Square well with finite potential

A square well potential with finite depth refers to a common model in quantum mechanics where a particle is confined within a potential well that has a rectangular shape.

Let us consider a finite depth potential well characterized by

V(x) = 0 ; -a< x< a

= V ; lx l > a ...i

Case I: Bound state ie E< V

a. The Schrodinger wave equation in the region x< -a is given by

u''(x) - Ko²u(x) = 0 ...ii where Ko² = 2m(V - E) /ℏ²

The general solution of equation ii is

As x-> -∞ then B-> 0

In the region x>a, the Schrodinger wave equation is

u''(x) - Ko² u(x) = 0 ....v where Ko² = 2m(V - E) /ℏ²

The general solution of equation v is

c. In the region -a<x<a, the Schrodinger wave equation is

u'' (x) + k²u(x) = 0 ....viii where K² = 2mE /ℏ²

Then the solution of eq vii is

u_II(x) = C cos kx + Dsin kx ....ix

Use the boundary conditions

These boundary conditions yield

Adding and subtracting equation xi and xiii we get

2C cos ka = (A + A') e^-k_oa ....xv

2Dsin ka = (A'- A)e^-k_oa.......xvi

Similarly adding and subtracting xii and xiv we get

2kD cos ka = (A - A') koe^-k_oa .....xvii

2kD sin ka = (A + A') koe^-k_oa .....xviii

Dividing xviii and xvi

kcot ka = - ko ....xix

Dividing xviii by xv

k tan ka = k_o ....xx

Now we introduce 𝛼 = ka and 𝛽 = k_o a in eqns xix and xx

k tan ka = k_o

ka tanka = k_oa

𝛼 tan 𝛼 = 𝛽 ....xxi

Also

k cot ka = - k_o

ka cot ka = - k_o a

𝛼 cot 𝛼 = -𝛽 ....xxii

It is impossible to make single general solution. Thus the solutions may be divided into two classes given by xxi and xxii. The equation xxi represents the solutions for even eigen states whereas the equation xxii represents the eigen value for odd eigen states. Now

𝛼² + 𝛽² = (k² + K_o²)a²

= [ 2mE/ℏ² + 2m(V-E)/ℏ² ] a²

= 2ma²V/ ℏ² = 𝜌²

𝛼² + 𝛽² = 𝜌² ....xxiii

This equation gives a circle of radius 𝜌 in cartesian 𝛼𝛽 plane.

a. The intersection of this circle in first quadrant (𝛼 𝛽 = positive) with 𝛽 = 𝛼 cot 𝛼 yield the energies of the even eigen value

b. The intersection of this circle in first quadrant (𝛼 𝛽 = positive) with -𝛽 = 𝛼 cot 𝛼 yield the energies of the odd eigen value

The number of eigen states provided by the number of intersection points on the corresponding circles

Fig: Graphical representation for the discrete energy states En of a potential box with finite walls. Here solid curve indicates 𝛽 = 𝛼 tan 𝛼 and dashed curve represents 𝛽 = - 𝛼 cot 𝛼

In the figure, energy states is drawn for these values of Va²

𝛼² + 𝛽² = 1 => 2mVa² / ℏ² = 1 => Va² = ℏ² / 2m

𝛼² + 𝛽² = 2² => 2mVa² / ℏ² = 4 => Va² = 4 ℏ² / 2m

𝛼² + 𝛽² = (√12)² => 2mVa² / ℏ² = 12 => Va² = 12 ℏ² / 2m

Smaller values of Va² we get one solution for 𝛽 = 𝛼 tan 𝛼. For Va²>3 and two solutions for 𝛽 = 𝛼 tan 𝛼 and for the solution for - 𝛽 = 𝛼 cot 𝛼. Thus as the values of Va² increases, number of solutions (for both odd and even eigen functions) increases.

As an example, for a proton (m = 1.67 x 10^-27 kg) inside a one dimensional well with Vo ⋍ 25 Mev and a ⋍ 3.65 x10^-3 cm. In this case, there will be one symmetric and one anti symmetric state. Their corresponding energy eigen values are 6.76 Mev and 22.56 Mev

The eigen values and eigen functions corresponds to 𝜌 ⋍ 2

Case II: When E = V

We have k = 2mE/ℏ² and k o = 2m(V-E)/ℏ² = 0

and tan 𝛼 = 𝛽/ 𝛼 = Koa / ka = 0

=> 𝛼 = n𝜋 / 2 (n: even)

and cot 𝛼 = - 𝛽/ 𝛼 = - Koa / ka = 0

=> 𝛼 = n𝜋 / 2 (n: odd)

Combining both we can write

𝛼 = n𝜋 / 2 for n = 1,2,3,4.....

ka = n𝜋 / 2

k² = n²𝜋² / 4a²

2mEn/ℏ² = n²𝜋² / 4a²

En = n²𝜋²ℏ² / 8ma² = V => Va² = n²𝜋²ℏ² / 8m ....1

Similarly there will be two energy states ( n=2) if Va² lies in between 𝜋²ℏ² / 8m and 4(𝜋²ℏ² / 8m). These two energy states have even and odd eigen function each. Thus as Va² increases the energy level appears successively.

Case II: When V -> ∞ (i.e infinite potential well)

For first class class

𝛼 tan 𝛼= 𝛽

tan ka = Koa / ka -> ∞ if ko-> ∞

ka = n𝜋 / 2 ( n =1, 2, 3, 4,.....)

k= n𝜋 / 2a

k² = n²𝜋² / 4a²

i.e 2m En/ ℏ² = n²𝜋² / 4a²

=> En = n²𝜋²ℏ² /8ma²

This expression shows that the energy levels are discrete and non degenerate. Hence the particle oscillates between two opaque wall. The corresponding different energy levels and their wave function is plotted below.

Square well potentials with finite depth are fundamental models used to understand various quantum phenomena, such as particle confinement, quantum tunneling, and scattering. They serve as building blocks for more complex systems in quantum mechanics and are extensively studied both theoretically and experimentally.

This note is a part of the Physics Repository.