Given an arbitrary time-independent Hamiltonian H, we want to approximate

1
12. Approx. Methods for Time Ind. Systems
12A. The Variational Principle
A Great Way to Find the Ground State

• Given an arbitrary time-independent Hamiltonian
  H, we want to approximate
• The eigenstates ?i?
• The eigen-energies Ei
• Idea behind the variational principle
• The ground state is the state with the lowest
  energy
• The first excited state is the lowest energy
  state orthogonal to the ground state
• Etc.
• The method
• Choose a large number of state vectors
• Measure their energy
• Pick the one with the lowest energy

2
The Ground State Has the Lowest Energy

• Imagine we knew the eigenstates and energies of
  the Hamiltonian
• These states are assumed to be complete and
  orthonormal
• Assume the energies are ordered
• For an arbitrary state ??, we use completeness
  to write
• Lets find the expectation value of the
  Hamiltonian for ??
• It follows that
• Consider the inner product
• We therefore have

3
The Variational Principle

• Select a wide range of trial vectors ??
• Calculate the expectation values at right
• Choose the one with the lowest ratio at right
• Use this ratio as an estimate of the ground state
  energy
• Use the corresponding state vector ??
  (normalized) as an estimate of ground state
  vector ?1?
• The variational part
• We want as many state vectors as possible,
  ideally infinitely many
• Best way to do this is to make ?? haveone or
  more variational parameters
• Calculate the energy as a function of these
  parameters
• Find the value of ?min that minimizes E(?)
• Then the estimate of the energy and wave function
  is

4
Theory vs. Practice

• For homework problems
• Pick a set of trial vectors described by a
  smallnumber (? 2) of parameters
• Find E(?) analytically
• Find the minimum using derivatives
• Substitute back in ?min
• For research problems
• Pick a set of vectors described by a large number
  (100s) of parameters
• Find E(?) numerically
• Find the minimum using multi-dimensional
  searchalgorithms (simplex method, for example)
• Substitute back in ?min

5
Good Trial Wave Functions

• How do you pick a good trial wave function?
• Discontinuous functions will have infinite
  derivative
• Can show this yields infinite ?P2? and hence
  infinite ?H?
• Dont use discontinuous functions
• Non-smooth functions are okay, but can be tricky
  to evaluate
• Discontinuous first derivatives have infinite
  secondderivative at a point
• This will contribute non-trivially to
• Unless ? vanishes at the non-smooth point
• When in doubt, you can avoid this problem with

6
Sample Problem (1)
A particle of mass m in 1D lies in potentialV
m?2x2/2. Estimate the ground-state energy.

• The potential rises suddenly
• Try a function thatdisappears suddenly
• No need to normalize in this formalism
• We now needto calculate
• Work out the pieces,one at a time

7
Sample Problem (2)
A particle of mass m in 1D lies in potentialV
m?2x2/2. Estimate the ground-state energy.

• Put the pieces together
• Minimize with respect to a
• Substitute back in to get E(a), an estimate of
  the energy

8
Some comments on how we did

• We picked a terrible trial wave function
• We still did pretty well (20 error)
• The error in the energy is caused by the square
  of the amount of bad wave functions in the wave
  function
• Small things squared are very small
• The state vector has first order errors
• Not as reliable
• We got an overestimate of the energy
• This will always happen
• With more parameters, you can do much better
• This method is powerful most realistic problems
  are solved with (advanced) variational approaches

9
Sample Problem Hydrogen Estimate (1)
A particle of mass m in 3D lies in potentialV
- kee 2/r. Estimate the ground-state energy.

• Do this in class
• Trial wave functions
• First function tried
• This is not normalizable, ???? ?
• Second function tried
• This looks very promising
• Things we need to find

10
Sample Problem Hydrogen Estimate (2)
A particle of mass m in 3D lies in potentialV
- kee 2/r. Estimate the ground-state energy.

• We now calculatethe energy function
• Find the minimum
• Substitute in to getthe minimum energy
• We got it exactly right!
• The wave function is
• Also exactly right

11
Can We Get Beyond the Ground State?

• Is there a way to get states beyond the ground
  state?
• Yes, if we pick states orthogonal to the ground
  state
• Removing the approximate ground state
• Assume you have an estimate of the true
  normalizedground state found by variational
  method
• Create a set of states that you estimate are
  close to the next excited state
• Remove the portion of these statesin the
  approximate ground state
• This state is, by construction,orthogonal to
  ?1?
• Calculate the energy for this state
• Minimize this energy and find ?min
• Then we approximate the first excited state
  energy as

12
Comments on Excited State(s)

• Is the resulting energy guaranteed to be an
  overestimate?
• In general no
• The state is guaranteed to contain none of the
  estimated ground state ?1 ?
• But it could contain a small mixture of the
  actual ground state ?1 ?
• Does this work as well for excited states as it
  did for the ground state?
• In general no
• Error from previous step gets compounded with
  this step
• Over many steps, errors accumulate
• An exception where it does work
• Suppose the problem has some symmetry
• Then all eigenstates can be classified by their
  eigenvalues under this symmetry
• Choose trial state vectors that have this
  symmetry eigenvalue
• The energies you find will be true overestimates
  of the energyof the lowest state with each
  symmetry eigenvalues

13
Sample Problem (1)
A particle of mass m in 1D lies in potential V
m?2x2/2. Estimate the first excited state
energy. Is it an overestimate?

