Lecture 40 Highlights
The Variational Method is another approximation method that is very useful for
estimating the ground state energy of a complicated Hamiltonian. The idea is very
simple. Given a Schrödinger problem to solve
Ψ
=
Η
Ψ
E
(which cannot be solved
exactly), make your best guess for the ground state wavefunction )(
,r
GuessGS
r
Ψ
(make sure
it is normalized: 1
,, =ΨΨ GuessGSGuessGS ) and calculate the expectation value of the
Hamiltonian with this wavefunction: GuessGSGuessGS ,, ΨΗΨ . The true ground state
energy is guaranteed to be less than or equal to this expectation value:
GuessGSGuessGSGS
E,, ΨΗΨ≤ . Basically this is true because your guess wavefunction is in
general a linear combination of the true ground state wavefunction and many excited
states. Hence the expectation of energy is bounded below by the true ground state
energy.
To improve the guess wavefunction, one can add many adjustable parameters to
it, call them K,,, 321
λ
λ
λ
These are often physically motivated quantities, such as the
width of the wavefunction in real-space, or the effective charge of the nucleus as seen by
an electron in an atom, or perhaps the distance between two nuclei in a molecule, etc.
Once again normalize the new guess wavefunction ),,,,( 321, K
r
λ
λ
λ
r
GuessGS
Ψ
and calculate
the expectation value of the Hamiltonian. Now we can minimize the expectation value of
with respect to variations in the parameter values. In other words, set Η
0
),,,,(),,,,( 321,321, =
∂
ΨΗΨ∂
i
GuessGSGuessGS rr
λ
λλλλλλ
K
r
K
r
for all parameters i
λ
. This
will give a closer estimate of the ground state energy. How close? Unfortunately there is
no way to estimate how far the result is from the true ground state energy.
In class we did the example of a 1D problem of an infinite delta function
well )()( xxV
δ
α
−= bound state. We guessed a parameterized ground state
wavefunction of the form: , which is a Gaussian centered on the
well. The parameter
2
),(
,bx
GuessGS Aebx −
=Ψ
b/1 is basically the width of the wavefunction in real space. We
found that
4/1
2⎟
⎠
⎞
⎜
⎝
⎛
=
π
b
Afrom normalization. The rest of the discussion followed pages
294-296 of Griffiths. Note that the expectation of kinetic energy of the
particle m
b
T2
2
h
=scales inversely with the square of the width of the wavefunction.
More narrow-in-space wavefunctions “force” the particle to have a greater uncertainty in
momentum and therefore a larger expectation value of kinetic energy. The expectation
value of potential energy is
π
α
b
V2
−= . The expectation value of the Hamiltonian
VT +=Η is minimized for the special value of 4
22
min
2
h
π
α
m
b=, giving a minimum
1
