TP computing lab - Integrating the 1D stationarySchrodinger equation

Dr. M. Mobius (mobiusm@tcd.ie)

September 21, 2010

Dr. M. Mobius (mobiusm@tcd.ie) TP computing lab - Integrating the 1D stationary Schrodinger equation

The stationary 1D Schrodinger equation

The time-independent (stationary) Schrodinger equation is given by

Eψ(x) = − ~2

2m

d2ψ

dx2+ V (x)ψ(x), (1)

ψ(x) is the wavefunction of the particle,E the corresponding energy,~ is Planck’s constant divided by 2π,m the mass of the particle,V (x) the potential under consideration.

Eigenvalue problem: Solutions that satisfy boundaryconditions (ψ(x → ±∞)=0) only exist for certain discretevalues of E .

Can be solved analytically for a few potentials such asharmonic potential (V (x) ∝ x2) or the infinite square well.

Dr. M. Mobius (mobiusm@tcd.ie) TP computing lab - Integrating the 1D stationary Schrodinger equation

Aims of the computing lab

Use shooting/matching method to find the Eigenstates andcorresponding energy Eigenvalues numerically for the infinitesquare well

How accurate can we get this? → Compare with analyticsolution.

Use numerical algorithm to find Eigenstates of a morecomplicated potential.

Check orthogonality of the Eigenstates.

Dr. M. Mobius (mobiusm@tcd.ie) TP computing lab - Integrating the 1D stationary Schrodinger equation

Some general remarks on Schrodinger equation

Eψ(x) = − ~2

2m

d2ψ

dx2+ V (x)ψ(x),

If ψ(x) is a solution, then C · ψ(x) is also a solution.

The probability distribution function is given byP(x) = ψ∗(x)ψ(x). Need to normalize such that∫∞−∞ ψ

∗(x)ψ(x) = 1

Whenever the potential V (x) has even parity (i.e symmetric)then the solutions have either even or odd (=antisymmetric)parity.

The normalized eigenstates are orthonormal:∫∞−∞ ψ

∗i (x)ψj = δi ,j

Dr. M. Mobius (mobiusm@tcd.ie) TP computing lab - Integrating the 1D stationary Schrodinger equation

Example: The infinite square well

The most simple potential:

V (x) =

{= V0 if 0 < x < L=∞ at x = 0 and x = L

Wavefunction ψ(x) has to vanish at x = 0 and x = L and obeySchrodinger equation:

ψn(x) =

√2

Lsin(knx) where kn = nπ/L for n = 1, 2, · · ·

Substituting back into Schrodinger equation we obtain the energyEigenvalues:

En =~2π2n2

2mL2+ V0

Dr. M. Mobius (mobiusm@tcd.ie) TP computing lab - Integrating the 1D stationary Schrodinger equation

Solutions of the infinite square well potentials

Since the potential is symmetric, we have even and odd paritysolutions (functions are shifted vertically for clarity)

Dr. M. Mobius (mobiusm@tcd.ie) TP computing lab - Integrating the 1D stationary Schrodinger equation

Numerically solving the Schrodinger equation

For simplicity, we put infinite wells at x = 0 and x = L, with anarbitrary potential between the wells.First step: Non-dimensionalize the equation to avoid computationswith small numbers:

d2ψ(x)

dx2+ γ2 (ε− ν(x))ψ(x) = 0, (2)

x = x/L is the non-dimensional spatial variable.

ν(x) = V (x)/V0: dimensionless potential energy with a rangeof values between −1 and +1.

ε = E/V0 is a dimensionless energy

γ2 = 2mL2V0~2

L is the physical size of the well.

m is mass of the particle.

Dr. M. Mobius (mobiusm@tcd.ie) TP computing lab - Integrating the 1D stationary Schrodinger equation

Numerically solving the Schrodinger equation

Equation is of the form:

d2ψ

dx2+ k2(x)ψ(x) = 0

an be integrated by the Numerov 3-point formula (see Appendix Iof handout for derivation):

Numerov algorithm

ψn+1 =2(1− 5

12 l2k2n)ψn −

(1 + 1

12 l2k2n−1

)ψn−1

1 + 112 l

2k2n+1

,

where l is the integration step size: l ≡ xi+1 − xi = 1/(N − 1) forN evaluation points.By specifying two neighbouring points, one can obtain thethird point.For the Schrodinger equation,

k2n = γ2 (ε− ν(xn))

Dr. M. Mobius (mobiusm@tcd.ie) TP computing lab - Integrating the 1D stationary Schrodinger equation

Numerically solving the Schrodinger equation

Now we can apply this algorithm to the infinite square wellpotential: We know that ψ(x) vanishes at both sides of the well:ψ(x = 0) = ψ(x = 1)Let xn = n · l , where n = 0 · · ·N − 1 and l = 1/(N − 1).NOTE: ψ(x1) is arbitrary!

Dr. M. Mobius (mobiusm@tcd.ie) TP computing lab - Integrating the 1D stationary Schrodinger equation

Numerically solving the Schrodinger equation

Now shoot from both sides. Matching point is in the middle. Dueto the symmetry of the potential, ψleft(x) and ψright(x) will meetin the middle. But slopes are discontinuous unless you have anenergy Eigenstate.

Dr. M. Mobius (mobiusm@tcd.ie) TP computing lab - Integrating the 1D stationary Schrodinger equation

Asymmetric potentials

Main task will be to write a routine that finds the energyEigenstate that will minimize the difference in the slopes.

Later on will look at an asymmetric potential. ψleft(x) andψright(x) will not meet at the matching point.

Need to rescale either ψleft(x) or ψright(x), such thatψleft(xmatch) = ψright(xmatch).

We are allowed to do that! Even though the ψ(x)’s may notsatisfy the boundary condition, they still satisfy theSchrodinger equation. Therefore, we can multiply the wavefunction by any constant.

After rescaling, you can apply the same procedure to makethe slopes continuous and thereby find the energy Eigenstate.

Dr. M. Mobius (mobiusm@tcd.ie) TP computing lab - Integrating the 1D stationary Schrodinger equation

Template

Don’t worry! You will start from a template that does theintegration from both sides for a given trial energy.

Details about the source code and the Numerov algorithm canbe found in the handout.

Read the code carefully and ask me or Steven Tobin if you arestuck.

Play around and enjoy!

This introduction, the handout with what you have to do andthe template can be found:

http://www.tcd.ie/Physics/People/Matthias.Moebius/teaching/

Dr. M. Mobius (mobiusm@tcd.ie) TP computing lab - Integrating the 1D stationary Schrodinger equation