• The potential is symmetric under parity
• The ground state, previously discussed, has even
  parity
• Lets find the lowest energy odd parity state
• Try an odd wave function
• Work out the pieces weneed, one at a time

14
Sample Problem (2)
A particle of mass m in 1D lies in potential V
m?2x2/2. Estimate the first excited state
energy. Is it an overestimate?

• Minimizethe energy
• Substituteit back in
• Since our state is odd, the lowest odd energy
  statemust be lower than this value
• Actual coefficient is 1.5 (15 off)

15
12B. The WKB Approximation
A Method For High-Energy States

• The variational method is good for ground state
  and other low energy states
• The WKB method is good for highly excited states
• Idea behind the WKB approximation
• If the energy is large, the wave function will be
  quickly oscillating
• The potential will then look like its slowly
  varying
• Start from Schrödingers equation in 1D
• Define
• Then Schrödingers equation becomes
• If we think of k(x) as a constant, this solution
  would be like e?ikx
• This suggests breaking ? into a magnitude and a
  phase
• A(x) and ?(x) are real functions

16
Breaking it into two equations

• Substitute in (denotederivatives with primes)
• Match real and imaginary parts
• Multiply second equation by A
• Anything with a zero derivative is a constant
• Rename ?? as W, then

17
0th Order WKB Approximation

• Expand out that second derivative
• Substitute it back in
• If energy is large, so is k(x)
• To zeroth order, assume that k2dominates the
  other two terms

18
Bound Problems with Steep Boundaries

• Suppose we want to consider bound states
• For now, assume the potential goes to
  infinitysuddenly at x a and x b
• For bound states, we generally prefer real wave
  functions
• Sines, cosines, or somelinear combination
• We have two additional constraints
• Wave function must vanish at x a
• Wave function must vanish at x b
• Vanishing at x a requires that
• Then vanishing at x b requires that
• Where n 0, 1, 2,
• Recall
• So we have

19
Bound Problems with Steep Boundaries

• Suppose we want to consider bound states
• For now, assume the potential goes to
  infinitysuddenly at x a and x b
• For bound states, we generally prefer real wave
  functions
• Sines, cosines, or somelinear combination
• We have two additional constraints
• Wave function must vanish at x a
• Wave function must vanish at x b
• Vanishing at x a requires that
• Then vanishing at x b requires that
• Where n 0, 1, 2,
• Recall
• So we have

20
Near a Soft Boundary Airy Functions

• Consider now the case where the potentialrises
  smoothly across the boundary
• This is the typical case
• We assumed k2 is large, but this is not true
  near x a, b
• By definition, k2 vanishes there
• Close to x a, Taylor expand k2(x)
• We are trying to solve
• Compare to the Airy equation
• General solution to Airy equation is
• Ai and Bi are called Airy functions
• The general solution for ? will be

21
Phase Shift Near One Soft Boundary

• Bi diverges as x ? ?
• We want only Ai, so
• The Ai function as x ? ? behaves like
• This implies as x goesabove a, we would have
• Compare this with the WKB wave function in the
  allowed region
• We thereforeconclude

22
Quantization Condition For Soft Boundaries

• When we had steep boundaries, we demanded
  wavefunction vanishes at x a and x b
• This implied
• The softening of the boundary at x a,changed
  the boundary condition there to
• Similarly, the softening of the boundary at x
  bchanges the boundary condition there to
• Recall thedefinition of k(x)
• We therefore have

23
Three Different Cases

• We can get quantization conditions forthree
  types of problems
• Hard boundaries
• Soft boundaries
• One hard, one soft

24
How to use these formulas

• Given a potential V(x), can we estimate the
  energyeigenstates?
• Pick the energy E
• Find the classical turning points a and b,
  thepoints where the particle would stop
  classically
• Write the relevant integral, usually
• Do the integral
• Solve for En as a function of n

25
Sample Problem
Estimate the eigenenergies of a particle of mass
m in the 1D Harmonic oscillator with V(x)
m?2x2/2

• We pick an energy E
• We solve for theturning points
• Substitute into the integral
• Make a trigonometric substitution
• By coincidence, we got it exactly right

