GITAM, Department of Engineering Physics

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The aim of this section is to demonstrate how Schrödinger's equation can be solved analytically for some simple potentials. This is followed by a discussion of a numerical approach for the more general case.

Particle in an infinite square well potential

The first problem we will solve is also sometimes called a particle in a box. The infinite square well potential confines a particle totally to a region of width L. Outside this region the potential is infinite and the wavefunction is zero.

Recall that E = T + V   and    p = ħk where k is the wave number. Thus

                k² = 2mT/ħ² = 2m(E-V)/ħ²

With V=0, we have

              k² = 2mE/ħ²

Then setting the potential energy inside the box to zero we can write the Time Independent Schrödinger equation as

           

This equation has two solutions

But since both solve the equation, so does the sum, ie. we can write

Since y(x) = 0    for    x < 0 and x > L,    the continuity condition tells us that

    y(0) = y(L) = 0

Putting y(0) = 0 gives c₁ + c₂ = 0 and hence

with A = 2ic

Putting y(L) = 0 gives sinkL = 0, i.e. kL = nπ  with n = 1, 2, .... (Note that n = 0 is not allowed since then k = 0 and y(x) = 0 for all x, meaning that there is no particle in the box.) Thus k is restricted to have values k_n = nπ =L and hence

and

Thus energy quantization arises naturally (but really this is happening because the particle has wavelike behavior). We show the energy levels diagrammatically ( figure 1(b)). Note carefully that for this potential the energy increases as n²   where n is an integer. The lowest energy level is called the ground state, the higher levels are excited states so that n = 2 is the first excited state. The ground state energy, E1, is not zero, i.e. a particle confined  to a box must have kinetic energy > 0. This is a pure quantum effect that does not occur in classical physics.

Fig - 1(a)	Fig - 1(b)

 

The solutions to the Schrödinger equation for this potential are the wave functions

There are an infinite number of them and we can sketch them as in figure 1(a). Notice that the solution have even parity for n odd and vice versa.

Since the probability of the particle being between x and x+dx is µ  ly(x)l²dx  we can sketch this also ( figure 2 (a) and (b)). Note that there is not the same probability of finding the particle at all points in the box. As the length of the box decreases the probability increases. Moreover there are some points at which the probability is zero, even though it can be found on either side of such points. Note carefully that the probability distribution depends on the energy.

Fig - 2(a)	Fig - 2(b)

A_n is determined from the normalization condition

i.e.

This is solved with the substitution

      thus     

so that the normalized wave functions are

 

If a particle is confined into a rectangular volume, the same kind of process can be applied to a three-dimensional "particle in a box", and the same kind of energy contribution is made from each dimension. The energies for a three-dimensional box are

This gives a more physically realistic expression for the available energies for contained particles. This expression is used in determining the density of possible energy states for electrons in solids.

 

Though oversimplified, this indicates some important things about bound states for particles

1. The energies are quantized and can be characterized by a quantum number n
2. The energy cannot be exactly zero.
3. The smaller the confinement, the larger the energy required.
 

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