Particle In A Box

The particle in a box (1-D and 3-D) is probably the most simple (and buttoned-down, as many students find it) and widely used example that one can find in any introductory quantum mechanics text. However, this example can be use to demonstrate some interesting concepts such as discreetness of energy levels, uncertainty, etc. in quantum mechanics which students often find difficult to comprehend when encountered in more complicated systems. I will use the expressions for wave-functions and energy levels for a particle in a 1-D and 3-D box without going into the drily derivation part, which can be readily found in any standard QM text.

Discrete energy levels and quantum numbers: It is important to note here that the solution of the Schrodinger equation is itself a continuous function. Schrodinger equation all by itself doesn’t demand the existence of discrete energy levels or of so called quantum numbers. The discreteness comes into picture when we try to impose the boundary conditions on the solution of Schrodinger equation. Boundary conditions arise when there are boundaries i.e. when we try to confine a particle in a finite region of space. Thus 

                           It is the confinement that causes discrete energy values.

For example, in case of particle in a 1-D box, the discreteness comes into picture when we require the wave-function to vanish at the right boundary i.e. at x = L. Mathematically,

where ‘n’ is called as the ‘quantum number.’

In case of a particle in a 3-D box, the particle is being confined in three dimensions. This is reflected when we try to solve the Schrodinger equation and end up with three equations similar to the Schrodinger equation for a particle in a 1-D box. Again, when we try to impose the boundary conditions on the solutions of each of these equations, we get three different quantum numbers, one for each dimension, which quantize the energy of the particle in a 3-D box. Thus quantum numbers can be regarded as an indication of the number of dimensions in which the problem is being solved.

If a problem is being solved in four dimensions we can expect to have four different quantum numbers.  For example, trying to solve the Schrodinger equation (actually it Dirac equation) in 4-D space-time, taking time as the fourth dimension, we would expect four quantum numbers and that’s exactly what we get : principle quantum number ‘n’, orbital quantum number ‘l’, azimuthal quantum number ‘m_l’, and the fourth quantum number spin ‘s.’ That’s how the spin quantum number comes into picture. These quantum numbers are given names according to the physical quantities that they quantize. The name ‘spin’ is, however, the result of historical misconception. It has nothing to do with the particle actually spinning.

Uncertainty : The idea that the act of observation affects the state of the system is at the heart of quantum mechanics and forms the content of the so called “Heisenberg’s Uncertainty Principle.” Mathematically it says, 

                                                                           

where  Δx  and  Δp are the “errors” in the measurement of position and momentum, when measured simultaneously. These “errors” mathematical jargon are known as standard deviations, defined in the following manner,

 
                                                                           

where angular bracets represent expectation values of x, x², p and p². It is instructive to derive the expressions for the standard deviations in position and momentum. On doing so we get the following result (for particle in a 1-D box),

                                                                           

It is interesting to note that the uncertainty product is minimum for the ground state (n = 1). In general, this is true for all the systems – the ground states are minimum uncertainty states.  The uncertainty product increases as the value of ‘n’ increases i.e. it is higher for higher excited states. But it is always greater than the minimum value h/4π given by the uncertainty principle . Thus the uncertainty products for different states in a system are different and are dependent on the values of quantum numbers of that state. There are systems in which the product turns out to be exactly h/4π  (one such system is a harmonic oscillator).

An interesting paradox to think about:

Knowing the energy of the particle in 1-D box, we can find the expectation value of the square of velocity v² using the relation E = (1/2)mv². The energy of particle in a box is inversely proportional to the length (or volume) of the box. Hence, as we decrease the length (or volume) of the box, the energy and the velocity increase. Can we increase the length to such an extent that the velocity of the particle becomes equal to or greater than the velocity of light, and hence contradict the theory of relativity? Consider an electron trapped inside a 1-D box of length L. Find the length of the box for which the electron’s velocity is equal to the velocity of the light. Also try to find out the velocity of an electron trapped inside a nucleus.

As an exercise, try to calculate the average velocity of a nucleon (a proton or neutron) trapped in a nucleus. You can approximate the nucleus with a 3-D box of side 1 fermi. Although this is a very simplistic model of a nucleus – not taking into account the intricate dynamics of nucleons – the exercise gives an idea of order of magnitude of velocities of nucleons in nuclei, which are quite huge.