26
Probability Density in WKB and Classical

• Consider the wave function in the classically
  allowed region in the WKB approximation
• The probability density is
• WKB works for high energy, k(x) large
• Rapidly oscillating sine function
• So we have
• k is like momentum, which is like velocity
• Classically, fraction of time spent in region of
  size dx is given by

27
12C. Time Independent Perturbation Theory
A Series Expansion

• Consider a situation where the Hamiltonian can
  besplit into a large, easily solved piece, and a
  small piece
• Replace W by ?W, where ? 1
• The large part is assumed to have known
  eigenvaluesand complete, orthonormal
  eigenvectors
• Let ?n? be the exact eigenstates of H with
  energies En
• In the limit ? ? 0, the ?n?s will be ?n?s and
  the Ens will go to ?ns
• It makes sense, therefore, to imagine a series
  expansion in terms of the parameter ? for ?n?
  and ?n?

28
The Idea Behind The Method

• We write out Schrödingers equation as follows
• Expand to some order
• Since this must be true for all ?, the
  coefficients of every power of ? must match

29
A Small Ambiguity Problem

• Schrödingers time-independent equation does
  notcompletely determine the eigenstates ?n?
• We can always multiply by an arbitrary complex
  number c
• Therefore these states are slightly ambiguous
• In the limit W 0, we know that
• One way to resolve the ambiguity is to simply
  demand
• The problem this means final states ?n? are not
  normalized
• Can always be normalized later
• Irrelevant for energy computation
• Only relevant in 2nd or higher order state vector

30
The Procedure

• For each order (p)
• Act on the left with ??n
• Let H0 act on ??n
• Act on the left with ??m for m ? n
• Reconstruct ?n(p)?using completeness
• Iterate order by order
• When done, reconstruct?n? and En using ? 1

31
First and Second Order

• First order
• Second order

32
Third Order Energy and Summary

• Third order energy
• Put it all together

33
Sample Problem
Find the ground state energy to second order and
eigenstate to first order for a particle in 1D
with mass m and potential V(x) m?2x2/2 ?x4
when ? is small.

• Split Hamiltonian into H0 and perturbation W
• Find eigenstates and energies of H0
• We now need to find

34
Sample Problem (2)
Find the ground state energy to second order and
eigenstate to first order for a particle in 1D
with mass m and potential V(x) m?2x2/2 ?x4
when ? is small.

• We therefore have

35
Validity of Perturbation Theory

• Perturbation theory can only be trusted if
  subsequent terms get smaller
• Lets compare first and secondorder energy
  contribution
• Let ? be smallest gapbetween ?n and ?m
• Then we have
• Compare to ?'n
• Roughly, perturbation theory works if
• So we need ? ? W

36
12D. Degenerate Perturbation Theory
The Problem and Its Solution

• Look at the second order expression for energy or
  first order for state vector
• If two states have the same unperturbed energy,
  we will get in trouble
• The problem disappears ifwe get lucky by
  having
• If we have such states, then the unperturbed
  eigenstates are in fact ambiguous
• We can use this ambiguity to change basis for our
  unperturbed states
• Thereby ensuring the problem goes away
• Then proceed with perturbation theory as normal

37
Procedure for Degenerate Perturbation Theory

• Suppose we have a set of degenerate states
• Define the reduced W matrix
• Find g normalized eigenvectorsand eigenvalues
  for this matrix
• Changebasis to
• Then in the new basis, we have
• And
• So we can use these as our new basis states!

38
Working in the New Basis States

• In the new basis, the matrix elements of W
  vanishfor any pair of distinct states with the
  same energy
• This gets rid of problem terms
• Note that the basis states we start with are
  determined partly by the perturbation
• Even to leading (0th) order, we need to include
  perturbation theory
• The perturbation breaks the degeneracy and
  determines our states
• Note that the first correction to the energies
  are just
• First order energy easy to find just from
  eigenvalues of

39
Sample Problem
A particle of mass m in two dimensions has
Hamiltonian as given at right, where b is
small (a) What are the eigenstates and energies
in the limit b 0? (b) For the first pair of
degenerate states, determine the eigenstates to
leading order and energies to first order in b.

• H0 is two harmonic oscillators
• The x-direction has frequency ?
• The y-direction has frequency 2?
• The unperturbed states and energies are
• The first two states are nondegenerate
• The second excited states are degenerate
• Need to use degenerate perturbation theory!

40
Sample Problem (2)
(b) For the first pair of degenerate states,
determine the eigenstates to leading order and
energies to first order in b.

• We place our degenerate states in some order
• We then calculate the perturbation matrix

41
Sample Problem (3)
(b) For the first pair of degenerate states,
determine the eigenstates to leading order and
energies to first order in b.

• We now find eigenstates and eigenvalues of this
  matrix
• The energies and eigenstates are therefore
• Note that to find the energies, all we needed was
  the eigenvalues