In this laboratory you will solve the one-dimensional, time-independent Schroedinger equation numerically and find the energy eigenvalues of an electron trapped in a finite square well, a harmonic well, and a triangular well.  You will use an Excel spreadsheet for the numerical work.

Theory:

We want to solve the eigenvalue equation

Hψ(x)  = Eψ(x),  
or  
(-ħ²/(2m))∂²ψ(x)/∂x² + U(x)ψ(x)  = Eψ(x),

or
∂²ψ(x)/∂x² - k²(x)ψ(x) = 0,

with k²(x) = (2m/ħ²)(E - U(x)).

We want to solve it in a finite region between x = -L and x = L.  We assume that in the middle of this region there exists a potential well of width a, with a/2 << L.  Because we cannot extend our numerical solution to infinity, we assume that U(x) = ∞ for x < -L and x > L.

We want to find the energies for which the wave function ψ(x) and its derivative ∂ψ(x)/∂x are continuous and the boundary conditions ψ(0) = ψ(L) = 0 are satisfied.  These are the energies of the bound states of the system.

We start by expanding ψ(x) in a Taylor series expansion.
ψ(x + ∆x) = ψ(x) + ∆x*∂ψ(x)/∂x + [(∆x)²/2]*∂²ψ(x)/∂x² + ... .
ψ(x - ∆x) = ψ(x) - ∆x*∂ψ(x)/∂x + [(∆x)²/2]*∂²ψ(x)/∂x² - ... .
Combining the two equations above yields
[ψ(x + ∆x) + ψ(x - ∆x) - 2ψ(x)]/(∆x)² = ∂²ψ(x)/∂x²,
or
[ψ(x + ∆x) + ψ(x - ∂x) -2ψ(x)]/(∆x)² = -k²(x)ψ(x).

ψ(x + ∆x) = (2 - (∆x)²k²(x))ψ(x) - ψ(x - ∆x).

For our numerical work, let {x_n} denote points on a grid defined in the region from x = -L to x = L, n = 0, 1, …, N.  Define ψ_n = ψ(x_n), k_n = k(x_n), ∆x = x_n+1 - x_n.  Then

ψ_n+1 = (2 - (∆x)²k_n²ψ_n - ψ_n-1.

We have a recipe for finding ψ_n+1, given ψ_n and ψ_n-1.  Integrating the wave function using this algorithm, i.e. finding its value on the next grid point from its value at the previous two grid points, is called integrating using the Numerov method.

To solve for the bound states of the system, we pick ψ₀ = 0, ψ₁ = c₁, where c₁ is some small number.  This number c₁ determines the overall normalization of the wave function.  We now calculate ψ_2, ψ₃, etc.  We start integrating in the classically forbidden region, where the magnitude of the wave function increases approximately exponentially.  As we reach the classically allowed region, the wave function becomes oscillatory at the classical turning point.  As we pass through the second turning point and enter again into the classically forbidden region, the integration becomes numerically unstable, because it can contain an admixture of the exponentially growing solution.  Integration into a classically forbidden region is likely to be inaccurate.  We therefore generate a second solution, picking ψ_N = 0, ψ_N-1 = c₂, and calculating ψ_N-2, ψ_N-3, etc.

ψ_n-1 = (2 - (∆x)²k_n²ψ_n - ψ_n+1.

For both integrations we pick the same value for the energy E.  To determine whether this energy E is an eigenvalue, we compare the results of our integrations at a matching point x_m in the classically allowed region.  The constant c₂ is chosen so that both integrations yield the same value ψ(x_m) at the matching point and we examine the slope ∂ψ(x)/∂x near x_m.  The values of E for which the slope is continuous across the matching point are the energy eigenvalues, because for these values of E the wave function is a solution to the Schroedinger equation which satisfies all boundary conditions.  We search for the eigenvalues by picking different values for the energy E.

Procedure:

Open the linked Excel workbook.  It can be used to solve the one-dimensional, time-independent Schroedinger equation for an electron trapped in various potential wells.  Sheet 1 contains the data and sheet 2 the user interface.

(a)  Examine sheet 1.

• Column A contains the position variable x in units of 10^-10 m.  We are going to solve the Schroedinger equation in the finite region between x = -5 and x = 5.

• Column B contains the potential energy function U(x) in units of eV.  We are starting with a "symmetric finite square well" extending from x = -1*10^-10 m to x = 1*10^-10 m with a depth of 500 eV.  The user interface on sheet 2 provides us with tools to change U(x).

• Cell E2 contains a trial value for the energy E.  This value can be changes with a slider located on sheet 2.

• Colum C contains k²(x) = (2m/ħ²)(E - U(x))in units of 1/(10^-10 m)².
  With E and U(x) measured in units of eV, (2m/ħ²) has the value
  2*9.11*10^-31kg/(1.05*10^-34 Js)² = 1.65*10^-38 J^-1m^-2 *(1.6*10^-19 J/eV)
  = 2.64 *10¹⁹ eV^-1m^-2 = 0.264 eV^-1(10^-10 m)^-2.

• Column D contains the wave function y.  It is calculated using ψ_n+1 = (2 - (∆x)²k_n²ψ_n - ψ_n-1 for x = -5 to x = 0.2 and using ψ_n-1 = (2 - (∆x)²k_n²ψ_n - ψ_n+1 for x = 5 to x = 0.2.

• Columns I, J, and K contain different potentials that can be copied into column B with the tools provided on sheet 2.

