The Potential Well

The time-independent Schrodinger Equation is

The wavefunction of a particle must obey certain conditions to be a physically realistic solution. These conditions are:

1. y(x) ® 0 as x ® ±¥
2. y(x)|_left = y(x)|_right at boundaries
3. y¢(x)|_left = y¢ (x)|_right at boundaries
4. 

Since we are dealing with square or step wells, we will have regions where V – E is positive or negative constant. The only real solutions for the Schrodinger equation in such regions is then

where . Note k will depend on the value of V in that region.

Example: The Finite Square Well

As shown in the Figure 1 below, we have three separate regions. In regions I and III, V > E will lead to exponential solutions. In region II, E > V will lead to sinusoidal solutions. The solutions are:

Figure 1 A finite potential well

Applying Condition 1 above to Region I, where x ® -¥ requires that B = 0. Similarly in Region III, where x ® +¥ requires that F = 0. So we now have

Continuity at x = 0 gives Ae⁰ = Csin(0) +Dcos(0), or D = A, while continuity at x = L gives Csin(pL) +Dcos(pL) = Ge^-kL, or G = [Csin(pL) +Dcos(pL)]e^kL. Continuity of the first derivative at each boundary gives kAe⁰ = pCcos(0) – pDsin(0), or C = (^k/_p)A, and pCcos(pL) –pDsin(pL) = -kGe^-kL ,or G = -(^p/_k) [Ccos(pL) – Dsin(pL)]e^kL. Thus all the prefixes can be expressed in terms of A. Summarizing:

• C = (^k/_p)A
• D = A
• G = [Csin(pL) +Dcos(pL)]e^kL
• G = -(^p/_k) [Ccos(pL) – Dsin(pL)]e^kL

The fact that we have two different constraints values for G actually fixes the value of E, so the energy constraint is

[Csin(pL) + Dcos(pL)] = -(^p/_k)[Ccos(pL) – Dsin(pL)].                        (1)

So the only undetermined variable is A, this is found using the last or normalization condition above. Condition 4 can be rewritten as

We need the value of A that sets the above to 1, that is

                        (2)

Both equation (1) and equation (2) are easily solved in MS Excel.

Caution: More Complex Wells

For a more complex well such as the one in Figure 2 below, extra care must be taken. The value of will be different in each region. Also, depending on the value of E, the solutions in region III can be either exponential or sinusoidal. Any spreadsheet must take this into account.

Figure 2 A Step Potential Well