quantum mechanics numerical solutions of the …physics.bu.edu/py502/slides/l09.pdf · quantum...
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Quantum MechanicsNumerical solutions of the Schrodinger equation
• Integration of 1D and 3D-radial equations• Variational calculations for 2D and 3D equations• Solution using matrix diagonalization methods• Time dependence
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Brief review of quantum mechanicsIn classical mechanics, a point-particle is described by itsposition x(t) and veloity v(t)• Newton’s equations of motion evolve x,v as functions of time
• The Schrödinger equation evolves in time• There are energy eigenstates of the Schrodinger equation
- for these, only a phase changes with time
Y(x,t)
In quantum mechanics, x and v cannot be precisely knownsimultaneously (the uncertainty principle). A particle isdescribed by a wave function Y(x,t)• the probability of the particle being in a volume dx is
Þ Finding the energy eigenstates (stationary states) is an important task
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Stationary Scrodinger equation in three dimensions
Similar to purely one-dimensional problems
Spherical symmetric potentials; separable
Radial wave function
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Numerov’s method (one dimension)Stationary Schrodinger equation
Can be written as (also radial function in three dimensions)
Add expansions for
Second derivative determined by the Schrodinger equationHow to deal with the fourth derivative?
Discretization of space: . Consider Taylor expansion
⇥(�x) = ⇥(0) +��
n=1
�nx
n!⇥(n)(0)
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Central difference operator
We can rewrite the previous equation
using the second central difference, giving
Approximate the fourth derivative
leads to the general result
g00(x) =1
�2x
�2g(x) +O(�2x
)
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Schrodinger equation gives
More compact notation:
Introduce function
Fortran implementationdo n=2,nx
q2=dx2*f1*psi(n-1)+2.d0*q1-q0q0=q1; q1=q2f1=2.d0*(potential(dx*dble(n))-energy) psi(n)=q1/(1.d0-dx2*f1)
enddo
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Boundary-value problemsThe Schrodinger equation has to satisfy boundary conditionsØ quantization, as not all energies lead to valid solutions
Example: Particle in a box (infinite potential barrier)
Boundary conditions:
How do we proceed in a numerical integration?
Using
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Choose valid boundary conditionas at x=-1
A is arbitrary (not 0); normalize after solution found
Pick an energy EØ Integrate to x=1Ø Is boundary condition
at x=1 satisfied?Ø If not, adjust E,
integrate againØ Use bisection to refine
“Shooting method”
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Solving an equation using bisection (general)We wish to find the zero of some function
First find E1 and E2 bracketing the solution
Then evaluate the function at the mid-point value
Choose new bracketing values:
Repeat procedure with the new bracketing values- until
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Bisection search for the ground stateØ First find E1, E2 giving different signs at x=+1Ø Then do bisection within these brackets
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More complicated example: Box with central Gaussian potential barrier
Ground stateSearch
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First excited state
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Potential well with non-rigid wallsLooking for bound state;
Asymptotic solution:
Use the asymptoticform for two points far away from the center of the well
Find E for which the solution decays to 0 at the otherboundary
A=0 for x>0B=0 for x<0
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Ground state search
Using criterion: