Approximation methods in Quantum Mechanics

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Approximation methods in Quantum Mechanics
Kap. 7/9/13
Introduction to
Time dependent
Time-independent methods
Methods to obtain an approximate eigen energy, E
and wave function
perturbation methods
Methods to obtain an approximate expression for
the expansion amplitudes.
Ground/Bound states
Continuum states
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Time independent perturbation theory
Assume the Hamiltonian can be written as a sum of
two parts
Exact (unknown) eigenstates
Approximate known eigenstates
try expansion
Assume
Insert
Rearrange
Energy Correction
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First order
Wave function correction
No problem with mn, since we have assumed
in the expansion
Summary -

• DISCUSS
• Validity
• Convergence
• Iterative solution procedure

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Second order
Energy correction
Final result
Energy to 2nd order

• Comments
• Second order correction involves matrix elements
  between the state in question and all
• other states. First order involves only matrix
  element between the the state in question.

• And from

We can iteratively continue to any order
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Example-1 1. Order
Infinite well with a small barrier in the
middle approximate ground state
Perturbation
Discuss validity
Discuss convergence
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Raleigh Ritz variational method
Theorem
Lowest energy state of system
System Hamiltonian
Proof
Application of theorem
Pick any reasonable function with respect to
the system, normalize it, and minimize the
expectation value of the Hamiltonian!
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Example-2 Triangular well
Minimize with respect to variation parameter
Compare with exact solution