E612: Harmonic oscillator with perturbation

1
E612: Harmonic oscillator with perturbation
Submitted by: Dan Bavli
The problem:
Adding to the Hamiltonian of a harmonic oscillator with frequency ω a pertubation of the form Ĥ1 = λx̂.
(1) Find the energy of the ground state up to the second order using the perturbation theory and by exact calculation.
compare the tow results.
(2) For perturbation of the form Ĥ1 = λx̂4 .
Find the first order correction to the n’th energy level.
The solution:
(1) The new Hamiltonian of the system will be:
where Ĥ0 | n(0) i = ~ω(n + 1/2) | n(0) i
Ĥ = Ĥ0 + λx̂
| n(0) i are the unperturbated eigenstates of Harmonic Oscillator.
The energy of the ground state expended to a series will be:
X
(k)
(0)
(1)
(2)
E0 =
λk E0 = E0 + λE0 + λ2 E0 + O(λ2 )
Using the perturbation theory formulas (there is no degenerecy in harmonic oscillator so we can use them) :
En(1) = hn(0) | V̂ | n(0) i
En(2) =
(1)
X |hm(0) | V̂ | n(0) i|2
(2)
(0)
(0)
En − Em
m(6=n)
where V̂ = x̂
and by using the lowering and raising operators â and ↠and their properties:
r
x̂ =
~
(â + ↠)
2M ω
;
(↠)n
| n(0) i = √
| 0(0) i
n!
As we expect the mean value of x̂ at the ground state will be zero therefore:
(1)
= h0(0) | x̂ | 0(0) i = 0
E0
(2)
To find E0
we will first calculate the terms :
xm0 = hm(0) | x̂ | 0(0) i
After some algebra one can see that :
r
r
~
~
(0)
†
(0)
(â + â ) | 0 i =
| 1(0) i
x̂ | 0 i =
2M ω
2M ω
Therefore:
xm0
1
=√
m!
r
~
h0(0) | âm | 1(0) i =
2M ω
r
~
δm1
2M ω
2
Substituting into (2) we get:
(2)
E0
=−
1
2M ω 2
Therefore the enrergy of the ground state up to second order in λ will be:
E0 =
1
λ2
~ω −
2
2M ω 2
We can see that the ground state shifts down when this perturbation is present.
The other conclution is that this perturbation couples only the nearest neighbour states.
By exact calculation we will have to find the eigenvalues of the Hamiltonian:
Ĥ =
p̂2
1
+ M ω 2 x̂2 + λx̂
2M
2
Changing variables to : x̂ = ŷ +
the ground state :
E0 =
λ
M ω2
yields the Hamiltonian of translated harmonic oscillator with the eigenvalue of
1
λ2
~ω −
2
2M ω 2
Therefore the result that we have got by the perturbation theory is exact, which means that higher order corrections
to the energy are all zero.
(2) Using (1) for the perturbation of the form Ĥ1 = λx̂4 we get :
r
~ 4
) (â + ↠)4 | n(0) i
En(1) = hn(0) | (
2mω
The only terms that are non-zero are:
hn(0) | â2 â†2 + â†2 â2 + 4↠â↠â + 4↠â + 1 | n0 i = 6n2 + 6n + 3
Therefore we get that the first order correction to the n’th energy level will be:
En(1) = (
~ 2
) λ(6n2 + 6n + 3)
2mω
We see that this perturbation couples to the n’th state the states (first order) :
| (n − 2)(0) i ; | (n + 2)(0) i ; | (n − 1)(0) i ; and | (n + 1)(0) i