of the atomic size.