(b)  Switch to sheet 2.  Fill in the tables and  keep a log of your answers to the question in the text below.

• (i)  Start with the symmetric square well with width a = 2*10^-10 m and a depth of 500 eV.
  Find the allowed energies for an electron trapped in this potential.  How many bound states do exist?
  Note the shape of the eigenfunctions.  Determine the parity of each eigenfunction.
  [In a symmetric well, i.e. a well that looks the same when reflected about a line through its center, bound-state wave functions are either symmetric or anti-symmetric when reflected about the same line.  We say that the symmetric wave functions have even parity and the anti-symmetric wave functions have odd parity.  In an asymmetric well  bound-state wave functions do not have a well defined parity.]. 
  Compare the eigenvalues you found with the energy eigenvalues of the infinite square well.
  For the infinite square well we have E_n = n²π²ħ²/(2ma²).
  If we measure a in units of 10^-10 m and E_n in units of eV, then E_n = 3.78*n²π²/a².
  Fill in the appropriate columns in the table below.  Extend the table downward when necessary.

  		Energy (eV)
  well with width a = 2	n	infinite well En = n2π2ħ2/(2ma2) = 3.78*n2π2/a2	symmetric well (500 eV depth) energy (eV)	parity (even or odd)	symmetric well (1000 eV depth) energy (eV)	parity (even or odd)	asymmetric well (1000 eV depth) energy (eV)
  ground state	1	9.327
  first excited state	2	37.307
  second excited state	3
  ...

• (ii)  Change the well depth to 1000 eV.  Find the allowed energies for an electron trapped in this potential.  How many bound states do exist now?  Fill in the appropriate column in the table.
• (iii)  Keep the well depth at 1000 eV and change the well asymmetry to 1.5.  Find the allowed energies for an electron trapped in this potential.  How many bound states do exist now?  Fill in the appropriate column in the table.  Note the shape of the eigenfunctions.

Comment on your results.  What have you learned about the stationary states of an electron in a finite square well?

• (iv)  Now choose the harmonic well.  U(x) = (1/2)kx².  Let k = 500 eV/(10^-10 m)².  For the harmonic oscillator potential the Schroedinger equation can be solved analytically.  The energy eigenvalues are E_n = (n + 1/2)ħω, n = 0, 1, 2, ... .  Here ω = (k/m)^1/2.  (Note:  The labeling of the energy levels for the harmonic oscillator starts with n = 0, while for the square well it starts with n = 1.)
  With k measured in units of  eV/(10^-10 m)² and E_n in units of eV we have E_n = (n + 1/2)*2.76 eV*k^1/2.
  Numerically find the five lowest allowed energies for an electron trapped in this potential.  Determine the parity of each corresponding eigenfunction.  Compare the eigenvalues you found with the energy eigenvalues found when solving the problem analytically.
  Fill in the appropriate columns in the table below.
  Note the shape of the eigenfunctions.  How does it differ from the shape of the eigenfunctions if the symmetric square well?
  [The harmonic well or harmonic oscillator with potential energy U(x) = (1/2)kx² = (1/2)mω²x² is one of the few system that can be solved exactly in quantum mechanics.  Most real systems that have an equilibrium position can be approximated by a harmonic well, as long as the average displacements from equilibrium are small and the energy is low.  It is therefore important to remember the energy levels of the harmonic well, E_n = (n + 1/2)ħω = E_n = (n + 1/2)hf.]
   

  		Energy (eV)
  spring constant k = 500	n	harmonic well analytical solution En = (n + 1/2)ħω = (n + 1/2)*2.76 eV*k1/2	harmonic well numerical solution energy (eV)	parity (even or odd)	triangular well numerical solution energy (eV	parity (even or odd)
  ground state	0	30.86
  first excited state	1	92.57
  second excited state	2
  ...

• (v)  Now choose the triangular well.  U(x) = kx.  Let k = 500 eV/(10^-10 m).  At x = 2, U(x) for the harmonic well above and U(x) for this triangular well have the same numerical value.
  Numerically find the five lowest allowed energies for an electron trapped in the triangular well.  Determine the parity of each corresponding eigenfunction.  Compare the eigenvalues you found with the energy eigenvalues you found for the harmonic well.  Compare the shape of the eigenfunctions.
  Fill in the appropriate columns in the table below.

Comment on your results. What have you learned about the stationary states of an electron in a harmonic well and in a triangular well?

Use your log to prepare a report.

Name:
E-mail address:

Laboratory 4 Report

• Summarize the laboratory exercise.

• Insert your tables and discuss your results.  In the discussion you should answer the following questions.

  • How do the energy eigenvalues of a finite square well compare to the energy eigenvalues of an infinite square well?

  • How do the shapes of the energy eigenfunctions of a finite square well compare to the shapes of the energy eigenfunctions of an infinite square well?

  • For a symmetric square well, what is the parity of the eigenfunctions?

  • Excluding nodes at infinity, how many nodes does the eigenfunction of a square well corresponding to the eigenvalue E_n have?

  • How do the numerically obtained energy eigenvalues of the harmonic well compare to the analytically obtained energy eigenvalues?

  • How do the shapes of the energy eigenfunctions of the harmonic well compare to the shapes of the energy eigenfunctions of a finite square well?  Describe similarities and differences.

  • How do the shapes of the energy eigenfunctions of the triangular well compare to the shapes of the energy eigenfunctions of the harmonic well?  Describe similarities and differences.